Convex Functions, Partial Orderings, and Statistical Applications (Mathematics in Science and Engineering)

A(a). If either A(X):s: A(a) for every x

E

(a, b)

(3.10)

A(X) ~ A(b) for every x

E

(a, b)

(3.11)

or holds, then (3.5) is valid.

A reversal of Theorem 2.30 is given in the same paper by Pecaric (1984g): 3.12. Theorem. Let f: I ~ ~ (I an interval in ~ k ) be a continuous function with increasing increments, and let p be a real n-tuple such that (3.1) is valid. If x;EI (i=I, ... ,n), An(x,p)EI where An(x,p) is defined as in (2.32), and if

Xl :s: Aix, p) :s: ... :s: An(x, p)

(3.12)

holds (or if the reverse inequalities in (3.12) hold), then (3.2) is valid.

3.2. Refinements of Inequalities

87

3.13. Remark. In Theorem 3.12 instead of considering a function with 0 increasing increments we can let f be a P-convex function.

3.2.

Some Refinements of Jensen's and Jensen-Stetfensen's Inequalities

Let f: U ~IR be a real-valued function and U be a convex set in an arbitrary linear real space M. Let I be a finite nonempty set of positive integers. If we define the index set function F by

where 1 A/(x; p) = P ~ P i X i ' liEf

then the following theorem-is valid: 3.14. Theorem. Let f be a convex function on U. Let I and J be finite nonempty sets of positive integers such that In J = 0, P = {pJiElUJ' and x = {Xi}iElU; are real sequences such that Xi E U (i E I U J), PlUJ > 0, and As(x, p) E U (5 = I, J, I UJ). If p/ > 0 and PJ > 0, then F(I U J)

:5

F(I)

+ F(J).

(3.13)

If PIP; < 0, then the inequality in (3.13) is reversed. Iffis strictly convex, then the equality in (3.13) holds iffA/(x, p) = A;(x, p).

Proof. For n = 2, this is a simple consequence of Jensen's inequality and of its reversal. Thus the proof follows by letting Xz= A;(x, p),

3.15. Corollary.

(a) If Pi;::::O (i = 1,

F(In):5 F(In-l) and

:5

and

pz

= P;,

o

, n) and Ik = {1, ... , k}, then :5

F(Iz) :5 0

(3.14)

88

3. Reversals, Refinements, and Converses

(b) If An(x, p)

E

Pl>O,

U and Pi :$ 0 for

i = 2, ... , n,

(3.16)

then the reverse of the inequalities in (3.14) is valid, and

3.16. Remark. Theorem 3.14 and Corollary 3.15 give generalizations of results in Vasic and Mijalkovic (1976), Vasic and Pecaric (1979b), and Pecaric (1986a) (see also Bullen, Mitrinovic, and Vasic (1987, pp. 24-25). D The following result is proved in Pecaric (1984g):

3.17. Theorem. Let I and J be defined as in Theorem 3.14. Let f : U ~IR be a continuous function with increasing increments (U is an interval in IRk), and let p and xbe defined as in Theorem 3.14. (a) Let PI > 0 and PJ > O. If (3.18)

i.e., if (3.19)

then (3.13) is valid, i.e., the function F is subadditive. (b) If PI>O and PJ < 0, and if (3.18) (i.e., (3.19» holds, then the inequality in (3.13) is reversed. 3.18. Corollary. Let the conditions of Theorem 2.30 be satisfied. Then (3.14) is valid. If the conditions of Theorem 3.12 are satisfied, then the reverse of the inequalities in (3.14) holds.

3.19. Remarks. (a) Theorem 3.17 can also be formulated for P-convex functions. (b) For k 2: 2, an interesting function with increasing increments is f(xl' ... ,Xk) = Xl' .. Xk (Xi 2: 0, i = 1, ... ,k). This function is also continuous, thus Theorem 3.17 and Theorems 2.28, 2.30 and 3.11, 3.12 give generalizations of several well-known results concerning Cebysev's inequality for monotonic functions and sequences; see, for example,

89

3.2. Refinements of Inequalities

Popoviciu (1959b), Pecaric (1980b, 1980d), Fink and Jodeit (1984), Vasic and Pecaric (1981b), Biernacki (1950, 1951), and Burkhill and Mirsky (1975). 0 A related refinement of the Jensen-Steffensen Inequality is given in Vasic and Pecaric (1984):

3.20. Theorem. Let x and p be two real n-tuples such that a s: Xl ::s ... ::s X n ::s band O::s Pk ::s P;

for

k

=

1, ... , n - 1,

Then for every convex function f : [a, b] Hn(XI' ..•

,xn )

(3.20)

~ ~

~ Hn-I(XI, •.. ,Xn-I) ~ ..•

~Hz(XI' X2)

~HI(XI) = 0

(3.21)

holds, where Hk(XI, .•• ,

Proof.

xd =

C?k(Xl' •.. , Xk, PI, ..• ,Pk-l,

Pk ) ,

The inequality Hk(XI, . . . , Xk) ~ Hk-I(XI, . . . , Xk-l)

(3.22)

is equivalent to (3.23) where

Note that Xk::S Xk and Xk-I::S Xk-l. Without loss of generality we may assume Xl < ... <x n and 0 < Pk < P; (k = 1, ... , n - 1). By the substitu> e have (3.22) from tions Y I ~ X k - I ' Y 2 ~ X k >X l ~ X k - I ' X 2 ~ X k w (1.7). 0 Another type of refinement of Jensen's inequality is given by Pecaric and Dragomir (1989a):

90

3. Reversals, Refinements, and Converses

3.21. Theorem. in M, and f : U

Let M be a real linear space, U a nonempty convex set a convex function. Then

~ ~

f (P ~ ~~ p xo ) : 5 (P-1 )kII

nl-l

n

0

1)2 :5 (P

0

n

1 :5 P n

'" LJ

'1.··.,'kE1

L

po11 ",pof(!(Xo Ik k II +",+xo)):5'" lk

PiIPiJ(2! (XiI + Xi2))

11,12E1

n

~ 1P;/(Xi)'

(3.24)

where 1= {1, ... , n}, Xi E U (i = 1, ... , n), and Pi> 0 (i = 1, ... , n).

The following theorem (Mitrinovic and Pecaric, 1987a) gives a result on Jensen's inequality via the monotonicity property of certain functions: 3.22. Theorem. Let f be a convex function defined on a convex set U c M (M is an arbitrary linear real space). Let the function g be defined by n 1 ( n ) g(x) = ~ l ~ f qixAi + (r -x) ~ 1A k ,

where qi>O (i=1, x) EZ=lAk E U (i = 1, (xy > 0, Y E I), then

,n) with ~ Z = 1 ( 1 / q k ) = 1 r, E ~ , qixAi+(r, n) for all x in an interval I (I ~ ~ ) . If Ixl :5lyl

(3.25)

g(x) :5 g(y).

Proof. Let s1 E [0, 1] (i, j = 1, ... , n) with E?=l s1 = 1 (j = 1, ... , n). By (3.13) we have (3.26) Using the substitutions n

ai=qiyAi+(r-y)

LA k=l

s{ = -1 ( 1 - X) - (i =1= j), o

b

qi

Y

and we obtain (3.25) from (3.26).

o

s:. = -qi1 ( 1 - Y-X) + Yx- ,

3.2. Refinements of Ineqnalities

91

3.23. Remarks. (a) The function g in Theorem 3.22 is also convex. Indeed, we have Ag(X) + Xg(y)

= ~ ~(Af(qiXAi+ (r-x) kt1 A k) + Xf(q.yA, + (r- y) ~ 1A k)) 2: g(h

+ Xy).

(b) Using the substitutions: lIqi-Wi, q;Ai-Xi, r=l, conclude that (3.25) is also valid if g(x) =

we may

~WJ(XXi + (1- x) ~ 1WkXk),

where Wi >0 (i = 1, ... ,n) are such that I:Z=1 Wk = 1 and XXi + (Ix) I : k ~ W 1 kXkE U (i = 1, ... , n) for all x in an interval I (I ~ ~ ) . D Note that for one-variable convex functions the result in Remark 3.23(b) can be obtained by using Fuchs' generalization of a theorem on majorization. Moreover, in this case we can obtain similar results for Jensen-Steffensen's inequality. The above results give refinements of related discrete inequalities. Next we give some similar results for linear isotonic functionals. Let E be a nonempty set, d be an algebra of subsets of E, and L be a linear class of real-valued functions g : E - ~ having the properties

+ bg) E L for all a, b E ~ ; that is if f(t) = 1 for tEE, then f E L;

L1: f, g E L=;,(af

(3.27)

L2: 1 E L,

(3.28) (3.29)

L3: f E L, E 1 Ed=;' fCEl E L,

where CEI is the indicator function of E 1 (i.e., CEl(t) = 1 for t E E 1 , and 0 for t E E\E 1 ) . It follows from L2, L3 that CEI E L for E 1 E d. We also consider isotonic linear functionals A: L - ~ by assuming:

AI: A(af + bg) A2: f

= aA(f) + bA(g) for f, gEL, a, b E E L, f(t) 2: 0 on E =;, A(f) 2: 0 (A is isotonic).

~ ;

(3.30) (3.31)

Furthermore, we make use of the fact that if L also satisfies L3, then for every £1 E d such that A( C E ) > 0, the functional Al defined for all gEL by A 1(g) = A(gCE)/A(CE) is an isotonic linear functional with A 1 (1) = 1. We observe that A(g) = A(gCEJ + A(gCE\E).

(3.32)

92

3. Reversals, Refinements, and Convenes

3.24. Theorem. Let L satisfy properties L1, L2, and L3 on a nonempty set E, and assume that

~ Fg(E1 )

+ Fg(E\E 1)

~ Fg(E 1 )

~0

(3.33)

for all E 1 E .sIl such that 0 < A(CE,) < 1, where

(3.34) If

Proof.

To prove the first inequality in (3.33), let p =A(CEJ, q = A(CE\E}) (hence p + q = 1 by the first inequality in (3.32», x = A(gCEJIA(CEJ, y = A (gCE\EJIA(CE\EJ in the convexity condition P
+ q
~
+ qy),

(p

+ q = 1),

(3.35)

and use the fact that px + qy =A(g) by the second inequality in (3.16); similarly, we can use A(
= A(
To prove the last inequality in (3.33), observe that it is just Jessen's inequality (see (2.6» applied to the isotonic linear functional AI: L I~R defined by gEL,

and having A 1(1) = 1. Equality holds in the first part of (3.33) precisely when it holds in (3.35) for a continuous and strictly convex function

3.25. Remarks. (a) The inequalities in (3.33) represent a refinement and improvement of the Jessen Inequality since (3.33) gives a lower bound ( ~ O o) n the difference A(
3.2. Refinements of Inequalities

93

The following two theorems are proved in Pecaric and Beesack (1987b): Let L satisfy properties Ll-L3 on a nonempty set E, and assume that 1> is a continuous convex function defined on a closed interval I c ~ . let A be an isotonic linear functional with A(l) = 1, J be an interval such that 1>(/) c J, and F : J z ~ ~ be an increasing function of its first variable. If E 1 E S'i such that A(CE\E) > 0, then for every gEL such that 1> (g) E L we have

3.26. Theorem.

F[A(1)(g)), 1>(A(g))]

2:

inf F[A(1)(gE 1 ' x)), 1>(A(gE1 , x))], (3.36) XEI

where

Proof. have

For brevity, let E z=E\E 1 and assume A(CEZ) >0. Clearly we

and

where

In addition, A(1)(g)) = A(1)(g)CE, + 1>(g)C E,) = A(1>(g )CE) + A(1> (g)C E,) 2: A(1>(g)C E)

+ o1>(z)

holds from the remark concerning (3.31) with E 1 replaced by E z . Now, Z E I holds because if 1= [Il', then Il' ~g(t) ~{3 for tEE since 1>(g) is in L (hence it is defined). Thus Il'Cdt) ~g(t)C E2(t) ~ {3CE2(t) for all tEE whenever Il'A(CEz) ~ A ( g C E z ~){3A(CE,), thus we have Il' ~ Z ~ {3. A simple modification of the argument shows that Z E I if either Il' = -00 or {3 = 00. It then follows from the above inequality and the increasing property of F(·, y) that

In

F[A(1)(g)), 1>(A(g))]

+ o1>(z), 1>(A(gCd + oz)] 2: inf F[A(1)(g)C E ) + o1>(x), 1>(A(gCE) + ox)]

2:

F[A(1)(g)C E)

XEI

= inf F[A( q,(gE" x», q,(A(gE XEI

"

x»],

94

3. Reversals, Refinements, and Converses

since A(gEt' x) = A(gC E) +XA(CE2) = A(gC E) + ox, <jJ(gEt' X)(t)

= <jJ(g(t)C Et(t) + xCE

2(t»

= <jJ(g(t»CEt(t)

+ <jJ(X)C E2(t).

0

3.27. Remark. Clearly, there are many variations and generalizations of Theorem 3.14 which have essentially the same proof. For example, if F(·, y) is decreasing for each y E J, then in place of (3.36) we have F[A( <jJ(g», <jJ(A(g»] :5 sup F[A( <jJ(gE t' x», <jJ (A(gE, , x»].

(3.36')

XEI

This also follows from (3.36) simply by replacing F with F1 = - F.

0

For another (more extensive) generalization, suppose E, E.sI1 for ' letting oi=A(C E), l:s;j:S;n with E inEj=0 (i*j) and E = U ~ E iBy zi=A(gCE)/A(CE) (where we assume all Oi>O), we find under the hypotheses of Theorem 3.14 that if x = (Xl' . . . ,xn ) , then F[A(<jJ(g», <jJ(A(g))]

2=

infn F[i 0i<jJ(X i), <jJ(i OiXi)]' 1

XEl

(3.37)

1

If we let n

gE,....,En.X(t) =

2: XiCe,(t), i ~ l

then the right-hand side of (3.16) can be written as

Note, however, that this value is independent of the function g and it provides a lower bound for the left-hand side of (3.37) which is valid for all admissible gEL. Similarly, if F(·, y) is decreasing, then instead of (3.37) we have F[A(<jJ(g», <jJ(A(g»]:s; sup F(i Oi<jJ(Xi), <jJ(i OiXi))' xe I"

1

(3.37')

1

There are other variations which follow immediately from (3.36) and (3.36'). For example, if 1:5 m < n, and if we let Xm = (Xm+l , .•. , xn ) and m

GEt .....En.Xm(t) = 2: g(t)Ce,(t) + ;=1

n

2:

j=m+l

XjCE/t),

3.2. Refinements of Inequalities

95

then under the hypotheses of Theorem 3.26 we can obtain

F[A(Ij>(g», Ij>(A(g»]:s;

sup F[A(Ij>(gEt .....En.xJ), Ij>(A(gEt .....En.xJ)].

xmEln-m

(3.38) Finally, we observe that in (3.36), (3.37), or (3.38) the lower bounds on the right-hand sides depend on the subsets E, c E (with E, E .sIl), and a possible sharper bound (hence a better result) might be obtained by allowing the sets E, to vary. For example, by (3.36) we have

F[A( Ij>(g», Ij>(A(g))]

2:

sup {inf F[A( lj>(gEt.X», Ij>(A(gEt. X»]}, (3.39) EtEstlt XEI

3.28. Remark. Inequality (3.36) is a generalization of the well-known inequality of Beckenbach (1966) (see also Mitrinovic, 1970, p. 52, and Bullen, Mitrinovic, and Vasic, 1987, pp. 156-57); i.e., for a discrete D functional A we get Beckenbach's inequality. Theorem 3.26 is related to Jensen's inequality. A generalization of Jensen's inequality for convex functions of several variables was given by McShane (1937) (see Theorem 2.10). The following theorem is a similar generalization of Theorem 2.6.

3.29. Theorem. Let Ij> be a continuous convex function on a closed, convex set U c IR n. Let L satisfy properties L1, L2, L3 on a nonempty set E, and let A: L I~R be an isotonic linear functional with A(l) = 1. Let L={G=(gl, ,gn):giEL for l:s;i:S;n}, and define A : L ~ l R by n A(G) = (A(gl), , A(gn»' Then A is a linear operator on the linear class L. Let J be an interval such that Ij> (U) c J and F: J2 ~IR be an increasing function of its first variable. If E 1 E.sIl such that A(CE\EJ > 0 then, for every GEL such that Ij>( G) E L, we have F[A(Ij>(G», Ij>(A(G))] 2: inf F[A(Ij>(G Et.x», Ij>(A(G E t . x » ], XEU

Proof. The proof is similar to that of Theorem 3.26, thus we merely outline the differences. Let E 2 = E \ E 1 and a=A(C E,). Then Ij>(A(G» = Ij>(A(GCEJ + az),

z = A(GCE2)/A(CEz)'

A(Ij>(G» =A(Ij>(G)CEJ +A(Ij>(G)C E,) 2:A(Ij>(G)C EJ + alj>(z),

96

3. Reversals, Refinements, and Converses

because Al(g) = A(gC EJ/A (C EJ is an isotonic linear functional on L with A 1(1) = 1. Consequently, McShane's inequality applies to the operator AI: L ~IR n defined by A 1(G) = (A 1(gl), ... ,A1(gn» = A(GCEz)/A(CEJ Moreover, by McShane's result, we have z = A 1(G) E U. The result of the proof remains unchanged except for obvious notational modifications. 0 Finally, we give a refinement of Jensen's inequality that is similar to (3.24). We shall use the notation kn = fk,n(x1' ... , xn) =

G)

-1

fG

15il<~
(Xii + ...

+ Xi.»)'

(3.40)

The following result can be found in Mitrinovic and Pecaric (1988c) and Pecaric and Volenec (1988): 3.30. Theorem. Let f be a mid-convex (J -convex) function defined on a convex set I in an arbitrary real linear space, and let XI' . . . , Xn E I. Then A.n 2::fk+l,n

for

k

= 1, 2, ... , n -1

(3.41)

holds. Iffis a strictly mid-convexfunction, then equality in (3.41) holds iff

3.31. Remark. Indeed, Theorem 3.30 is an interpolation inequality for Jensen's inequality, This is so because from (3.41) we have ) 1 n 1 n f (Xi = fn,n ::; ... ::;kn ::; ... ::;An = f (X;). ni=1 ni=1

L

L

(3.42)

o

3.32. Theorem. Let f and Xi be defined as in Remark 1.16. Then (3.41) (and hence (3.42» is also valid.

3.2. Refinements of Inequalities

97

The following definition was given by Gabler (1979):

3.33. Definition. Let f: (a, b) ~IR, and let A,n be defined as in (3.40). f is said to be sequentially convex if {A,n} is a convex sequence in k for all n > 2 and all XI' . . . , X n E (a, b). Gabler (1979) showed that the following results hold for sequentially convex functions:

3.34. Theorem. (a) Iff: (a, b ) ~IR is convex of orders 2 and 4, then it is sequentially convex. (b) The function f(x) = X (x E (0,00)) is sequentially convex for A ~ s < 3, where A= log 3/(log 3 -log 2). S

3.35. Remark. Inequality (3.41) is also due to Gabler (1979). He stated his result for sequentially convex functions, but his proof depends only on 0 the fact that sequentially convex functions are convex functions. The following result is also valid (Rasa, 1988b, and Pecaric and Dragomir, 1989b):

3.36. Theorem. Let ¢: I£; IR ~IR be a convex function, a be an increasing function on [0, 1] such that f6 da(t) = 1, and let f: [0, 1] ~I be a-integrable on [0, 1]. Then I

¢(Jf(t) da(t)) o

f... ¢(_1 ± IT + ~ f... f ¢(! ~ IT I

I

~

0

k

0

I

I

o ~

kl

J

1 ,~l

k

k

0

I~l

I

~J

J

o

0

k

f(t;))

¢G ~ I J1 2

I~l

da(t;)

2

f(t;))

I

~ J ¢(f(t)) da(t) o

da(t;)

,=1

... I

for all positive integers k.

kl

f(t;))

da(t;)

98

3.3.

3. Reversals, Refinements, and Converses

Converses of Jensen's Inequality

In this section we again consider the positive linear functional A : L ~IR, where L is a linear class of real-valued functions g: E ~IR having properties L1, L2 stated in (3.27) and (3.28). The following result is proved in Beesack and Pecaric (1985a): 3.37. Theorem. Let be convex on 1= [m, M] (-00 < m < M < (0); let L satisfy conditions L1, L2 and A be any isotonic linear functional on L with A(1) = 1. Then for every gEL such that (g) E L (so that m ~ get) ~ M for all tEE), we have A((g))

Proof.

~ {(M -

A (g))(m)

+ (A(g) - m)(M)}/(M - m). (3.43)

From the definition of convex functions we then have w-v w-u

v-u w-u

< I > ( v ) ~ - - < I > ( u ) + - - < I (> u( w~ )v ~ u<w). w ,

Now let u

= m, u = get), and (g(t))

w ~ M; we have

~ M - get) (m) + get) - m (M), M-m

M-m

t

e E.

(3.44)

Since A satisfiesA1, A2 defined in (3.30)-(3.31) and A(k) = k holds for all constants k, (3.43) follows. 0

3.38. Remarks. (a) Theorem 3.37 is a generalization of Theorem 1 of Lab and Ribaric (1973), which concerns the case A(g) = Hg du] Hdu and the equality in (3.43). (b) The right-hand side of (3.43) is an increasing function of M and a decreasing function of m. This follows by writing this expression in the form cp(m) + (A(g) - m) cp(M) - cp(m) M-m

= cp(M) _

(M _ A(g)) cp(M) - cp(m) , M-m

and noting that m ~ A ( g ) ~ while M , (cp(M)-cp(m»/(M-m) is an 0 increasing function of both M and m by the convexity of cp. The following two theorems are also proved by Beesack and Pecaric (1985a).

3.3. Converses of Jensen's Inequality

99

3.39. Theorem. (a) Let L satisfy condition L1, L2 and A satisfy conditions A l , A2 with A ( l ) = 1. Assume that @ is convex on I = [m,MI (--oo < m < M < m) such that @"(x) 2 0 with equality for at most isolated points of I (so that @ is strictly convex on I ) . Assume further that either (i) @(x) > 0 for all x E I, or (if) @(x) > 0 for m < x < M with either @(m)= 0 , @'(m)# 0, or @ ( M )= 0, @ ' ( M )# 0, or (ii) @(x) < 0 for all x E I , or (ii') @(x) < 0 for rn < x < M with precisely one of @(m)= 0, @ ( M )= 0. Then for all g E L such that @(g)E L (so that m 5 g(t) 5 M for all t E E ) ,

A(@(g)5 ) A@(A(g))

(3.45)

holds for some d > 1 in cases (i), (i') or d E (0, 1) in cases (ii), (ii'). More precisely, a value of A (depending only on m, M , @) for (3.45) may be determined as follows: Define p = ( @ ( M ) - @ ( m ) ) / ( M- m). If p = 0, let x = 1 be the unique solution of the equation @'(x) = 0 (m <.2 < M ) ; then d = @ ( m ) / @ ( 1satisfies ) (3.45). Zf p # 0, let x = 1 be the unique solution in [m, MI of the equation

+

p @ ( x ) - @ ' ~ x ) { @ ( m )pfx - m)} = 0; then d = p / @ ( i ) satisfies (3.45). Moreover, we have m <x' < M in the cases (i), (ii). (b) Let all the hypotheses of (a) hold except that @ is concave on I with @'('x) I0 with equality for at most isolated points of I. Then the reverse inequality in (3.45) holds, where d is determined as in (a). Furthermore, A > 1 holds if @(x) < 0 on (m, M ) and 0 < A < 1 if @(x) > 0 on (m, M).

Proof. (a) As in MitrinoviC and VasiC (1975) and Beesack (1983), we consider the points B(m, @(m)),C(M, @ ( M ) ) on the convex curve y = @(x). The equation of the chord BC is y

+ p(x

= @(m)

- m ) = h(x).

By Theorem 3.37, we obtain A ( @ ( g ) ) % h ( A ( g ) )If. we consider the family of convex curves with equations y = d@(x) (d > 0), we show as in MitrinoviC and VasiC (1975) that there is a unique d > 0 which satisfies the conditions stated in the lemma, such that the curve will be tangent to the chord BC. Thus we have

100

3. Reversals, Refinements, and Converses

By elimination of

A we obtain

+

G(x) = P@(x) - @ r ( ~ ) ( ~Y) = 0.

The solution of the above equation is the abscissa (call it xo) of the contact point, where xo E (m, M ) . First, we have G ’ ( x )= -(px + Y)@”(x) < 0, i.e., the graph of G can cut the x-axis in at most one point of (m, M ) . Furthermore,

G(m)G(M)= @ ( m ) @ ( M ) ( @ ’ ( m-)P ) ( @ r ( M )- P> holds and, by the mean value theorem and the fact that @ r is an increasing function (since $”(x) > 0), we have G(m)G(M)5 0. Thus the assertion is true. Consequently, h ( y )IA@(y) holds for all y E I , and taking y = A ( g ) yields

A(@k)) Ih ( A ( g ) )5 A@(A(t?)h which establishes (3.45). (b) This follows from (a) by applying to the convex function

= -@.

0 3.40. Remarks.

(a) It is clear that the last inequality in the proof of Theorem 3.39 constitutes a refinement of (3.45). (b) Equality in (3.45) is valid iff the point of contact of the segment BC and the curve y = A@@) coincides with A ( g ) . In the case of a discrete positive functional with n

AV) = C p i f ( x i ) (pn i=l

= 1,pi

’01,

equality holds iff there exist two subsequences ( x i , , . . . , xJ and (xikc,,. . . , x i , ) of the sequence ( x l , . . . , x,) such that every element of the first subsequence is equal to m, every element of the second subsequence is equal to M , and

+

xo= MB m ( 1 -

01,

k

@(xo)= @(M)B

+ + ( m > ( l - B),

where xo is a unique solution of the equation G(x)

=0

6=

C pi,,

r=l

on (m, M ) .

0

Similarly, the following theorem can be obtained:

3.41. Theorem. (a) Let L , A , and g be as in Theorem 3.39, and let be a differentiable function on I = [m, MI such that @ ‘ ( x ) exists and

).(a

3.3. Converses of Jensen’s Inequality

101

si strictly increasing on I . Then we have

A(@(g)5 ) A + @,(A(g))

(3.46)

for some A satisfying 0 < A < ( M - m ) { p - @‘(m)},where p = ( @ ( M )@ ( m ) ) / ( M- m ) . More precisely, A may be determined as follows: Let x = 1 be the unique solution of the equation @ ‘ ( x ) = p ( m < 1 < M ) , then il= @(m)- @ ( i f ) + p ( 1 - m )

satisfies (3.46). (b) Let all the hypotheses of (a) be satisfied except that @ ’ ( x ) is strictly decreasing in I. Then 5

A + A(@k))’

where 0 < A < ( M - m){@’(m) - p } with p defined as in (a). In fact, we have A = @(if) - @ ( m )- p(2 - m ) with 2 given in (a). A further generalization of the above results is given in PeEariC and Beesack (1987a): 3.42. Theorem. Let J be an interval such that J =I @ ( I ) . If F(u, v ) is a real-valued function defined on J X J and increasing in u, then

(= max

F[O@(m) + (1 - O ) @ ( M ) ,#(Om + (1 - O ) M ) ] ) . (3.47)

ec[m,Ml

Furthermore, the right-hand side of (3.47) is an increasing function of M and a decreasing function of m.

Proof. By (3.43) and the increasing property of F ( - ,y ) , we have

102

3. Reversals, Refinements, and Converses

Thus the first part of (3.46) follows. As in Remark 3.38(a) we have, for m % x and m < M‘ c: M ,

{ ( M - x)Cp(m)+ ( x - m ) @ ( M ) } / ( M- m ) 2 { ( M ’ - x ) # ( m ) + (x - m)Cp(M‘)}/(M’ - m). Hence, by the increasing property of F ( . , y ) ,

d ( x ; m , M , C p ) ? d ( x ; m , M ’ , Cp),

msx,

m<M’<M.

(3.48)

By (3.48) and the fact [m,M ’ ] c [m,M I , it follows that max d ( x ; m, M, Cp)

2

xclm,Ml

max d ( x ; m, M’, Cp) 2 max d ( x ; m, M ’ , Cp). xe(rn,M ‘ ]

xc[m,MI

Similarly, we can show that max d ( x ; m, M , Cp) xc[m,M]

5

max d ( x ; m’, M , Cp)

for m’ 5 m < M .

x e [ m ’ ,M ]

Finally, the identity in (3.47) follows immediately from the change of variable 8 = ( M - x ) / ( M - .m>, so that x = Om (1 - 8 ) M with 05851. 0

+

In a similar fashion (simply by replacing F by -F in the above theorem) we can prove 3.42’. Theorem. Under the same hypotheses as in Theorem 3.37 except that F si decreasing in its first variable, we have

F[A(Cp(g)), $ w g ) ) l L

min

0; m, M , $1

x~[m,Ml

(= min F[eCp(m)+ (1

-

8elO.ll

8 ) @ ( M ) ,@(Om+ (1- e ) M ) ] ) . (3.47’)

Furthermore, the right-hand side of (3.47’) is a decreasing function of M and an increasing function of m. 3.43. Remark. Theorem 3.42 is a generalization of a result of Knopp (1935, Satz 1). For an equivalent form of Knopp’s result see Popoviciu (1944, p. 34). We now show that Theorems 3.39 and 3.41 are simple consequences of Theorems 3.42 and 3.42’. We first establish Theorem 3.38 by applying Theorems 3.42 and 3.42’. For case (i) we apply Theorem 3.42 and for case (ii) we apply Theorem 3.42’, both with F ( x , y ) = x / y , and

3.3. Converses of Jensen’s Inequality

103

J = (0, 00). We proceed only with case (i) since the proof for case (ii) is similar. In case (i) the inequality in (3.47) becomes

where

f

( x ) -f(x; m, M ,

@I= { ( M - x ) @ ( m +) ( x - m ) @ ( M ) ) / ( M- m ) @ ( x ) .

Clearly we have f’(x) = G ( x ) / @ ( x ) * where G ( x )= p @ ( x )- ($(m) + p(x - m ) ) @ ’ ( x ) )The . equation G ( x )= 0 (see proof of Theorem 3.38 has exactly one solution. Thus G ( x )= 0 holds for a unique x = Z(m, M). Since @ is convex and positive, it follows that f ( x ) L 1, with equality for x = m and x = M . Hence the maximum value on the right-hand side of (3.49) is attained at x = i. Next we prove Theorem 3.39 by applying Theorems 3.42 and 3.42‘. In Theorem 3.42 take F ( x , y ) = x - y. Then (3.46) becomes

where

Y ( x )= Y ( x ;m, M , @) = { ( M - x ) @ ( m )

+ (x -m)@(M))/(M - m ) - @(XI. Then Y ’ ( x )= p - @ ’ ( x ) is strictly decreasing on Z with Y ’ ( i )= 0 for a unique Z E (m, M ) . Consequently, Y ( x ) has its maximum value at x=i.

0

A similar converse of Theorem 3.26 is given by PeEariC and Beesack (1987b):

3.44.

Theorem. Let all the conditions of Theorem 3.26 be satisfied except that for a compact interval Z = [m,MI, m 5 g ( t ) IM holds for all t E E. Zf u = A ( C E , E ,> ) 0, then

104

3. Reversals, Refinements, and Converses

Proof. As in the proof of Theorem 3.26 we denote £2 = £\£1' Now let d(t) = (M - g(t»/(M - m), so that g(t) = md(t) + M(l- d(t», and let (3 = A(dCEJ Then ep(A(g» = ep(A(gCEJ + A[(md + M(l- d»CEzD = ep(A(gCEJ + m{3 + M( a - (3». Also, by the convexity of ep on I we have

A(ep(g»

= A[ep(g)CEt + ep(g)CE,) = A[ ep(g)CE1+ ep(md + M(l - d»CEzl :5A[ep(g)C E1+ {dep(m) + (1- d)ep(M)}CEz] = A {(ep(g)CEt + ep(m)A(dCEz) + ep(M)A«l-d)C Ez) = A(ep(g)CEJ + (3ep(m) + (a - (3)ep(M).

Since 0:5d(t):51, we have O:5{3=A(dCEz):5A(CEz)=a. The result in (3.50) now follows from this fact, the increasing property of F(·, y), and 0 the equality for ep(A(g» and the inequality for A(ep(g». 3.44'. Theorem. If F(·, y) is decreasing on J for every y EJ and all the other conditions of Theorem 3.44 are satisfied, then

F[A(ep(g», ep(A(g))]

~

inf F[A(ep(g)CEJ

+ 8ep(m)

OS(J::s:a

+ (a - 8)ep(M), ep(A(gCE1) + Bm + (a - 8)M],

(3.50')

where a=A(CE\EJ Theorem 3.44' follows by applying (3.50) to the function FJ. = - F. We note that in the special case E 1 = 0, Theorem 3.44 reduces to Theorem 3.42. If we also note that a = a Et and denote the right-hand side of (3.50) by H(E 1 ) , we obtain the best (least) upper bound for F[A( ep(g», ep(A(g))] given by

F[A(ep(g», ep(A(g»]:5 inf H(E 1 ) E1Ed,

under the hypothesis of Theorem 3.44, where slJ. 1 = {E 1 E sIJ.:A(CE\EJ = aEt>O}. 3.45. Remarks. (a) Mitrinovic and Vasic (1975) conjectured that .It. in Theorem 3.39 (Theorem 3.41) for discrete functionals are increasing

3.3. Converses of Jensen's Inequality

105

functions (decreasing functions) of m. A geometric proof of this conjecture is given in Vasic and Pecaric (1979b). Theorem 3.42 and its proof provide an analytical proof for a more general result. (b) It is obvious that Theorem 3.37 is an important result in the theory of convex functions. This inequality has several generalizations. For example, we can similarly give its generalization for g-convex functions (for the discrete case, see Vasic and Pecaric, 1979a). Furthermore, Vasic and Pecaric (1979b) proved their result without the assumption that 4J is a convex function, i. e., they showed that we need only assume either (4J(x) - 4J(m»/(x - m) or (4J(M) - 4J(x»/(M -x) is an increasing function of x on (m, M] and [m, M), respectively. Furthermore, with the same assumption, Theorem 3.42 is proved for discrete functionals (see Pecaric and Crstici, 1986). Several similar results are also given in those two papers. (c) The following interpolation of Theorem 3.41 for the discrete case is given in Pecaric and Mesihovic (1989): Let P; > 0 (i = 1, ... , n), (i = 1, ... , n) andf"(x) >0 on [m, M]. Then P; = 1, m ~ x ; ~ M

~pJ(x;) - · f ( P~ iX;) ~ 1 J 1 c ~ ~WeI) ~A, where In

= {I, ... , n}, A is defined as in Theorem 3.41, and W(I) = Pd(M) + (1- Pj)f(m) - f(PjM + (1- Pj)m).

(d) The following converse of McShane's inequality can be found in Andrica and Drimbe (1988): n

A(f) - f(A(Pl)' ... ,A(Pn» = K,

2:

(A(pD - A

2(pd),

k=l

where A is an isotonic linear functional, Pk: ~ ~ ~ is the k-projection defined by Pk(Xl,'" ,xn ) = xi , k = 1, ... ,n, f E C 2(E), and K, = max IIH(f)(x)1l where IIH(f)(x)lI is the Frobenius norm of Hessian D matrix H(f)(x).

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Chapter 4

4.1.

Applications of Jensen's Inequality to Means and Holder's Inequalities

Inequalities for Means

We shall again consider positive (isotonic) linear functionals as in Section 3.2. Let I = (a, b), -00::5 a < b ::5 00, and let 1jJ, X: I ~IR be continuous and strictly monotonic. Suppose that L and A satisfy the conditions L1, L2 and AI, A2 stated in (3'.27)-(3.31) with A(l) = 1, on a base set E, and that 1jJ(g), X(g) E L for some gEL. We define the generalized mean with respect to the operator A and 1jJ by

M1J'(g, A) = 1jJ-l{A(1jJ(g»},

gEL.

(4.1)

Observe that if a::5 1jJ(g(x»::5 f3 for x E E, then by the isotonic charact of A, we have a::5A(1jJ(g»::5f3, so that M1J' is well defined by (4.1). We also note that the above assumptions imply that g(x) E I for x E E. In the rest of this chapter we assume that gEL satisfies the above conditions, so that the theorems hold only for such g.

7r

4.1. Theorem.

Under the above hypotheses we have M1J'(g, A)::5 Mx(g, A),

(4.2)

provided either X is increasing and = X0 1jJ-l is convex, or X is decreasing and is concave. Proof. For gEL we have both 1jJ(g) ELand X(g) E L by assumption. Hence <1>( 1jJ(g» = X(g) E L for gEL; thus if is convex, then it follows from Theorem 2.4 that (A(1J!(g)))

::5 A(X(g».

107

108

4. Applications of Jensen's Inequality

Consequently if X is increasing, then X-I is also increasing, and we obtain

which is (4.2). If is concave, then - is convex, and we obtain the first inequality above with the direction reversed; since now X-I is decreasing with X, we again obtain (4.2). D

4.2. Remark. Theorem 4.1 is a generalization to functionals of the general mean value inequality found in Hardy, Littlewood, and P6lya (1934, 1952, p. 75, Theorem 92). D As an application of Theorem 4.1, we consider the generalization of the classical means M[rl(g, A) for isotonic functionals A, defined for r E IR by M[rl(g A)

,

= {(A(gr»lIr exp(A (log g»

for r *0, for r = 0,

(4.3)

where g(x) > 0 for x E E, gr E L for r E IR and log gEL. From Theorem 4.1 it follows as a special case that M[rl(g, A) ::; M[sl(g, A)

if

-00

< r ::; s < 00.

(4.4)

4.3. Theorem. Let L, A, 1jJ, and X be as in Theorem 4.1, but with 1= [m, M], -00 < m < M < 00. Then for every gEL such that m ::; g(t)::; M and for tEE, we have (1jJ(M) -1jJ(m»A(x(g» - (X(M) - x(m»A(1jJ(g» ::; 1jJ(M)x(m) - x(M)1jJ(m),

(4.5)

provided that = X 0 1jJ -1 is convex. The inequality in (4.5) is reversed when is concave.

Proof.

If 1jJ is increasing on I, we have m,

= 1jJ(m)::; 1jJ(g(t» ::; 1jJ(M) = M 1

for every tEE.

Thus by Theorem 3.37 with m, M replaced by ml, M 1, we have A( <1>( 1jJ(g))) :s {( 1jJ(M) - A(1jJ(g )))x(m) + (A( 1jJ(g» - 1jJ(m »X(M)}

x [1jJ(M) -1jJ(m)r 1 ,

4.1. Inequalities for Means

109

which reduces to (4.5). If 1jJ is decreasing on J, we have M 1 ~ 1jJ(g(t» ~ ml for every tEE and, with an obvious modification of the proof, the result follows as before. D

In the case of classical means, (4.5) gives Goldman's inequality for isotonic functionals (see, for example, Bullen, Mitrinovic, and Vasic, 1987, p. 203): (M r _ mr){MIsJ(g;A)Y- (M S - mS){MI'I(g;A)Y ~ M'rn" - MSm' (4.6) for 0 < r < s, or r < 0 < s, and the inequality is reversed for r < s < O. Similarly, for r = 0 we obtain

M m

log - {MIsl(g; A)Y - (M S

-

mS)log{MlOI(g; A)} ~ m'

10gM - M log m S

(0 <s).

(4.7)

4.4. Theorem. Let L, A, 1jJ, X be as in Theorem 4.1 and gEL with m ~ 1jJ(g(t» ~ M for tEE.' Assume I = (a, b) = ~+ = (0, (0), X ( + ~) = ~ + , and let =x o 1jJ - l. (a) Suppose "(x) 2: 0 with equality for most isolated points of [m, M], and let A> 1 be determined as in Theorem 3.39(a). Then (4.8)

holds if Xis increasing and supermultiplicative on ~", while the inequality is reversed if X is decreasing and supermultiplicative. Moreover, (4.9)

holds if X is increasing and submultiplicative, with the inequality reversed if X is decreasing and submultiplicative. (b) Suppose "(x) ~ 0 with equality for most isolated points of [m, M], and let A (0 < A < 1) be determined by Theorem 3.39(b). Then Mx(g; A) 2: X-1(A)M",(g; A)

(4.8')

holds if X is increasing and submultiplicative, with the inequality reversed when X is decreasing and submultiplicative. Moreover, (4.9')

holds if X is increasing and submultiplicatioe, while the inequality is reversed if X is decreasing and supermultiplicatiue.

110

4. Applications of Jensen's Inequality

Proof.

The proof follows from Theorem 3.39 in precisely the same manner as in Beesack (1983, Theorem 3). We illustrate by proving the reverse inequality in (4.8') when X is decreasing and supermultiplicative. The inequality to be proved is

which, since X is decreasing, is equivalent to

Since X is supermultiplicative, this holds provided A (( 1JJ(g))):5 A (A (1JJ(g))),

because gEL, 1JJ(g) E L, and <1>( 1JJ(g)) = X(g) E L hold by the assumptions preceding (4.1). Therefore the last inequality follows from Theorem 3.39(a) with g replaced by 1JJ(g). 0

4.5. Remark. Theorem 4.4 is a generalization to isotonic functions of the result in Beesack (1983, Theorem 3). As Beesack noted, the previous factors X-\A), {X- 1(r 1)} -1 are equal only when X is multiplicative. 0 For the generalized classical means M[r1(g, A) defined in (4.3), Theorem 4.4 gives the following results: Let 0 < X 1:5 g(x) :5 X 2 < 00 for all x E E, and for r, S =f= 0 let Il = ( X ~- XD(X; - x;rl, XsX r _ xrX s }(lISl-(lIrl B ( / )lIr I 2 1 2 { r,s = W s [1 - (s/r)](X; - XD

Then we have (Beesack, 1983, p. 334) M[s](g, A):5 Br,sM[r](g, A)

(r < s, r, s =f= 0).

(4.10)

For the cases involving r = 0 or s = 0, let

= ( X ~- XD/log(X2/X1) , B, = (1l,/(et))1I'X 1 1 e x p ( X ~Il,)/ Il,

for t =f= O. Then M[S](g,A):5BsM[OJ(g,A) if O<s,

(4.11)

M [ U ) ( g , A ) ~ B ; 1 M [ r ) (ifg , r
(4.12)

111

4.1. Inequalities for Means

4.6. Theorem. Let g, L, A, 'ljJ, X be as in Theorem 4.4, with m z: 'ljJ(g(t)):o; M for tEE, and = XO 'ljJ-l. (a) Let '(x) exist and be strictly increasing on [m, M], and A> 0 be determined as in Theorem 3.41(a). If X is superadditive on 1= IR+, then (4.13) when X is increasing, while the inequality is reversed when X is decreasing.

(b) If '(x) exists and is strictly decreasing on [m, M], A is determined as in Theorem 3.41(b), and X is superadditive, then (4.14) when X is increasing, with the inequality reversed when X is decreasing.

4.7. Remark. As in Beesack (1983, p. 336), we obtain the following results for the generalized classical means: M[sl(g, A)::; C r. s + M[rl(g, A) for

provided that 0 < Xl::; g(x) -s X 2 for x tl

C

=

E

X ~ X ; X

S -

r

s

2= 1,

(4.15)

E, where C r,s is defined by

XD- 1,

( X ~- XD(X; -

= { X; r,s

0 =1= r < s,

(W)s/(S-r)}lIS.

+r- -s ~

'

and MIsl(g, A):o; Cs + M[OI(g, A) for

s

2=

1,

(4.16)

where C, is defined by tl

Cs

=

{

=

( X ~-

XD/log(X 2/X 1) ,

tl[ log (tl)- - 1]}lIS.

X~ I O g X2 - X~ l o g Xl + log(X 2/X1)

s

As noted by Beesack (1983), the constants not, in general, the best possible. 0

s

c.S'

C, in (4.15), (4.16) are

4.8. Remark. Similarly, we can give a generalization of Theorem 5 of 0 Beesack (1983). The above results are given in Beesack and Pecaric (1985a). A further generalization of Theorems 4.4 and 4.6 is given in Pecaric and Beesack (1987a).

112

4. Applications of Jensen's Inequality

4.9. Theorem. Let g, L, A, 1jJ, X be as in Theorem 4.4, where I is an arbitrary interval. If F(u, v) is a real-valued function defined on [m, Mf and is increasing in u, then F(M",(g, A), Mx(g, A»:s max F[1jJ-l(81jJ(m) 8e[O.l]

+ (1-

8)1jJ(M», X- 1(8x(m) + (1- 8)X(M))]

(4.17)

holds provided that 1jJ is increasing, 1jJ 0 X- 1 is convex, and F(u, v) is a real-valued function defined on I

X

I and is increasing in u.

Suppose that X is increasing on 1. Let Fl(x, y) = F(1jJ-\x), 1jJ-l(y», epl(X) = 1jJ(X-1(x», gl = xog, ml = x(m), and M1 = X(M). Then the conclusion follows from Theorem 4.1 when applied to F1 , epl' gl' If X is decreasing on I, we need only define ml = X(M) and M1 = x(m). Then (4.4) implies

Proof.

F(M",(g, A), Mx(g, A»:s max F( 1jJ-l( 81jJ(M) 8e[O,1]

+ (1 which is equivalent to (4.17).

~ 8)1jJ(m», X-

1(8x(M)

+ (1- 8)x(m»),

0

4.10. Remarks. (a) In the special case F(x,y)=x-y, X(x)=x, and A(g) = Hgdt, (4.17) yields an inequality in Knopp (1935, Satz 2). (b) Theorem 4.9 is a generalization of a result of Beck (1969), who considered quasi arithmetic mean values Mq,(r.7 Piep(Xi» (see also Bullen, Mitrinovic, and Vasic (1987, pp. 256-258». (c) Similarly, we can apply Theorem 3.24 to prove generalizations of index-set type inequalities (see Bullen, Mitrinovic, and Vasic, 1987, pp. 94-98, 103-105, 173-176, and 234-242. 0

4.2.

Holder's and Minkowski's Inequalities

We begin this section with a lemma for non-normalized isotonic linear functionals which includes the corresponding versions of Jensen's inequality and Theorems 3.37,3.39, and 3.41. 4.11. Lemma. Let L satisfy conditions L1, L2, and A satisfy conditions AI, A2 (defined in (3.27)-(3.31» on a base set E. Suppose that k E L with

4.2. HOlder's and Minkowski's Inequalities

113

k 2= 0 on E and A(k) > 0, and that

(4.18) If, in addition, I = [m, M] where

-00

< m < M < 00, then

A(k
+ [A(kg1) -

mA(k)]
(4.19)

Moreover, when

(4.20)

or

(4.21) where A is determined as in Theorems 3.39 and 3.41, respectively.

4.12. Theorem (Holder's inequality for isotonic functionals). Let L satisfy conditions Ll, L2, and A satisfy conditions A1, A2 on a base set E. Let p > 1 and q = p/(p - 1) so that p-l + q-l = 1. If w, f, g 2= 0 on E and wf", wgq , wfg E L, then we have (4.22) In the case 0 0 (or p
Proof. First assume that A(wgq ) > 0, and p> 1. Then (4.22) follows from (4.18) by substitutions k

=

wg q ;

(4.23)

this is so because k E L, kg, = wfg E L, and k 0, we may apply (4.22) with p, q, f, g replaced by q, p, g, f to again obtain (4.22). Finally, suppose that A(wgq ) = 0 and A(wjP) = O. Since 1 1 O:5wfg:5-wjP +-wg q p q

on E,

it follows that A(wfg) = O. This completes the proof for the case p > 1.

114

4. Applications of Jensen's Inequality

°

For the case

1, and may apply (4.22) with p, q, f, g replaced by P, Q=(l-p)-l, fl=(fg'f and gl=g-P; in this q, case wff= wfg, wgf = wg and Wflgl = wJP are all in L. Thus we obtain A(wfP) :5 AP(wfg)A1-P(wgq),

which is (4.22) with the inequality reversed provided A(wg q ) > 0. Finally, if p < 0, then 0< q < 1 and we may apply a similar argument with p, q, f, g replaced by q, p, g, f provided A(wfP) > 0. 0 4.13. Theorem (Minkowski's inequality for isotonic functionals), Let L and A be as in Theorem 4.12. If P > 1 and w, f, g ~ on E with wi", wgP, w(f + g'f

°

E

L, then AlIp(w(f + g'f):5AlIP(wJP)

°

°

+ Al/P(wgP).

(4.24)

If < p < 1 or if p < and A(wfP) > 0, A(wgP) > 0, then the reverse inequality in (4.24) holds.

Proof.

This is an immediate consequence of Theorem 4.12.

4.14. Theorem.

0

Let L and A satisfy conditions Ll, L2 and AI, A2 on a q, base set E. Let p > 1, q = pl(p -1), and w, f, g ~ on E with wf", wg wfg E L. If < m :5 f(x)g-q/P(x) :5 M for x E E, then (M - m)A(wJP) + (mMP - MmP)A(wgq):5 (MP - mP)A(wfg). (4.25)

°

°

q) If P < 0, then (4.25) also holds provided either A (wfP) > Oar A(wg > 0; if o

Oar A(wg > 0.

Proof.

First we note that if p > 0, then we have 0:5 mrwg"

:5

wfP :5 M'wg"

on E,

and the inequalities are reversed if p < 0. In particular, this shows that q) A(wfP), A(wg are either both zero or both positive for all p. If q A( wg ) > 0, then, since (x ) = x'' is convex for either p > 1 or p < 0, (4.25) follows from (4.19) by the substitutions in (4.23). If P > 1 and A(wgq) = 0, then (4.25) also holds because it reduces to the case 0:5 (MP - mP)A(wgf). q) q) If p < and either A(wfP) > or A(wg > 0, then A(wg > 0; thus (4.25) holds. (Note that (4.25) does not hold if A(wJP) = A(wgq) = 0

°

°

4.2. Holder's and Minkowski's Inequalities

115

unless A (wfg) =0, which need not be the case.) If 0 (x ) = x" is concave on [m, M] and the reverse inequality in (4.19) holds. The substitutions in (4.22) then yield the reverse inequality in (4.25) provided A(wgq ) > 0, and the proof is complete. (Again we note that the reverse inequality in (4.25) does not hold if A(wfP) = A(wgq ) = 0 D unless A(wfg) = 0.)

4.15. Remark. Theorem 4.14 is a generalization of the Diaz-Metcalf Inequality, which is the special case p = q = 2 with w == 1 and A(t) = 'l.7A or AU) = f ~ f d x . See, for example, Bullen, Mitrinovic, and Vasic (1987, p. 109), and Mitrinovic (1970, pp. 61-63). D 4.16. Theorem. then

Let L, A, p, q, w, f, g be as in Theorem 4.14. IfP > 1,

A(wfg) ~ Ip!lIp \qlllq

( M - m)lIP ImMP - M IMP -mPI

m

PllIq

A lip (wfP)AlI q( wg q).

(4.26) If p < 0 or 0 < p < 1, then the reverse inequality in (4.26) holds provided either A(wfP) > 0 or A(wgq ) > O.

In the case p > 1 or p < 0, (x) = x'' is convex on 1= [m, M]; thus when applying (4.20) with the substitutions in (4.23) we have

Proof.

or (4.27) In the case p > 1, (4.26) dearly holds when A(wgq ) = O. As noted in the proof of Theorem 4.14, we may assume that A(wfP) > 0 and A(wgq ) > 0 for the rest of the proof. If p > 1 and A(wgq ) > 0, then from (4.27) we obtain A(wfg)

p(wfP )A lI q( wg q),

~A-lip A II

where A is determined as in Theorem 3.39(a) with (x) =x P on 1= [m, M]. Using the same notation in Theorem 3.39(a), we find Ii

= (MP

x

- mP)/(M - m), = q(mMP - MmP)/(MP - mr], p(mMP A = p-lql- MmP)I-p(M - m)-l(MP - mP)p;

116

4. Applications of Jensen's Inequality

thus (4.26) follows for p > 1. Now suppose that 0 < P < 1 and P = P -1 > 1. We apply the result in (4.27) with p, q, t. g replaced by P, Q = (1- P )-1, it = (fgr, and gl = g-P, for which wfi = wfg, wgF = wgq, Wf1g1 = wI;,. Moreover, f1g1Q1P = holds, and since m 5,fg-q1p 5, M on E, it follows that

re :

m1 = m" 5, f1glQIP 5, MP

= M1

on E.

Thus the modified (4.27) yields

A(wfg)5, AlA1Ip(wr)A1Iq(wgq), where

A1 = p-1Q1-P(m1Mf- M1mD 1-P(M1 - m1)-1(Mf- mDP. This reduces to the constant term on the right-hand side of (4.26). Finally, if p < 0, then 0 < q < 1 and we may use a similar argument and, with P1 = q, q = p, f1 = g and gl = f We then have f1g1q,lp, = si since plq <0,

?"

The reverse inequality in (4.26) is still valid since A(wgi1 ) = A(wfP) > 0, consequently we obtain

A(wfg) 5, s'" Ipillp

M 1- m1)lIq I:;1- Mq ~m

(M

1

M1

m1qillp

AlIq(wgq)AlIP(wfP).

1

By writing -plq = 1- p, the factor on the right-hand side of this inequality reduces to that in (4.26) after an elementary calculation. This completes the proof of the theorem. 0

4.17. Remark. It is also possible to obtain (4.26) (or the reverse inequality in (4.26)) from (4.25) by using the arithmetic-geometric means inequality. Thus, for example, if p > 1 and p-1 + q-1 = 1, then the left-hand side of (4.25) is bounded below by {p(M - m)A(wr)} IIp{q(mMP - MmP)A(wgq)}lIq, and this reduces to the right-hand side of (4.26), with a factor of (MP - m'']. This shows that the inequality in (4.25) is sharper than that in 0 (4.26). 4.18. Theorem. Let L, A, p, w, f, g be as in Theorem 4.14, and let 0< m < F(x) 5, M and 05, G(x) 5, M for x E E, where F = f(f + g)-qlp,

4.2. Holder's and Minkowski's Inequalities

G

= g(f + g)-qIP.

117

If p > 1, then

A IIP(w(f + gY) 2:: K(p, q, m, M){AIIP(wfP ) + A IIP(wg P)}

(4.28)

holds where K(p, q, m, M) is the constant on the right-hand side of (4.26). If 0 < p < 1 or p < 0, then the reverse inequality in (4.28) holds provided that A(w(f+ gY) > 0 for p < o.

Proof.

This follows immediately from Theorem 4.16.

0

4.19. Remarks. (a) By writing s = ar + bt, where a + b = 1, Holder's inequality yields the following Liapunov inequality for isotonic functionals: (4.29) If we define the means of order r by

M[r](g, A) = A(gr)lIr for

r

=1=

0

and A(1) = 1, then (4.29) can be written in the form M[sl(g, A) :0:; M[r](g, A)(rls)«t-s)/(t-r))M[t](g, A )(tls)«s-r)/(t-r))

for

0 < r <s

< t. (4.30)

Hence by the arithmetic-geometric means inequality we have

r(t - s) -[

t(s - r) -[

-[I I I MS(g,A):O:;Mr(g,A)+Mt(g,A), s t-r s t-r

or equivalently, M[t](g, A) - M[rl(g, A) <: s(t - r) M[t](g, A) - M[s](g, A) - r(t - s)

for

0 < r < s < t.

This inequality was proposed by Hsu (1955) (as a problem for the classical means); see Beesack (1983), and Pecaric and Wang (1988). (b) In Beesack (1983), the last inequality in (a) for the classical means M(r) (Hsu's inequality) was proved by showing that the function M(r- 1 ) is convex for r > O. Here we give a shorter and simpler proof for a more powerful result: The function f(s) = M[lIs](g, A) is log-convex (hence convex) for s > O. Indeed, this follows from the Liapunov Inequality (4.30) by the substitutions t=x- 1, r=y-l, and t(s-r)/[s(t-r)]=A,

118

4. Applications of Jensen's Inequality

r'.

which yields 1- A = r(t - s)/[s(t - r)], s = [Ax + (1- A)y inequality f(AX+(1-A)y):;f(x)y(y)1-A for

O<x
and the

0<1.<1.

Consequently, the log-convexity property of f(s) holds. (c) Holder's inequality for isotonic functionals can be stated in the following form: Let L satisfy conditions L1, L2, and A satisfy conditions A1, A2 on a base set E. If O
E

=f~/A,

IR then (4.31) yields (4.32)

~ Hence the function G(r) = by using the substitutions f ~ ! ' and g f". A(!') is log-convex (thus it is convex) on IR. (e) Let us consider the function

g(x) =

DA(fr Dl f I ~ - x )

l Iqj

(4.33)

where A is a positive linear functional (we assume that all expressions exist, i.e., if we write A(s) we assume that s E L), fk'S are positive functions, qi > 0 (i = 1, ... , n) with 1:1 (l!qk) = 1. By using Remarks 4.19 and Theorem 2.26, we have that g is a log-convex function (thus it is convex), and for IxI:; Iy I (xy > 0) we have g(x):; g(y). This result is proved in Mitrinovic and Pecaric (1987), and it is a generalization of results in Callebaut (1965), Daykin and Eliezer (1968), Godunova and Cebaevskaja (1971), McLaughlin and Metcalf (1967), and Pecaric and Beesack (1986). As a special case we have the following generalization of the well-known Callebaut's inequalities (1965): If 1:; a:; f3 :; 2 or f3 :; a:; 1, then A(fg)Z:;A(fcxgz-a')A(fz-cxg CX) :; A(ff3g Z- (3)A(f z- f3 g f3 ) :; A(fz)A(gZ). (4.34)

(f) Some further generalizations of the previous results are given in 0 Pecaric (1989f).

4.3. Dresher's Inequality

4.3.

119

Dresher's Inequality

First, we consider a two-parameter family of means, Bp,q(f, A), defined by

Bp,if, A) = {A(fp)/A(rnl/(p-q) for

p"* q,

(4.35)

Bp,p(f, A) = exp{A(fP10gf)/A(fP)},

for p, q E IR. We assume that f(x) > 0 for x E E, JP E L, JP logf E L for all p E IR, that L satisfies Ll, L2, and that A: L ~IR satisfies AI, A2. When applying the known result for convex functions

(xz -Xl)-I{cp(XZ) - cp(Xln:s; (yz - Yl)-I{cp(yZ) - CP(Yl)} Xz:s; Yz,

Yl "*Yz

to the convex function cp(x) = 10gA(r), we can obtain

Bxz,x,(f,A):s; B)Iz,y,(f, A) for

Xl:S; Yh Xz:S; Yz and XI "* Xz,

Yl "*Yz.

(4.36)

We now show that (4.36) holds even if Xl = Xz or Yl = Yz. To prove this we use the fact that M[r1(g, A) is an increasing function of r on IR and, in particular, that

A(fx,-xZ)lI(x,-xz) :5 exp(A(logf):s; A(fx2-X,)1!(xz-x,) for

Xl < Xz. (4.37)

We then apply (4.37) to the isotonic linear functional AI: E ~IR defined for certain gEL (see Lemma 4.11) by Ai(g) = A(rig)/A(r') for i = 1, 2. By taking i = 1, the right-hand inequality (4.37) reduces to

Bx"x,(f, A):s; Bxz,x,(f,A) for

Xl < Xz.

Similarly, by taking i = 2, the left-hand inequality of (4.37) reduces to

Bx2,x,(f, A) :5 Bx2,xz(f, A) for

Xl < Xz.

These two inequalities show that (4.36) is valid even when Xl = Xz or Yl = Yz· Note that (4.36) implies that Bp,if, A) is an increasing function of both p and q for all p, q E IR.

4.20. Remarks. (a) As in Stolarsky (1975), let us take E = [x, y] for fixed x, Y with 0 < X < Y < 00, P = s - 1, q = r - 1, f(t) = t, and

f y

A(g) =

x

get) dtl(y - x),

120

4. Applications of Jensen's Inequality

where L is the set of all bounded measurable functions on E. Then the means in (4.35) become, when writing E(r, s) = Bs-I,r-I(t, A),

r (yS _ XS)}II(s-r) E(r,s)= {;(yr_xr) for E(r, 0) = E(O, r) E(r, r)

yr

xr

= { r(lny -lnx)

rs =1= 0,

r=l=s, }llr for

r =1= 0,

= e- lIr(x 'l yY') II(x' - Y') for r =1= 0, X

and E(O, 0) = yXy. By (4.36), the function E(r, s) is increasing in both r and s. These means were introduced by Stolarsky (1975), and have been considered by Leach and Sholander (1978,1983). The proof of the monotonicity property of E(r, s) given in Leach and Sholander (1978) appears to be more difficult than that given above. (b) In (4.35), take E={1,2, ... ,n}, f(k)=ak>O(kEE), and A(f) = I : Z ~ lakin. Then

B p . q ( f , A ) = D ( P , q ) = { * a ~ / ~ a kforr ( pp=l=q, -q) D(p, p)

= exp{

~ a ~log s, / ~af },

becomes the Gini-Dresherrnean (for sums), introduced by Gini (1926) and Dresher (1953). Again we can conclude that D(p, q) is increasing in both p and q; see Brenner (1978, Section 3), Bullen, Mitrinovic, and Vasic (1987, pp. 189-190) and the references given there. 0 For a second result related to the Gini-Dresher means we prove the following generalization of Dresher's inequality (1953).

4.21. Theorem. Let A, B: L- ~ be isotonic linear functionals. If fi, u;:E-[O, oo) with f'[, (1:lfiY, u ~ , (1:7u;yEL, wherep2=l>r>O and A(uD > 0 for 1:5 i :5 n, then n

{

)p}lI(p_r)

~fi B( ~ u;)' A(

:5

{A(ff)}II(p-r).

n

~

B(uD

4.3. Dresher's Inequality

Proof.

121

By Minkowski's inequality for isotonic functionals, we have

Hence

={ x

n

~ [(A(ff))lIp-r)](p-r)!p

}p!(p-r)

n {

~ [(B(uD}-lI(p-r)](p-r)!(-r)

}-r!(p-r)

n

:S

2: A(ff)lI(p-r)B(uD-lI(p-r) 1

follows from the basic Holder's inequality for sums with the conjugate indices P = (p - r)/p and Q = (p - r)/( -r). D Dresher (1953, Section 7) dealt with the special case n = 2, B = A, E = [0, 1], and A(t) = fbi du. A simpler proof, also for this case (essentially the proof given above), can be found in Danskin (1952). A special case for 1 :S P :S 2 and r = p - 1 was treated earlier by Beckenbach (1950). See also Beckenbach and Bellman (1961, 1965, pp. 27-28) for other proofs of the results of Beckenbach and Dresher. The following is a simple consequence of the above result: Let a, b, p, q be positive n-tuples, a E A, b e B, A x B be a convex set. If 0 < r < 1 < S, then

is a convex function on A x B. The proof of this result can be found in Godunova (1967a).

122

4. Applications of Jensen's Inequality

4.4.

Beckenbach's Inequality

Let E be a nonempty set, .sI1 be an algebra of subsets of E, and let L be a linear class of real-valued functions satisfying conditions L1-L3 defined in (3.27)-(3.29). The following two results are given in Pecaric and Beesack (1987b): 4.22. Theorem. Let L satisfy L1, L2, L3 on a nonempty set E and let A be an isotonic functional on L. Suppose E 1 E.sI1 has A(C E,) > 0 where E 2 = E\E l' Then for every nonnegative gEL such that gP E L (p > 1) and A(gCEJ >0, we have A(gP)1/P/A(g) 2 : : A ( g ~ Y / P / A ( g E t ) '

(4.38)

where

First observe that A(g) 2:: A (gC Et) > 0 and A(gE1) > A (gCEJ > 0; thus both sides of (4.38) are well-defined. Applying Theorem 3.26 with A replaced by A 1(g)=A(g)/A(1), F(x, y) =X llP/y1/P, ep(x)=xP, and with 1= J = [0, (0), we have

Proof.

A(gP)1/p / A(g) 2:: inf A ( g ~ ' . X ) l /A(gE IP t' x),

(4.39)

XE[

where gE,.x(t)

= g(t)CE,(t) + xCE,(t).

Hence we have By elementary calculus we find that the minimum value of k(x) = {A(gPCE,) + x PA (C Ez)}lIP/ {A(gC E,) + XA(CE2)},

x> 0

is attained at x = {A(gPCEJ/A(gCEJ}lI(p-l). Thus (4.38) follows from (4.39). 0 4.23. Theorem. Let L satisfy Ll, L2, L3 on a nonempty set E, and let A be an isotonic linear functional on L. Suppose the nonnegative functions t. g: E ~'U? are such that [P, gq,fg E L, where p > 1, p-1 + «:' = 1. If E 1 E .sI1 has A(fgCEJ > 0 and A(gqCE2) > 0 where E 2 = E\£1, then A (fP)lIPjA(fg) 2 : : A ( j ~ , ) l I P j A ( j g ) ,

(4.40)

4.4. Beckenbach's Inequality

123

where

Proof. We apply Theorem 4.22 to the functional A1(gl) defined for certain g l : E ~ ~by A1(gl)=A(kg1)/A(k), with k=gqEL. Since A ( k ) ~ A ( g q C E 2 by » 0 Lemma , 4.11 and the fact that <jJ(u)=u P is convex on 1= [0,00), we have

for all functions gl: E ~ ~ for which gqgl ELand gqgf E L. But this is precisely the inequality corresponding to (2.6) for the functional A1(gl) and <jJ(u) = u", and this in turn implies the validity of Theorem 3.31, and hence that of Theorem 4.22 with A = AI. Consequently, we may apply Theorem 4.22 with the function g replaced by gl = fg- q /p because both A1(gl C E) > 0 and A1(CE,} > 0 hold. Now gqgl = fg and gqgf = JP; thus

It is easy to verify that (4.38), with A and g replaced by Al and gl, respectively, reduces to

(4.41) where

gdt)

= gl(t)CE,(t) + {A(JPCE)/A(fgCE)}1I(p-IlCE2(t).

Hence by 1/(p -1) = q -1 = q/p, we have

gqgE , = g{fCEI + [gA(JPCE,/A(fgCE)]q/PC E,} = fgE " g q g ~ =, JPC E, + [gA(JPC E,/A(fgCE,WCE2 = It, and so (4.40) follows from (4.41).

0

4.24. Remark. (a) Beckenbach's inequality (1966) is the special case of Theorem 4.23 corresponding to E = {1, 2, ... , n}, £1 = {1, 2, ... ,m} (where 1:s: m < n), L = ~ n , the vector space of all real n-vectors a=(al, ... ,a n ) , and A(a)=I:7a;. Let a=(al, ... ,a n ) , b= (b 1 , . . . , b n ) be two n-tuples of positive real numbers, and p, q be real numbers such that p-l + q-l = 1 (p > 1). If 0 < m < n, then P ( n~af )l/P( n~a.b, )-1 ~ (n~ iif)lI ( n~ii;b; )-1 ,

(4.42)

124

4. Applications of Jensen's Inequality

where

a, ii, = I

{ (b i

m

m

1=1

1=1

2: a)/2:

for

1 ~ i ~ m,

for

m

qlp

ajbj)

+ 1 ~ i ~n,

and the equality in (4.42) holds iff iii = a, for all i. The inequality in (4.42) is reversed if p < 1 and p *0. Furthermore, for m = 1 (4.42) reduces to HOlder's inequality. (b) Some related results are also given in Pecaric and Beesack (1987b). (c) In the same fashion we can give generalizations of Theorem 4.23 and Corollaries 3, 4 in Bullen, Mitronovic, and Vasic (1987, pp. 258-260). 0

4.5. Aczel's and Related Inequalities We noted that from Jensen's inequality we can easily obtain Holder's inequality. Similarly, from Theorem 3.1 (the reverse Jensen's inequality) we can obtain Aczel's (1956) (Mitrinovic, 1970, pp. 57-58) and Popoviciu's (1959b) inequalities, and their proofs can be found in Vasic and Pecaric (1982b). The ideas can be used to prove more general results. First, we note that the following generalization of Theorem 4.22 is valid:

4.25. Lemma.

Let E, L, A, t/J be defined as in Lemma 4.11 and assume that pEL with p ~ Oon E and O < A ( p ) < U (ua-A(pg))/(uE ~ , A(p)) E I (a E I), pg ELand pt/J(g) E L. Then t/J (

Proof.

ua - A(pg)) > ut/J(a) - A(pt/J(g)) . u -A(p) u -A(p)

(4.43)

By letting p=u,

in Theorem 3.1 for n

q = -A(p),

= 2,

b = A(pq)/A(p),

i.e., from

A..(pa+qb»_pt/J(a)+qt/J(b) ." p+q p+q

for

q < 0, P

+ 1 > 0, pa+qb p+q

E

I,

4.5. Aczel's and Related Inequalities

125

we have 4J(ua - A(pg)) ~ u4J(a) - A(p)4J(A(pg))/A(p)). u-A(p) u-A(p)

By applying Lemma 4.11 we have (4.43).

0

4.26. Theorem (Aczel's inequality for isotonic functionals). Let L satisfy Ll, L2, and A satisfy A1, A2 on a base set E. IfF, s'. fg ELand g ~- A(g2) > 0 (or f ~ - A(F) > 0), where go, fo are real numbers, then (fogo - A (fg))2 ~ ( f ~- A ( f 2 ) ) ( g- ~ A (g2)).

(4.44)

4.27. Theorem (Popoviciu's inequality for isotonic functionals). Let A and L be as in Theorem 4.26. If P > 1, q > 1, p-l + «: = 1, f, g ~ 0 on E, I", s". fg E L, and fo, go are positive real numbers such that (4.45) then

(f£ - A (jP))lI P(gg - A(gq))lIq 75, fogo - A(fg).

(4.46)

In the case 0 < P < 1 and A(g!l) > 0 (or p < 0 and A(fP) > 0), the reverse inequality in (4.46) holds.

Proof.

By applying the substitutions

4J(x) =xP (p > 1),

in (4.43), we obtain (fogo - A(fg)Y ~ (f£ - A (fP))(gg - A(gq)Y-l

(4.47)

if the first condition in (4.45) is satisfied. If the second condition is also satisfied, then from Holder's inequality we have A(fg) < A (fP)lIp A(gq)lIq < fogo, i.e., fogo - A(fg) > O. Thus from (4.47) we obtain (4.46). 0

4.28. Remark. Note that for p = 2, we obtain Aczel's inequality from (4.47). Thus (4.47) is a generalization of A C Z ( ~ I ' sand Popoviciu's inequalities. 0 4.29. Theorem (Bellman's inequality for isotonic functionals). Let A and L be as in Lemma 4.25. Let f, g ~ 0 on E with I", gq, (f + g) E L, and fa, go be positive real numbers satisfying (4.48)

126

4. Applications of Jensen's Inequality

If p > 1, then «(fg - A(fP»lIP(gg - A(gP»lIPY:S (fo + goY - A«(f + gY). (4.49)

If 0 < p < 1 or p < 0 and A(fp) > 0, then the inequality in (4.49) is reversed.

Proof. We give proof for p > 1 only; the proof for the other case is similar. Using (4.48) and the Minkowski Inequality we have (fo + goY> (A(fP)lIp

+ A(gP)lIPY

From the discrete Minkowski Inequality for n «a1

+ b 1Y + (az + b zy)lIp :s (af +

~ A « ( +f gY)·

= 2, i.e.,

a ~ ) l I P+ (M + bf)lIP ,

the substitutions

and the Minkowski Inequality, we have

«(fg - A(fp»llp + (gg - A(gP»lIPY :s (fO + goY - (A(fp)llp + A(gP)lI PY :s (10 + goY - A«(f + gY).

0

4.30. Remark. The last inequality in the above proof is an interpolation of (4.49), and it is a generalization of a result in Mitrinovic and Pecaric (1988a). Similarly we can prove Theorem 4.27 by using only Holder's inequality, i.e., we can obtain a similar interpolation of Popoviciu's inequality. Of course, we can also obtain other results similar to those that can be obtained from HOlder's inequality. 0

4.6.

Further Generalizations of Holder's and Minkowski's Inequalities

Let us further consider the generalized means defined in Section 4.1, and assume that I = (a, b), -00:S a < b < 00, 1J11"'" 1J1n: I ~ ~ are continuous and strictly monotonic, X: I ~ ~ is continuous and increasing; L and A satisfy the conditions Ll, L2 and AI, A2 defined in (3.27)-(3.31), with A(l) = 1 on a base set E; gl" .. , gn : E ~ ~ and f:I1 x··· x I n ~ ~ are real-valued functions such that gl(E) c 11, ... , gn(E) c In, 1J11(gl), ... , 1J1n(gn), x(f(gl, ... , gn» E L.

4.6. Further Generalizations

127

We consider the inequality of the form

MX(f(gl, ... ,gn), A) =Sf(M"'l(gl' A), ... ,M"'n(gn, A)), (4.50) and observe

4.31. Theorem. A necessary and sufficient condition for (4.50) to hold is that the function H(Sl' ... , sn) = x(f(1J11 1(Sl)' ... , 1J1;;\sn))

is concave. If H is convex, then the inequality in (4.50) is reversed. Proof. McShane's inequality (see Theorem 2.6 and Remark 2.11(b)) for the function H becomes x(f(1J111(A(fr)), ... , 1J1;;l(A(fn)))) 2':A(X(f(1J111(fl), ... , 1J1;; 1 (fn))))' (4.51) Thus if we let t.

= 1J1i(gi) (i = 1, ... , n), then (4.51) becomes

X(f(M"'l(gl, A), ... ,M"'n(gn, A))) 2':A(X(f(gl' ... ,gn))), (4.52) which is equivalent to (4.50).

0

4.32. Remark. For the special case of discrete functionals with n = 2, this result is given in Beck (1970) (see also Bullen, Mitrinovic, and Vasic (1987, pp. 246-255). 0 The following two corollaries can be proved as in Beck (1970):

4.33. Corollary. Assume that f(x,y)=x+y, and let H(s,t)= X(1J11\s) + 1J12 1(t)), E = 1J1U 1 J 1 ~ , F = 1 J 1 ~ / 1 J 1 ~ , G = X'/ X", and all of 1J1;, 1 J 1 ~ , X', x", 1 J 1 ~ , 1 J 1 ~ be positive. Then Mx(gl + gz, A) =s M"'l(gl, A) + M"'2(gZ, A) holds iff G(x + y) 2': E(x) + F(y). 4.34. Corollary. Assume that f(x, y) =xy, and let H(s, t) = X(1J11 1(S)1J12 1(t)), A(x) = 1J1;(x)/(1J1;(x) + x 1 J 1 ~ ( x ) ) B(x) , = 1 J 1 ~ ( x ) / ( 1 J 1 ~ (+x ) X1J1~(x))C , (x) = X'(x)/(X'(x) + xx"(x)), and 1J1;, 1 J 1 ~ , X', X", 1 J 1 ~ , 1 J 1 ~ be positive. Then

MX(glgZ,A) =s M"'l(gl, A)M"'2(gZ'A) holds iff C(xy) 2':A(x) + B(y).

128

4. Applications of Jensen's Inequality

Several other results related to Holder's and Minkowski's inequalities will be given in other parts of this book. We complete this section by observing the following general result in Bourbaki (1952, pp. 9-14) (see Mitrinovic, 1970, pp. 355-56, and Kuczma, 1985, pp. 203-4): Let P be the set of all mappings of a set S into the nonnegative reals. Let M be a mapping of P into the nonnegative real numbers satisfying (i) M(O) = 0, M(At) = AM(t), where A> 0 and f E P; (ii) f(x)::5 g(x) for all XES and f, g E P implies M(t)::5 M(g); and (iii) M(f + g)::5 M(t) + M(g) for all f, g E P. Let h(t l , . . . , tn) be a real-valued function of n real variables t l , . . . , t n which is defined and continuous for t, 2= 0 (i = 1, , n). Let h have the following properties: (iv) inequalities ti > 0 (i = 1, , n) imply , Atn) = that h(tl>.'" tn) > 0; (v) if A> 0, then h(Atl, Ah(tl, ... , tn); and (vi) the set of all points (t l, ... , tn) in En satisfying t, 2= 0 (i = 1, ... , n) and h(tl>' .. ' tn) ~ 1, is convex. Then (a) fl , ... .t; E P implies

M(h(fl' ... ,fn))::5 h(M(fI), ... , M(fn))'

(4.53)

(b) O
2= 1

(4.54)

and f, g E Pimply

M«f + g)P)lIP::5 M(fP) lip + M(gP)lIP.

(4.55)

4.35. Remark. Some related results were obtained by You (1989).

o 4.7.

Some Inequalities

fOl"

Complex Functionals and Norms

As in Sections 2.1 and 3.2, we assume that L satisfies cOnditions L1, L2 (defined in (3.27) and (3.28)) on a base set E, and that A is an isotonic linear functional on L. We consider the class

L= and the functional

A:L

{ f : E ~ C Ref , E

L, Imf E L},

~C defined by

A(t) = A(Ret) + iA(lmt) = Re(A(t)) + i Im(A(t)). Clearly

A is a complex linear functional on L; that is, A(af + {3g) = aA(t) + {3A(g) for f, gEL.

In the following we give some simple properties of A.

a, {3

E

(4.56)

C. (4.57)

4.7. Some Inequalities for Complex Functionals and Norms

Theorem 4.36.

(a) Iff ELand If I E L, then (4.58)

IA(f) I :5A(lfl)·

(b) Iff E L, If I E L, and a - 8:5 argf(t):5 a 8 are real numbers such that 0 < 8 < rr/2, then

+ 8 for all

tEE, where a,

IA(f)I2= (cos 8)A(lfl).

(c) If in (b) we have a < argf(t):5a then

+8

(4.59)

for all tEE (0< 8 < rr12),

Vi IA(f)I2= 2: A(lfl)·

Proof.

129

(4.60)

(a) For arbitrary but fixed 8 E IR we have Re(eiOA(f» = Re(A(ei0f) = A (Re(ei0f) :5A(lfl).

Suppose that A(f) = Reit • Take 8 = -t to obtain IA(f)1 = r = Re(e-itA(f) :5A(lfl).

(b)

IA(f) I = le- iaA(f)I2= Re e-iaA(f) = A[lfl Re

ei(arg/-a)]

= A(Re(e-iaf)

= A(lfl cos(argf - a»

2= A(cos 8 If I) = (cos 8)A(lfl).

(c) For an arbitrary but fixed complex number z = x + iy, we have Izl2= (Vi12)(lxl + Iyl). Hence IA(f) I = le-iaA(f)I 2= Vi

V;

{IRe(e-iaA(f)1 + IIm(e- iaA(f»1}

= 2: {A(lfl cos(argf -

a»

.

+ A(lfl sm(argf -

a»},

or Vi IA(f)I 2= 2: A(lfl [cos(argf - a)

+ sin(argf -

am·

Since the function cosx+sinx=h(x)2=h(0)=h(rr/2)=1 rr/2, (4.60) follows. 0

for O:5x:5

Note that it follows from (4.59) and (4.60) that if f E L, If I E L, and + 8 for all tEE, where 0 < 8 < rr/2, then

a :5 argf(t) :5 a

V IA(f)1 2= max { 2' cosi8} A(lfl)·

(4.61)

130

4. Applications of Jensen's Inequality

4.37. Remark. (a) The inequality in (4.59) is a generalization of an inequality of Petrovic (1933), and (4.61) is a generalization of a result in Vasic, Janie, and Keckic (1971), namely, the inequality n

I

~ Zk

I

2:

max

{Vi T' cos ()}

n

~ IZkl

for

a s: arg Zk

:5

a

n + () < a +"2'

o

4.38. Theorem. If g: E -- ~ and f: E -- C are such that g2 E L, Ifl 2 E L, Ifgl E L, fg E L, and FE L, then

IA(fg)12:5 ~ A(g2){A(lfI2) + IA(f2)1}. Proof.

II = eiaf,

First we show that there exists a real number a then

A(gRelI)2:0 and

(4.62)

E

~

such that if

A(g Imh) =0.

It then suffices to prove the second of the two identities because, if necessary, we may replace f1 with (-f1) (that is, a by a + n), to yield the first. Let

k(a)

= A(g Im(eiaf) = A(g If I sin(a + argf).

Then k(O) = -k(n). Thus it suffices to prove that k is continuous on [0, n]. But this follows from

Ik(a) - k(ao)1

= IA(g If I [sin(a + argf) - sin(ao + argf)]) I :5IA(lgl . If I . la - aol)! = la - aol A(lgfl)·

With this choice of a, we have

Thus we have

by the special case p = q = 2 of the generalized Holder's inequality. Since 2(Re 1I)2 = Ifd2 + Re(fi) we have

= If/ 2 + Re(fi),

4.7. Some Inequalities for Complex Functionals and Norms

131

But A(Re(fi» = ReA(fi) ~ IA(fDI = IA(F)I;

hence (4.62) follows.

D

Note that the inequality in (4.62) is a generalization of an inequality involving sums due to De Bruijn (see, for example, Mitrinovic, 1970, pp. 313-314). It is an improvement of the generalized Cauchy Inequality IA(fgW ~ A ( l f I 2. A(g2), )

since we have IA(F)I ~ A ( l f I 2by ) (4.58). Our next result concerns a generalization of the well-known inequality of Bohr (see Mitrinovic, 1970, p. 312), or an extension of the inequality 2 n

~ Zk

1

n

n

~ ~ ak IZkl2 for

Zk E

and

1[,

2: aJ;l = 1. 1

I

4.39. Theorem. Let f: E ~I[ and p: E pll(l-r), IfI, plfl', and f E L, then IA(f)lr

~[0,

00). If for r > 1 we have

~ A ( p l l ( l - r ) y - I Ifn. A ( p

(4.63)

Proof. We use (4.56) and HOlder's inequality (Theorem 4.12) with conjugate exponents r, r/(r - 1) to obtain IA(f)lr ~ A ( l f I=YA(p-lIrplir IflY ~ A ( p l l ( r - l ) y - I Ifn. A ( p Similarly, using (4.62) we obtain a refinement of this result for r

D

= 2: (4.64)

provided that f E L, pF E L, and r:'. If I, plfl 2 E L. This follows as above with r = 2, except that (4.62) is used in place of Holder's inequality (with the substitutions g ~p-112 and f ~p1l2f). The inequalities (4.63) and (4.64) are generalizations of results given in Vasic and Keekic (1971b) which deal with the case E = {1, 2, ... ,n} and A(f) = 'L.Uk' 4.40. Theorem. Let
(4.65)

132

4. Applications of Jensen's Inequality

If tjJ is concave and decreasing, then the reverse of the inequality in (4.65) holds.

Proof. By (4.58) we have IA(!)I =sA(lf!). Thus (4.65) follows from the properties of tjJ and Jessen's inequality (2.6). For the reverse inequality, just apply (4.65) to - tjJ. 0 Now for given f : E -- C with f cos e

E

L, let us define the constant

if a -

e =s argf(t) =s a + e for e E (0, ~ ) ,

some a

c, = m a x { ~cos , e}

E

IR, and all tEE,

if a =s argf(t) =s a + e for some a

o

C, by

E

e E (0, ~ ) ,

IR, and all tEE,

otherwise.

Then we have 4.41. Theorem.

(a) A and

A satisfy

1 A(lf!) =s -IA(!)I for f

c,

E

L,

If I E L.

(4.66)

(b) If cjJ: [0, oo]__ 1R is concave and increasing and if A(l) = 1, then A(cjJOfl» =s

c j J ( ~ IAU)I)

(4.67)

f

provided that f E L, IfI E L, and cjJ(lf!) E L. If cjJ is convex and decreasing, then the reverse of the inequality in (4.67) holds.

Proof. Inequality (4.66) follows from (4.59) and (4.61), while (4.67) follows from Jessen's inequality (2.6) and (4.66). 0 4.42. Theorem. Let L, A satisfy conditions Ll, L2 and Ai, A2 defined in (3.27)-(3.31), respectively, and let gEL. If O=sg(t) =sA(g) for all tEE, tjJ: [0, oo) __ 1R is convex with tjJ(O) = 0, and tjJ(g) E L, then A(tjJ(g» =s tjJ(A(g».

(4.68)

4.7. Some Inequalities for Complex Functionals and Norms

133

Proof. This follows from Theorem 12 of Beesack and Pecaric (1985a) with Xo = O. 0 Note that in the case A(1) = 1, equality in (4.68) holds by (2.6). 4.43. Theorem. and iff E

L, IfI E

If ep is concave and increasing on [0,00) with ep(O) L, ep(lfl) E L, and If(t)1 :o:;A(lfl) for all tEE, then

ep(I.4(f)I) :o:;A(ep(lfl))· Proof.

=0

(4.69)

Since -ep is convex, it follows from (4.68) and (4.58) that A(-ep(lfl):o:;

-ep(A(lfl»:o:; -ep(I.4(f)I).

0

4.44. Remark. By applying (4.61) and (4.62) we can obtain additional generalizations of Bohr's inequality similar to those given in Vasic, Janie, and Keckic (1971), and Kocic and Maksimovic (1973). 0 Using (4.65) together with Lemma 4.11, we can prove, as in Theorem 2 of Kocic and Maksimovic (1?73), the following result: Let p(t) > 0 for tEE and ep: [0, o o ) ~IR be strictly convex, with ep(uv):o:; ep(u)ep(v) for u, v> O. Let 1jJ(t) = ep(t)/t for t > 0 and 1jJ(0+) = 0, 1jJ(00) = 00. If X(t) = 1/1jJ-\t), then

4.45. Theorem.

ep(I.4(f)I) :0:; 1jJ(A(X(p)))A(pep(lfl)) holds provided that f E L, IfI E L, x(p) E L, x(p )ep(x(p) IfI) E L, and A(X(p))> O.

pep(lfl) E L,

Note that the inequality in (4.63) is a special case of this result. In the same fashion we can prove converse inequalities similar to those in Vasic, Janie, and Keckic (1971), Kocic and Maksimovic (1973), and Pecaric and Janie (1988) by using the other parts of Lemma 4.11. Similar results for norms can be obtained by using the same arguments. First, we observe the following theorem (Pecaric and Dragomir, 1989a): 4.46. Theorem.

Iffis an increasing convex function for x

2:

0, then (4.70)

where Pi 2: 0, P; = ~ 7 ~ Pi' 1 Xi E V (i = 1, ... , n), and V is an arbitrary normed vector space with norm 11·11.

134

4. Applications of Jensen's Inequality

As a special case of this inequality we have

I I ~xl :s ( ~ 1p}I(1_,»)'-1 ~Pi Ilxill'

for

r> 1.

(4.71)

Similarly, by the reverse of Jensen's inequality, the reverse of the inequality in (4.70) holds under the same conditions on f for the following values of the p/s:

Pi :s 0 (i

Pl>O,

= 2, ... , n) and P; > O.

(4.72)

Also note that if

qi:S 0 for i = 2, ... ,n, and q1:s

( ~ z l q i l l l ( l - ' »1-', ) (4.73)

then (4.74) As a consequence of (4.71) and (4.74) we then have

Ilxl + x211' II xllI' II xzll' - , , - - - ~ - = . : . c . . . . ~_ _ + __ u+v

u

v

for

uv(u+v»O,

II x1 + xzll':s /./X111' + Ilxzll' for uv(u + v) <0, u+v

u

(4.75)

v

where 1 :s r ~ 2. Another generalization of (4.71) is given in Kocic and Maksimovic (1973). A similar result is also given in Pecaric and Janie (1988). A special case of those results is the following result of Delbosco (1980): (4.76) where Xi EX (i = 1, ... , n), and the constant cp,n = nP - 1 for P ~ 1 and cp,n = 1 for O:s P < 1 is the best possible. If z and ware complex numbers and A ~ 2, then van der Corput and Beth (see Mitrinovic, 1970, p. 322) showed that [z + wl A+ [z -

W I A ~ 2 ( l z I IwI A +A).

(4.77)

Klamkin (1975) extended this inequality by showing that z,

2: IDIVI + DzVz + ... + O n V n I A ~ 2 IVilzt n(~

(4.78)

4.7. Some Inequalities for Complex Functionals and Norms

135

where A> 2 or <0, D; = -lor 1 (i = 1, ... , n), the V;'s are vectors in Em and the summation on the left-hand side of (4.78) is taken over all 2n points of (D1 , • . . , Dn ) . The inequality is reversed for 0 < A < 2, while for A= 0,2 it becomes an identity. A geometric interpretation of this result corresponding to the case A = 1 is that: For all parallelotopes of given lengths, the rectangular one has the largest sum of the lengths of the body diagonals. Its proof follows from an identity for A= 2 and some known inequalities (see also Shore, 1980). In Pecaric and Janie (1982) further generalizations are given by using Jensen's inequality and Petrovic's inequality for convex functions. Also, Shore (1980) and Pecaric and Janie (1982) gave some inequalities for norms. However, the well-known Clarkson inequalities (see, for example, Mitrinovic and Vasic, 1977) are closely related to the above results. 4.47. Theorem. If x, yare elements of the space t p (or Lp ) , where p and «: + «:' = 1, then the inequalities 2(llxll P + lIyIlP)q- 1:s IIx + yllq 2(jjxjjP

+ lIyllP):s

jlx

+ Ilx -

yllq,

~2

(4.79)

+y

jjP + Ilx - yjjP:S 2P- 1(jjxjjP + Ijy liP),

(4.80) IIx + yliP

+ IIx -

Y II p :s 2(lIxll

q

+ lIyllqy-l

(4.81)

hold. If 1 < P :s 2, then the reverse inequalities are valid. For some similar results, see Dragomir and Sandor (1987) and Koskela (1979).

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Chapter 5

5.1.

Hermite-Hadamard’s and JensenPetrovik’s Inequalities

Hermite-Hadamard’s Inequality

On November 22, 1881, Hermite (1822-1901) sent a letter to the journal Mathesis. An extract from that letter was published in Mathesis 3 (1883, p . 82). I t reads as follows (see Mitrinovii. and Lackovii., 1985): “Sur deux Iimites d’une inte‘grale de‘finie. Soit f ( x ) une fonction qui varie toujours dans le m h e sens de x = a, ri x = b. On aura les relations

ou bien

a

suivant que la courbe y =f ( x ) tourne sa convexite‘ ou sa concavite‘ vers l’axe des abcisses. En faisant dans ces formules f ( x ) = 1 / ( 1 +x ) , a = 0 , b = x il vient XZ x - -< 2+x

log(1 + x ) < x -~ x 2 2(1+x)‘

3,

It is interesting to note that this short note of Hermite is nowhere mentioned in mathematical literature, and that these important inequalities (of Hermite) are not widely known as Hermite’s result. His note is recorded neither in the authoritative journal Jahrbuch iiber die Fortschritte der Mathematik nor in Hermite’s collected papers, which 137

138

5. Hermite-Hadamard's and Jensen-Petrovic's Inequalities

were published "so US les auspices de l'Academie des sciences de Paris par Emile Picard (1905-1917), membre de l'Institut." In the booklet on Hermite by Jordan and Mansion (1901), Mansion published a bibliography of Hermite's writings, but this note in Mathesis was not included. Beckenbach (1948, p. 441), a leading expert on the history and theory of complex functions, wrote that the first inequality in (5.1) was proved by Hadamard (1893, see, in particular, pp. 174-76, 186) and apparently was not aware of Hermite's result. It should be mentioned that Fejer (1880-1959), while studying trigonometric polynomials (1906), obtained inequalities which generalize that of Hermite, but again Hermite's work was not acknowledged. In its original form, Fejer's result reads: "Consider the integral f ~ f ( x ) g ( xdx, ) where f is a convex function in the interval (a, b) and g is a positive function in the same interval such that g(a

+ t) = g(b - t),

1 2

O ~ t ~ - ( a + b ) ,

i.e., y = g(x) is a symmetric curve with respect to the straight line which contains the point (Ha + b), 0) and is normal to the x-axis. Under those conditions the following inequalities are valid: ~

f(a;b)

J a

~

~

g ( x ) d x J~ f ( x ) g ( X ) d X ~ f ( a ) ; f J( g(x)dx." b) a

(5.2)

a

Clearly, for g(x) == 1 and x E (a, b) we obtain Hermite's inequalities. Therefore, Hermite's important result in (5.1), which provides a necessary and sufficient condition for a function f to be convex in (a, b), has not been credited to him in mathematical literature. In fact, the term "convex" also stems from a result obtained by Hermite in 1881 and published in 1883 as a short note in Mathesis, a journal of elementary mathematics. There are results of lesser importance which have received more attention in the area of inequalities, but unfortunately this fundamental work of Hermite has been frequently cited without the correct identification of its original author. It is obvious that (5.1) is an interpolating inequality for (5.3)

5.1. Hermite-Hadamard's Inequality

139

More than twenty years after Hermite's work was published, J. L. W. V. Jensen (1905, 1906) defined convex functions (i.e., J-convex functions) using inequality (5.3). His remark, which we cite here, was shown to be justified: "Il me semble que la notion de fonction convexe est a peu pres aussi fondamentale que celles-ci: fonction positive, fonction croissante. Si je ne tromp pas en ceci, la notion devra trouver sa place dans les expositions elementaires de la theorie des fonctions reelles" (see Jensen, 1906, p. 191). Indeed, it is not easy to give a complete treatment of the literature when studying convex functions, but the importance of Hermite's result is obvious. Since the inequalities in (5.1) have been known as Hadamard's inequalities, in this volume we shall call them the Hermite-Hadamard's inequalities. In the classic book of Hardy, Littlewood, and P6lya (1934, 1952, p. 98) the following result is given:

5.1. Theorem. A necessary and sufficient condition that a continuous functionf(x) be convex in (a, b) is that x+h

f ( x ) ~ 2J ~ f(t)dt

for

a ss x r-h-c x-r h ss b.

(5.4)

x-h

It can be shown that this result is equivalent to the first inequality in

(5.1) whenfis continuous on [a, b]. However, it remains unclear by who and when the transition from the inequality (5.1) to the convexity criterion in (5.4) was made. Note that a generalization of Theorem 5.1 is already given in Theorem 2.14. In the following we shall give some other generalizations. For f E C(I), h >0, and x E 11 (h ) = {t:t - h, t + h e I}, the operator Sh defined by x+h

ShU, x) =

2~

J f(t) dt

(5.5)

x-h

is often called a Steklov function, although it is an operator mapping e(I) into C(I1 ) . For a finite interval I = [a, b], the maximum value of h can be (b - a)/2. In this case 11 contains a single point and Sh becomes a functional. The Hermite-Hadamard inequality (5.4) now has the form f ( X ) ~ S h U , for X ) xEI1(h) and is equivalent to the convexity of the function. The iterated Steklov operators (with the step h > 0) S ~ (n E .N)

140

5. Hermite-Hadamard's and Jensen-Petrovlc's Inequalities

are defined by x+h

Sh(f, h) =

s2(f, x) =f(x),

2~

f

Sh-1(f, x) dt,

(5.6)

x-h

where n E X, x E In(h) = {t:t - nh, t + nh E I}. For convenience we write S; instead of sL and (5.4) becomes s2(f, x)::5 Sh
E

C(I) is convex iff for every h > 0 and

(5.7)

f(x) ::5 Sh(f, x) holds for every fixed n.

5.3. Theorem. A function f x E In(h) the inequality

E

C(I) is convex iff for every h > 0 and (5.8)

holds for every fixed n.

5.4. Remarks. (a) Theorem 5.2 can be found in Horova (1968). (b) It is known (see, for example, Timan, 1963) that f(x) can be uniformly approximated by Sh(f, x) as h ~o. (c) The operator Sh can be used for the following characterization of convex functions (see Kocic, 1984): Let I = [a, b]. Then the function f E C(I) is convex iff for every HE [0, (b - a)/2) and x such that [x - h, x + h] E I and for every h E (0, H) the inequality SH(f,X)2=Sh(f,X)

holds.

(5.9)

0

It is easy to see that Theorem 5.3 generalizes the convexity criterion based on the inequality (5.4) (we obtain (5.4) by letting h ~0). Theorems 5.1-5.3 are all related to the first inequality in (5.1). Note that the first inequality is stronger than the second inequality in (5.1); i.e., the following inequality is valid for a convex function f: b

b

_1_ ff(x) dx _f(a +2 b) ::5 f(a) +2 f(b) __ 1_ ff(x) dx. b-a b-a a

a

(5.10)

5.1. Hermite-Hadamard's Inequality

141

Indeed, (5.10) can be written as b

b: a

f

d X : 5 (f(a) ~ + f(b) + 2f(a: b)),

f(x)

a

which is 2 b-a

f

(a+b)/2

f a

b

f(x)dx+_2b -a

f(x)dx

(a+b)/2

This immediately follows by applying the second inequality in (5.1) twice (on the interval [a, (a + b )/2] and [(a + b )/2, b D. By letting a = -1, b = 1, we obtain the result due to Bullen (1978). However, the second inequality in (5.1) can also be used as a convexity criterion. In Roberts and Varberg (1973, p. 15), the following result is given:

f

5.5. Theorem. A function [a, b] we have

-l-f f

E

qa, b] is convex iff for every s < t in

I

(x ) dx :5 f(s)

t-s

+ f(t).

(5.11)

2

More general results are given by Rado (1935). In the following we state some characterization results given in his paper. Let f (x) be a positive and continuous function on (a, b) and let u, v E ~ . We define 1 ( 2h

I(f,x,h,u)=

fh -h

{

f(x + tt dt

)1IU

h

f

eXP(2~

logf(x + t) dt)

for

u=O;

-h

1 )1IV ( If(x - h)V + f(x + h)V

A(f, x, h, v) = {

(f(x + h )f(x - h

»112

for

v

=

O.

Further, let E denote the set of all pairs (u, v) such that I(f, x, h, u) :5A(f, x, h, v)

(5.12)

142

5. Hennite-Hadamard's and Jensen-PetroviC's Inequalities

holds for all x and h satisfying a < x - h < x + h < b and all positive, continuous, and convex functions f on (a, b). Similarly let E be the set of points of (u, v) such that (5.12) holds for all such x, h and all positive and continuous functions f on (a, b). The main result in Rado's paper concerns the explicit determination of the sets E and E, and the following theorems is proved:

5.6. Theorem. (a) (u, v) belongs to E iffone of the following conditions is satisfied: (i) u $ -2 and v 2': 0; (ii) -2 $ u $ -! and v 2': (u + 2)/3; (iii) -! $ u $1 and v 2': (u log2)/(log(1 + u»; and (iv) 1 $ u and v 2': (u + 2)/3. (b) (u, v) belongs to E iff3v - u - 2 $ O. In Theorem 5.5 if we replace the word "convexity" by "concavity," the inequality "/$A" in (5.12) by "/2':A" and E, E by E* and E*, respectively, then the following theorems is true (Rado, 1935):

5.7. Theorem. (a) (u, v) belongs to E* iff one of the following conditions is satisfied: (i) u $ -2 and v $ (u + 2)/3; (ii) -2 $ u $ -1 and V$O; (iii) -15U5-! and V2':(ulog2)/(log(l+u»; (iv) -!5u51 and v 5 (u + 2)/3 and (v) 15 u and v $ (u log 2)/(log(1 + u ». (b) (u, v) belongs to E* iff 3v - u - 22':0. As a simple consequence of Theorems 5.6-5.7, Rado (1935) proved the following result:

5.8. Theorem. Let f be a positive and continuous function on (a, b). Then (a) the inequality in (5.12) is equivalent to the convexity of f iff (u, v) satisfies the conditions: (i) 3v - u - 2 = 0 and (ii) either -2 $ u $ -! or 1 $ u < 00; (b) the reverse inequality in (5.12) is equivalent to the convexity of f iff (u, v) satisfies the conditions (i) 3v - u - 2 = 0 and (ii) either u 5 -2 or -! $ u $1. The following result of Pitenger (1980) is a consequence of Rado's results (see Lupas 1983):

143

5.1. Hermite-Hadamard's Inequality

5.9. Corollary. Let 0 < a

< b,

-m

< r < ~0 and define

]

log2 log(r 1)

+

for r>-l,r+O, for r =0,

r+2

for r s - 1 .

Let r, be defined similarly with min replaced by max. Then M['"(a, b ) 5 L,(a, b) 5 M[rzl(a,b)

(5.13)

holds, where L,(a, b ) is the generalized logarithmic mean given by L,(a, b ) =

{

- l(bbl a a ) l l ( b -0) for r=O ( b - a)/(log b - log a ) f o r r = -1, [(br+'- ar")/((r l)(b - a))]", f o r r # 0, -1,

+

and L,(a, a ) = a. The equality in (5.13) holds iff a = b or r = 1, -2. Moreover, the values of r, and r . are sharp.

-4,

or

5.10. Remarks. (a) LupaS (1983) also proved (5.12) by using (5.13). (b) Rado's (1935) results are further generalized in Hartman (1972).

Namely, Hartman considered the monotonicity of the difference A - Z as a function of x and h when f is positive, increasing, and convex. He also showed that Rado's inequalities are valid under some weaker conditions. This, of course, is natural because the classes of functions he considered are narrower than those in Rado's paper; i.e., positive convex functions are also monotonic. Another generalization of Hermite-Hadamard's inequalities is given in Vasi6 and LackoviC (1974, 1976) and LupaS (1976):

5.11. Theorem. Let p , q be given positive numbers and a , 5 a < b 5 b,. Then the inequalities

144

5. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

hold for A = (pa + qb )/(p tions f: [aI' b d IR ~ iff

y

+ q), Y > 0, and all continuous convex funcb-a p+q

~ - - m i n { pq}. ,

(5.15)

5.12. Remarks. (a) Observe that (5.14) may be regarded as a refinement of the definition inequality for convex functions. (b) For p = q = 1 and y = (b - a)/2, (5.15) is the HermiteHadamard's inequality. We now show that under the same conditions Hermite-Hadamard's inequality yields the following refinement of (5.15):

f

A+y

f(pa p

+ qb) ~ ~ f(t) dt ~ ~{f(A- y) + f(A + y)} ~ p f ( a +) qf(b) +q 2y A-y 2 p +q (5.16)

First, observe that if O < y ~ [ ( b - a ) / ( p + q ) ] m i n { p then , q } by , considering two cases (0 < P ::5 q and 0 < q < p ), we can easily verify that a ~ A - y < A + y so ~ bthat , f is defined on [A-y,A+y]. By Hermite-Hadamard's inequality in (5.1) with a, b replaced by Ay, A + y, we obtain

f

A+y

f(A) ~ 12 y

f(t) dt ~ 12[f(A - y) + f(A + y)].

(5.17)

A-y

By the definition of convexity, we have for a

Hence, taking

Xl

= a and X3 = b,

~Xl < X 2 < X 3 ::5 b

we obtain (5.18)

f(A+y)::5

b - (A + y) A +Y- a b-a f(a)+ b-a f(b).

(5.19)

5.1. Hermite-Hadamard's

Inequality

145

From (5.17)-(5.19) we then have A+y

f(A) ~1

2y

-

J

f(t) dt

A-y

~ 1-{f(A-

y) + f(A + y)}

2

~ !{b -Af(a) + A - a f(b)} =pf(a) + qf(b) , 2

b-a

b-a

p+1

proving (5.16). (c) Note also that (5.14) can be proved from the integral analogues of Jensen's and Lah-Ribaric's inequalities, i.e., from the inequalities b

f(

b

b

b

(J p(x)g(x) dx)/ (J p(x) dX)) ~((J p(x)f(g(x» dX) / (J p(x) dX)) a

a

a

a

~ M - g f(m) + g - m f(M), M-m

M-m

where m ~ g ( x ) ~forM all !E[a, b], g = ( f ~ p ( x ) g ( x ) d x ) / ( f ~ p ( x ) d x ) . Indeed, if we make the substitutions p(x)=l, g(x)=x, a=A-y, b = A + y, m = a, and M = b, we obtain the result that (5.14) is valid if the following condition holds:

a

~ A-

y

~b

and

a

+y

~ A

~ b.

(5.20)

We can then easily show that conditions (5.15) and (5.20) are equivalent. Therefore, condition (5.15) is sufficient. Now we show that it is also necessary. Let p >q and assume that (5.15) is not valid, i.e., y> (q(b - a»/(p + q). Since the function f(x) = a - x for x < a and f(x) = 0 for x2:a is convex, (5.14) becomes: 0 ~ ( a - A - y ) / 2 y ~a 0 con, tradiction. Therefore we must have y ~ q(b - a)/(p + q). Similarly, for the case p
146

5. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

(e) Inequalities (5.14) can also be used as convexity criteria. Of course, the first inequality is the same as (5.4), and the related result for the second inequality (i.e., the generalization of Theorem 5.4) is given in Kocic (1984). (f) Using a proof similar to that given in (b), we can prove the following generalization of (5.10): Under the same conditions as stated in Theorem 5.11,

f

A

~ 2y

A

~

f

A

f(t) dt - f(pa p ~

+ qb) :spf(a) + qf(b) - ~ +q p +q 2y

A

~

f(t) dt. (5.22) ~

o

Generalizations of Theorem 5.11 for positive linear functionals are given in Pecaric and Beesack (1986):

5.13. Theorem.

Let f be a continuous convex function on an interval I ~ [m, M], where -00 < m < M < 00. Suppose that g: E - [R satisfies m:sg(t):SM for all tEE, gEL; and f(g)EL. Let A:L-[R be an isotonic linear functional with A(l) = 1, and let p = Ps» q = qg be nonnegative real numbers (with p + q > 0) for which A(g) = (pm

+ qM)/(p + q).

(5.23)

Then f(pm + qM) :SA(f(g» :spf(m) + qf(M). p+q p+q

(5.24)

Proof. Observe first that since m :s A(g) -s M, there always exist p ;::: 0, q ;::: 0, (p + q) > satisfying (5.23). The first inequality in (5.24) is just (2.6), while the second of (5.24) is (3.43). 0

°

5.14. Theorem.

Suppose that L satisfies conditions Ll-L3 defined in (3.27)-(3.29) on a nonempty set E and that f is a continuous convex function on an interval I, while g, h E L with f(g),f(h) E L. Let A, B be isotonic linear functionals on L for which A(l) = B(l) = 1. If A(h) = B(g), E[ E.s4 satisfies A(CE) > 0 and A (C Ez) > 0 where E 2 = E\E 1 , and if A(hCE)/A(CE) :sg(t) :sA(hCEz)/A(CE,) for all tEE,

(5.25)

f(A(h»:S B(f)(g) :sA(f(h».

(5.26)

then

5.1. Hermite-Hadamard's Inequality

Proof.

147

By Jessen's inequality we have i

= 1, 2.

(5.27)

Applying Theorem 5.13 «5.24» to the isotonic linear functional Band the function g with m=A(hCE,)/A(CE,), M=A(hCE,)/A(CE,), for p= A(CE,), q = A(CE,), we have P + q = A(CE) = A(1) = 1 and B(g)=A(h)=A(hCE,)+A(hCEz) =

pm+qM . p+q

Hence, by (5.24) we obtain f(A(h» = f(B(g» :s B(f(g» :s A(CE,)f(

~ ~ h ~ ~ +/ )A(CE,)f( ~ ~ ~ : ) )

:sA(f(h)CE,) +A(f(h)CEz) (by (5.27»

= A(f(h», proving (5.26).

0

Note that again the inequality (5.26) is a refinement of Jessen's inequality and is also a generalization of an inequality given in Vasic, Lackovic, and Maksimovic (1980). Wang and Wang (1982) proved the following generalization of Theorem 5.11:

5.15. Theorem. Let f: [a, b ] ~IR be a convex function, Xi E [a, b], and = 0, 1, ... , n). Then the following inequalities are valid:

Pi> 0 (i

n-l

+

2: xi1 -

n

tj + 1)t1 ••• tj

j=1

IT dt,

+ X n t 1 t 2 ••• tn )

i=1

(5.28) where

(£1'i

n

+ f3;)/2 = ~ 1Pk

/

n

k ~ - Pk 1 for i = 1, ... , n

(5.29)

148

5. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

and

O:s a i < f3i :s 1 for

i

= 1, ... , n.

(5.30)

Proof. (Pecaric, 1989a). By the integral and discrete versions of Jensen's inequality, we have

f(~PiXi/itPi)

f{n (f3j - aj)-l f

fJ,

n

=

)=1

f ... f fJ,

n

:s)] (f3j - a'j}-l

a1

fJ,

n

-s

I ~(f3) -

aJ-1

n

/

n

~Pi-

f(xo(1- (1)

an

fJn

f ---f eXt

= ~pJ(Xi)

fJn

(f(xo)(l - (1)

(1'"

o

5.16. Remarks. (a) In Pecaric (1989a) the condition in (5.30) is weakened and a generalization to Theorem 5.15 for convex functions of several variables is given. Another extension of Theorem 5.15 is given in Hu (1986). (b) Generalizations of Hermite-Hadamard's inequalities are also given in Lackovic and Stankovic (1973) and Lackovic (1969). Note that the results in Lackovic and Stankovic (1973) are simple consequences of the well-known results for support line of convex functions. For example, by integration and Theorem 1.6 we can obtain the following generalization

5.1. Hennite-Hadamard's Inequality

149

of the first inequality in (5.1): b

f(c)

b

+ f ~ ( c ) ( xf p(x) dX) /

(f p(x) dX) -

a

cf~(C)

a b

b

~ ( fp (x)f(X)dx)/(JP(X)dx) a

where

for

a
(5.31)

a

f is convex and p is a positive integrable function.

D

Another generalization of the first inequality in (5.1) can be found in ~ u ~ v and a, E IR) be an Neuman (1986). Let x(t) = I : ~ ~ ua .t' (for algebraic polynomial of degree not exceeding v and let a = min{x(t): c ~

°

t ~ d } , b = m a x { x ( t ) : c ~ t ~ d } .

5.17. Theorem.

Let f be a convex function on (a, b). Then d

f ( ~ uarm r) ~ J M n ( t ) f ( ~artu r) dt,

(5.32)

c

where M; is a B-spline and m, the rth generalized symmetric mean of to, ... , tn (see Section 1.3).

The following generalization of the first inequality in (5.1) for convex functions of several variables is given in Neuman and Pecaric (1989). 5.18. Theorem. Let f be a convex function vol([Xo, ... ,xd) > 0, Xi E IRk, i = 0, 1, ... ,n. Then f(me, , ... , mek)

on

IRk

and

let

f(xf', ... , Xfk)M(xI Xo, ... , xn ) dx, (5.33)

~J IRk

where f i

= 1, 2, ... , ; i = 1, 2, ... , k.

Proof. Consider Jensen's inequality for multivariate continuous convex functions on IRk, ( ~ 1 " ' " ~ k ) E C(lRk), and let du be a probability measure on IRk. Then

f(

f ~ 1 du, ... , IRk

J ~ kdp:) ~ J f( IRk

IRk

1>1' ... ,

~ k )d u,

(5.34)

150

S. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

and the equality in (5.34) holds iff f E III (IRk) (the space of all k-variate polynomials of total degree :51). Let df.-l(x) = M(x I Xo, ... ,xn ) dx. Then du is a probability measure on IRk. To establish (5.33) we let t(x) = Xi; i = 1, 2, ... ,k in (5.34). Hence the inequality follows from Theorem 1.6(e). 0 5.19. Theorem.

Under the assumptions of Theorem 5.18, we have

[ ( -n +1 1 2:n) xi:S i=O

f

IRk

[(x)M(x I Xo, ... , Xn ) dx:s - 1 n [(Xi), n + 1 ;=0

2:

and the equality in (5.35) holds iff[

E

(5.35)

III (IRk).

Proof. The first inequality in (5.35) follows immediately from (5.33) and (1.58) by letting t l = ... = t k = 1. To obtain the right-hand side inequality in (5.35) we apply (1.56) and

f

Ago ... APn dA = r(po + 1) ... f(Pn + 1) n

S"

r(Po+"'+Pn+n+1)'

where Po, ... , P« > -1 and f(-) is the gamma function. We then obtain

As a special case we obtain: 5.20. Theorem. Let a = [x,, ... ,xd where k::::: 1 and volk( a) > O. If [ : a ~IR is a convex [unction, then (5.36) and equalities hold iff[

Proof.

E

III (IRk).

Immediate from (5.35) and (1.57).

0

5.21. Remark. Using a proof similar to the proof of Theorem 5.18 we can give a generalization of Theorem 5.17 which includes Theorem 5.18 as a special case (see Neuman and Pecaric, 1989). 0

5.2. Jensen-PetroviC's Inequalities

151

Let X = {Xo, ... , X n } (n > k 2= 1), and assume that any subset that consists of k + 1 points spans a proper simplex. let Xj = X \ {x, : ::5 j ::5 n}. Then a multivariate B-spline can be written as M(·I X) (with knot set X). Similarly let M(·\ X) denote the multivariate B-spline with knot set Xj . For real numbers Ao, ... , An with ~ 7 = Ao j = 1, let y = ~ 7 ~ A o jXj and let z= ( ~ 7 = o x j ) / +( n1). The following generalization of Theorem 5.11 is a special case of a more general result of Neuman (1990):

°

5.22. Theorem.

f

Let f be a convex function on ~ k . Then f(x)M(x I X) dX::5

IRk

t

f

Aj

]-0

f(x)M(x I X) dx

(5.37)

IRk

holds iff y = z, and equality in (5.37) holds ifff

E

TIl

(~k).

Note that Neuman's general inequality is also a generalization of Fejer's inequality given in (5.2).

5.23. Remark. Other generalizations of Hermite-Hadamard's inequalities were obtained by Zhang (1987), Neuman (1988), Sandor 0 (1988), and Alzer (1989c).

5.2.

Jensen-PetroviC's Inequalities

5.2. 1.

Inequalities for Starshaped Functions

The following theorem is proved in Vasic and Pecaric (1979a): 5.24. Theorem.

Let x, p and q, be nonnegative n-tuples. Then (5.38)

holds for every n, x, p, and q such that

X; E

[0, a] (i

n

2. p.x, 2=Xj

i=1

= 1, ... , n) and

n

for

j = 1, ... ,n

and

2. PiX; E [0, a]

i=1

(5.39)

iff ep : [0, a] ~ ~ is a decreasing function. Similarly, the reverse inequality holds iff ep is increasing. (i) If ep is decreasing, then e p (5.38). (ii) For n = 2, XI = X, Xz = h,

Proof.

( ~ 7 =PiXi) 1 ::5

PI

ep(x). Thus we obtain

= pz = 1, ql =

1,

and

qz

= p,

152

S. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

(5.38) becomes (1 + P )¢(x + h) ::; ¢(x) + p¢(h). As p ~0, we have ¢(x + h) $¢(x). The proof for the reverse inequality is similar. 0

5.25. Remark. For Pi = 1 (i = 1, ... , n) the first condition in (5.39) is satisfied; thus we can obtain a result of J. L. W. V. Jensen (1906) from Theorem 5.24 (see also Hardy, Littlewood, and P6lya, 1934, 1952, p. 83): We say that f(x)/(x - x o) is a decreasing (increasing) function if ...J...J--'>.,. f(xt) :> Xt-rXO -r- X2 ....,... Xt-XO

Letting ¢(x) = f(x)/x and qi = p.x, (i have:

= 1, ...

«) -

f(x2) • X2- XO

o

, n) in Theorem 5.24, we

5.26. Theorem. Let x and p be two nonnegative n-tuples such that (5.39) is satisfied. If f(x )/x is a decreasing function, then (5.40)

If f(x)/x is an increasing function, then the reverse of the inequality in (5.40) holds. 5.27. Corollary. If the conditions of Theorem 5.26 are satisfied and X2 - ... - Xn > 0 (2:0 iff is defined at 0), then

Xt -

(5.41) The following generalization of Theorem 5.26 is given in Vasic and Pecaric (1979a): 5.28. Theorem. such that

Let p be a nonnegative n-tuple and x be a real n-tuple

(5.42)

5.2. Jensen-PetroviC's Inequalities

153

If (f(x) - f(xo»/(x - xo) is an increasing function, then

%1 pd(Xk):5 Af(%1PkXk) + B(%1 Pk -

1)f(xo)

(5.43)

holds, where A

=

(%1 Pk(Xk- xo»)/ (%1 pix; -xo), (5.44)

B=

(%1 PkXk)/(%/kXk-xo).

Proof. This is a simple consequence of a result given in Remark 3.45(b) by letting either m = Xo and M = ~ ? = P1 iXi or m = ~ ? = P1 iXi and M = Xo. 0 5.29. Theorem.

Let f(x)/x be an increasing function.

(a) IfO<xl:5'" :5xn and if there exists an m (:5n) such that

Pm + 1 = ... = Pn = 0, where P k = ~ 7 = I P i P' k = Pn - Pk - 1 (k=2, ... ,n), and (5.40) holds. If there exists an m (:5n) such that

P1 = Pn ,

then

Pm + 1 = ... = Pn = 0, then the reverse of the inequality in (5.40) holds, i.e.,

(5.45)

(b) Let Xl:5"':5x m:50:5Xm+l:5"':5X n and f(O) =0. Then the inequality (5.40) is valid if either (i) there exists a i. 1:5 j :5 m + 1, such that PI = ... =

Pm + 1 = ... = Pn = 0,

~ + 1 '

or (ii) there exists a k, m s; k :5 n, such that PI = .. ·=Pm=O, The inequality in (5.45) holds if there

Pk + 1 = ... = Pn = O. exist integers j and k, 1:5 j:5 m + 1,

m :5 k :5 n, such that PI = ...

=

= 0,

~ - 1

0:5 Pm + 1:::s;· .. :::s; Pk:::s; 1,

1

~ ~ ~

Fk+ 1 =

~ Pm

~ 0,

= Fn = O.

154

S. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

5.30. Remarks. (a) A new proof of Theorem 5.29 is given in Vasic and Pecaric (1979a). In fact, they proved Theorem 5.29 under weaker conditions. (b) Vasic and Pecaric (1979a) also proved that inequality (5.40) holds for every n and every f such that f(x)/x is a decreasing function iff Pi ~ 1 (i = 1, ... , n). (c) If inequality in (5.40) holds for all positive n-tuples x and p, then f(x) = cx where c is a real number (Hardy, Littlewood, and P6lya, 1934, 1952, p. 84). 0

5.2.2.

Inequalities for Convex Functions

First we give the following result:

5.31. Theorem. Let f be a convex function on [0, a). If the conditions of Theorem 5.26 are satisfied, then (5.46)

Proof. 5.26.

The function f(x) - f(O) satisfies the conditions of Theorem 0

5.32. Remarks. (a) For Pi = 1 (i = 1, ... , n). (5.46) becomes (5.47) This is the well-known Petrovic's inequality for convex functions (see Petrovic, 1932, or Mitrinovic, 1970, p. 28). (b) Inequality (5.45) was first given by Pipoviciu (1944) under the incorrect assumptions that Pi > 0 and Xi >0 (i = 1, ... ,n). Vasic (1968b) noted that this result is true for Pi ~ 1 and Xi ~0 (i = 1, ... , n). In fact, Vasic's conditions are necessary and sufficient for the inequality in (5.46) to hold for every n and every convex function f (see Vasic and Pecaric, 1989).

5.2. Jensen-PetroviC's Inequalities

155

(c) Vasic (1968b) considered the following generalization of (5.46):

~ 1Pk!(Xk) :5 r f ( ~ ~ 1PkXk) -

,n and where Xk E [0, a) for k = 1, , n) and conditions are Pk;:==: 1 (k = 1, nt- 1

nz-l

2: PkXk = k=l

2:

~ 1Pk)[(0),

(r -

1/r I : k ~ l P k EX [0, k a).

(5.48) His

n

PkXk = ...

=

2:

PkXk'

k=nr-l

k=nt

For Pk = 1 (k = 1, ... , n), we obtain an inequality of Marshall, Olkin, and Proschan (1967) which includes, as a spcial case, the inequality

-1 2:n

n k=l

s

(n)S-l( 2:

lak - al :5 -

2

n

k=l

lak -

)S

al ,

where a = (lin) I: k=l ai; ak;:==: 0 and s;:==: 1. Moreover, Vasic's conditions can be replaced by 1 n O:5Xi:5-2:PkXk for r k=l

i=l, ... ,n.

(5.49)

o

We note that the special case Pi = 1 (i = 1, ... , n) and r = 2 was considered by Petrovic (1938) with an aspect of applications in geometric inequalities (Pavlovic, 1961). Also note that inequalities (5.46) and (5.48) are equivalent. To see this, by the substitution P i ~ p J (5.48) r follows from (5.46). Similarly, for r = 1 (5.48) yields (5.46). A similar consequence of Theorem 5.28 is: 5.33. Theorem. Let f: I ~IR be a convex function where Xo and I:k=lPkXk are in 1. Let the n-tuples x and p satisfy the conditions of Theorem 5.28. Then (5.43) holds.

Proof.

The function

f satisfies the conditions of Theorem 5.24.

0

5.34. Remarks. (a) Inequality (5.43) for convex functions is proved by Giaccardi (1955) under stronger conditions. Theorem 5.33 is proved in Vasic and Stankovic (1973). In fact, they proved a generalization of Theorem 5.33 for g-convex functions. Vasic and Pecaric (1979a) showed that this result for g-convex functions can be obtained from a similar generalization of the discrete case in Theorem 3.37 (see Remark 3.45(b».

156

5. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

(b) A generalization of Theorem 5.33 for positive linear functionals is given in Beesack and Pecaric (1985a): Let L, A satisfy the conditions Ll, L2 and AI, A2 defined in (3.27)-(3.31) and let gEL. If xo*A(g) and either xo:S g(t) :s A(g) for all tEE or A(g) :s g(t) :s Xo for all tEE, and if (jJ is convex on [xo, A(g)] (or on [A(g), xoD and (jJ(g) E L, then A((jJ(g»:s ([A(g) - A(I)xo](jJ(A(g» + [A(I) - 1]A(g)(jJ(xo)}/[A(g) - xo].

o The following generalization of the well-known Brunk-Olkin Inequality (see, e.g., Mitrinovic, 1970, pp. 47-48, or Beckenbach and Bellman, 1961, 1965, pp. 47-49) is given in Barlow, Marshall, and Proschan (1969): 5.35. Theorem. Let xl:s·· ·:Sxs:sO:SXs+I:S··· :sxn (SE{O, 1, ... , n}), X; E 1(1 :s i :s n; 0 E I), and p be a real n-tuple. (a) The inequality in (5.46), i.e.,

~ lp J(x;) ~ f ( PiX;) ~ + ( ~ 1p; -1 )f(O) holds for every convex function f : I

o:s Pk :s 1

~[ffi

1:s k -s sand

for

(5.50)

iff

0 :s Pk :s 1 for s + 1 :s k :s n. (5.51)

(b) Let r..?=IP;X;

E

I. Then the inequality

(5.52) holds for every convex function f : I f}:SO

for

i<m,

f } ~ 1 for

~[ffi

iffthere exists an m :s s such that

m:Si:ss,

and

P;:s0

for

i ~ s + l ;

(5.53) or there exists an m f}:SO

for

i :ss,

~s

such that

P; ~ 1 for

s

+ 1 :s i :s m, and P;:s 0 for

i

< m.

(5.54)

5.2. Jensen-PetroviC's Inequalities

157

Proof. Let Theorems 2.19 and 3.3 (i.e., Remarks 2.20(a) and 3.4) hold for n + 1 with Xi = Xi and Pi = Pi (1 ~ i ~ n + 1). By the substitutions Xi=X;

and

P;=P;

for

and P;=P;-I

X;=X;_I

Xm + 1 = 0,

1 ~ i ~ m ;

we obtain Theorem 5.35.

for

m + 2 ~ i ~ n + 1 ;

0

5.36. Remark. We have shown that Theorem 5.35(a) is a special case of Theorem 2.19, and that Theorem 5.35(b) is a special case of Theorem 3.3. Now we show that the reverse implications are also valid (Pecaric, 1984h). Without loss of generality we may assume that P; = 1. Then (2.19) and (5.50) are equivalent and (3.2) and (5.52) are equivalent. In this case (5.51) becomes O ~ P k ~ P for n

1 ~ k ~ s and

Fk-I;:::O

for

O ~ F k ~ P for n

Pk;:::O

for

Fk;:::O,

1 ~ k ~ s and

Pk -

I ;:::

0 for

Fk;:::O

1 ~ k ~ n - 1a nd

which is equivalent to (2.20). Similarly, for m ~ ~ Ofor

i<m,

~ ; : : : P n

for

s + 1 ~ k ~ n ; i . e . ,

s

+1

for

P;-I ~0 for

i<m,

~0

for

i<m

and

becomes

i e:s

+ 1; i.e.,

i e.s

+ 1; i.e.,

m ~ i ~ sand

P ; ~ O for

P,

i.e.,

and

P ; ~ O for ~ ~ Ofor

~ n,

2 ~ k ~ n ,

~ s (5.53)

m zi i zs s

~k

P; ~ 0 for i > m,

which is the condition in (3.3). By a similar argument for m ;::: s (5.54) becomes ~ ~ Ofor

i

P;;::: t; for

~ s ,

s + 1 ~ i ~ m and

P ; ~ O for ~ ~ Ofor

F.-I

i zs s,

~0

for

s + 1 ~ i ~ m and

P ; ~ O for

P, ~0

for

i

<m

and

i<m;i.e.,

P;

~0

for

i > m.

i>m; i.e.,

158

5. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

Thus we again obtain the condition in (3.3). Consequently, we have proved Theorems 2.19 and 3.3 with the assumption that 0 E 1. Suppose that 0 $ I, and let I = [a, b]. If f is a convex function on [a, b], then F(t) = f(t + (a + b )/2)) is a convex function on [- (b - a)/2, (b - a)/2]. Thus in the case P; = 1, from (5.50) (and (5.52)) we have

i.e., f

n (

a

(

+

b)) ::s;("?:.)t:/ J

-

(a

n

~ P ; t;+-2-

a

(

+

b)

t;+-2- ,

which is

where we have used t, = X; 0 completes the proof.

+ b )/2 for

[a, b] and 1::s; i ::s; n. This

X; E

Note that similar assertions also hold for some of the results given later. To obtain an analogue of Theorem 5.35 from Theorems 2.21 and 3.6, we let I\.(x) = g(x) = g(x) + sex), where sex) = 0 for X E [a, to] and sex) = 1- G(b) for X E (t a , b], and

f

G(t)

=

f b

t

dg(x),

G(t)

a

=

dg(x).

t

5.37. Theorem. Let to E [a, b] be fixed and f be continuous and monotonic with f(to) = O. Let g be a function of bounded variation. (a) If O::s; G(t) ::s; 1 for

a s: t ::s; to,

4J: I

then for every convex function we have b

O::s; G(t) ::s; 1 for ~IR

such that f(x) E I for all x

b

a

(5.55) E

[a, b]

b

J4J(f(t» dg(t)"?:. 4J(J f(t) dg(t») + (J dg(t) a

to < t s; b,

a

1)4J(0).

(5.56)

5.2. Jensen-Petrovic's Inequalities

(b) If J ~ f ( t dg(t) )

E

159

I and either

there exists an s ::s to such that G(t)::s 0 for t < s, G(t)

~1

for

s ::s t s: to, and G(t)::s 0 for t > to,

(5.57)

or there exists an s ~ to such that G(t) ::s 0 for t < to, G(t) ~ 1 for to
b

b

J¢(f(t» dg(t) ::s ¢ (J f(t) dg(t») + (J dg(t) - 1) ¢(O). a

a

(5.59)

a

5.38. Remarks. (a) For Pk = (-1)k- 1hk> 1 ~ h, ~ ... .2: 0, we obtain the Brunk-Olkin Inequality from Theorem 5.35. It is known that this inequality is a generalization of results given by Szego (1950), Weinberger (1952), and Bellman (1953) (see also Mitrinovic, 1970, p. 1; Beckenbach and Bellman, 1961, 1965, pp. 47-48, or Karlin and Studden, 1966, p. 431). (b) A generalization for functions with increasing increments was given in Pecaric (1984b) with conditions (5.57) and (5.58) replaced by, respectively, ~1

for

a s; t < to,

G(t) ::::; 0

for

to::::; t s: b;

(5.57')

G(t) ::::; 0

for

a::::; t < to,

G(t) ~0

for

to::::; t s: b.

(5.58')

G(t)

(c) In the case to = a, inequality (5.56) and a similar result for functions with increasing increments was proved by Brunk (1956, 1964). (d) Some additional generalizations can be obtained from Theorems 2.21 and 3.6 if we let A(X)=g(x)+S(x), where S(x)=O for a::sx
= Xo·

(c) Pecaric (1983a) used Fuchs' generalization of a theorem on majorization to obtain some similar results. He also obtained some refinements of Petrovic's inequality. These results are extensions of the results in Vasic and Stankovic (1973). 0

160

5. Hermite-Hadamard's and Jensen-Petrovic's Inequalities

5.39. Corollary. Let to E [a, b] be fixed and f be continuous and monotonic with f(to) = O. Let 1J be continuous and convex on I such that f(x) E I for all x E [a, b] and 1J(0) ~O. Let A and 1t be functions of bounded variation such that: 1t(b»1t(a), 1 t ( b ) - 1 t ( a ) ~ A ( b ) - Aand ( a ) t

b

O ~J d A ( X ) J~ d1t(x) a

for

a~t~to,

for

to~t~b.

a b

b

o ~J d A ( x ) J ~ d1t(x) a

t

Then b

b

b

b

(J f(x) dA(X»)/ (J d1t(X»)} ~ (J 1J(f(x» dA(X»)/ (J d1t(X»).

1J{

a

a

a

a

(5.60)

Proof. Immediate by letting g(x) = A(X)/ f ~ d1t(y) in the BarlowMarshall-Proschan Inequality (5.56). 0 5.40. Remark. Inequality (1970a). 0

(5.60)

was

also

considered

by

Boas

5.41. Corollary. Let to, f, and 1J satisfy the conditions of Corollary 5.39. Let A and 1t be functions of bounded variation such that A ( a ) ~ A ( t ) ~ A ( b ) for + d

a ~ t ~ t o ,

A ( a ) - d ~ A ( t ) ~ A for ( b ) t o ~ t ~ b , b

A(b) > A(a),

b

Jd1t(x) ~ J dA(X) + d a

for

d > O.

a

Then (5.60) is valid.

Proof.

This is an immediate consequence of Corollary 5.39.

5.42. Remark. For to = band f decreasing, the condition f(b) 0 be replaced by f(b) ~0 (see Boas, 1970).

0

=0

can

,

5.2. Jensen-Petrovic's Inequalities

161

In a similar fashion we can obtain: 5.43. Corollary. Let to, f, and 4> satisfy the conditions of Corollary 5.39. Let A be a function of bounded variation such that b

t

f dA(t)"'?' 0 for

a s: t :s, to,

a

f dA(t)"'?' 0 for

to < t s: b,

t

b

f IdA(t)1 > O.

and

a

Then the following inequality holds: b

b

b

b

4>{(f f(x)dA(x»)/(fIdA(X)I)}:s,(f4>(f(x»dA(x»)/(f IdA(X)I). a

a

a

a

5.44. Remarks. (a) Corollary 5.43 is a generalization of the Ciesielski Inequality (1958); see also Boas (1970a) and Brunk (1956). (b) Corollaries 5.39, 5.41,_ and 5.43, are proved in Pecaric (1983b). Related generalizations for functions with increasing increments are also 0 valid (see Pecaric, 1984b). Let us consider once again the inequalities with alternative signs (see Remark 5.38(a». First we show that the well-known inequality of Szego (see Mitrinovic, 1970, p. 112; Beckenbach and Bellman, 1961, 1965, p. 47; and Karlin and Studden, 1966, p. 431) is valid under weaker conditions: 5.45. Theorem. Let Xl"'?' X2"'?' ... ",? X2m+l, Xi E I for i = 1, ... , 2m + 1, and let f be a Wright-convex function on 1. Then the following inequality is valid: (5.61)

Proof.

For m = 1, the inequality f(xI) - f(x2) + f(x3) ",? f(x I

-

X2 + X3)

(5.62)

holds if XI ",? X2 ",? x 3. This inequality is a simple consequence of the definition inequality for Wright-convex functions D"'?'O,

162

5. Hermite-Hadamard's and Jensen-Petrovie's Inequalities

by letting al = X3, az = Xz, () = Xl - Xz. Suppose that the theorem is valid for m - 1, and that the conditions of the theorem are satisfied for m. Then we have

by first using the inductive hypothesis for m - 1, then applying (5.62), because Zm+l

2:

(-l)i-l xi;:o:xZ m;:O:XZm+I'

0

i=l

Similarly, we can prove the following: 5.46. Theorem. and let

Let XkE I, k

= 1, ... ,2m + 1 (I is

Zk+l

2:

(-l)i-l Xi E I for

k

= 1, ...

an interval in IR),

, m.

i ~ l

(a) If Zk

2: (-l)i-l Xi

;:0:

0 for

k = 1, ... , m,

(5.63)

i=l

then the reverse of the inequality in (5.61) holds for every Wright-convex functions f : I ~IR. Further, if the reverse ofthe inequalities in (5.63) hold, then the reverse of the inequality in (5.61) is also valid.

(b) If instead of (5.63) the following conditions hold Zk

2: (-ly- lxi

~ o for

k

=

1, ... , m,

(5.64)

i=l

then (5.61) is valid. If the reverse inequalities in (5.64) hold, then (5.61) is also valid. 5.47. Remarks. (a) Theorem 5.46 is a generalization of an inequality of Opial (1960) (see also Mitrinovic, 1970, p. 351). For some related results see Vasic and Janie (1970a) and Pecaric (1980f).

5.2. Jensen-Petrovie's Inequalities

163

(b) Theorems 5.45 and 5.46 can be generalized for functions with increasing increments. The same is true for results given in Vasic and Janie (1970a) and Pecaric (1980f); i.e., these results can be generalized for continuous functions with increasing increments. (c) Let the conditions of Theorem 5.35 be fulfilled with Xl> 1. Then for the convex function -log(1 +x), (5.50) becomes a generalization of the well-known Bernoulli inequality n

n

II (1 + Xi)"'::; 1 + L p.x, , i=l

i ~ l

and (5.52) gives the reverse inequality.

5.2.3.

0

Combination Convexity Inequalities

Hwang and Yang (1985) gave some generalizations of results of Beckenbach (1969). First they proved: 5.48. Theorem. If f E K(b) (K(b) is the class of real-valued functions that are continuous and nonnegative on [0, b] with f(O) = 0) is convex on [0, a] and starshaped on [0, b], where a ::; b, then for all real numbers Xi in [0, b] (i = 1, ... , n) and weights Ai> such that r . 7 ~ 1Ai::; alb, we have

°

(5.65)

Furthermore, the constant alb is the best possible. By letting Ai

= aw.lb

we obtain Beckenbach's (1969) inequality.

5.49. Theorem. Iff E K(c) is starshaped on [0, b] and superadditive on [0, c] where b ::; c, then for every X in [0, c] and every A E [0, blc] we have f(Ax)::; (clb )Af(x). 5.50. Theorem. If f E K(c) is convex on [0, a], starshaped on [0, b], and superadditive on [0, c], where a z: b -s c, then for all real numbers Xi E [0, b] (i = 1, ... , n) satisfying r . 7 ~ 1Xi = co::; c, we have f«A/n)co»::; (A/n)f(co), where O::;A::;alb. This theorem is also equivalent to Beckenbach's inequality.

164

5. Hermite-Hadamard's and Jensen-Petrovie's Inequalities

5.51. Theorem. Iff E K(c) is convex on [0. a]. starshaped on [0, b] and superadditive on [0, c], where a s; b :s: c. then for all real numbers Xi E [0, c] (i = 1, ...• n) satisfying ~ 7 = X1 i E [0. c] and all weights Ai> such that ~ 7 = A 1 :S: a]c, we have

°

(5.66)

Proof. Denoting A= ~ 7 = A1 i we have Axi:S: (alc)x i E [0, a] for i = 1•...• n; and from Jensen's inequality we have n )... (C b ) -; f(Axi) = 2: -; f -b A- Xi . 1=1 1=1 C

f ( 2: AiXi :s: 2: n

1=1

)

n)... I\.

I\.

Since (blc)Xi E [0, b], (clb)A < 1, and f starshaped on [0. b], we obtain the first inequality in (5.66) from the fact that f c I b)A(b I c )x;) :s: (clb)Af«blc)x;). To prove the second inequality in (5.66). note thatfis increasing on [0, c] which implies

«

and

-bc

i

1=1

i

c AJ(!!-Xi) :S:-b f(i Xi) c 1=1 1=1

A i : S : ~ 1=1 b f (Xi)' i

Also. from f«blc)Xi):S:f(b) for i=l, ...• n, we have (clb) ~ 7 = AJ«blc)x;):s: 1 (alb )f(b); and from ~ 7 = 1AiXi:S: ~ 7 = 1AiC:S: a. we 1 :s: f(a). Since f is starshaped on [0, b]. we have obtain f ( ~ 7 = AiX;) f(a) =f«alb)b):S: (alb)f(b). 0 Note that when b = c. Theorem 5.48 is a special case of Theorem 5.51.

5.2.4.

Inequalities for Sums of Order p

Let us denote and

5.2. Jensen-Petrovic's Inequalities

165

where a and p are positive n-tuples such that Pi 2= 1 (i = 1, ... , n). The well-known inequality for sums of order p states that (5.67) for s>r>O (see Hardy, Littlewood, and Polya, 1935, 1952, pp. 28-30). But this inequality is also valid for r < s < 0 and s < 0 < r (see Vasic and Pecaric, 1979a). Furthermore, Vasic and Pecaric also proved that (5.68) for s > r > 0 and a ~> 0; and that

a ~-

vs(a,p)::svr(a,p)

a2 - ... for

> 0 or r < s < 0 and

a ~

s>r>O,

or

r<s
or

a ~-

a ~-

... -

s
(5.69) Additional generalizations of the above results are given in Hardy, Littlewood, and Polya (1934, 1952, pp. 84-85) (a generalization of (5.67» and Vasic and Pecaric, 1979a and of (5.68) and (5.69». Similarly we may define "sums" involving an arbitrary function . We write vcj>(a) =

- 1 ( ~ 1(a;)),

ocj>(a)

= -1((a1) -

Vcj> (a, p)

=

-1 ( ~ / i < / > ( a ; ) ) ,

(az) - ... - (a n

».

Here (x) is a continuous and strictly monotonic function, positive for all positive x and tends to 00 either as x ~0 or as x ~00. We shall also assume that the components of a are all positive and that Pi 2= 1 (i = 1, ... , n). The following two results are simple consequences of Theorem 5.26 and Corollary 5.27 (Vasic and Pecaric, 1979a, and Hardy, Littlewood, and P6lya, 1934, 1952, pp. 84-85). 5.52. Theorem. If 'l/J and X are continuous, posttuie, and strictly monotonic, then v1J1 and V x (v, is either vcj>(a) or vcj>(a, p» are comparable whenever (i) 'l/J and X vary in the opposite directions, or (ii) 'l/J and X vary in the same direction and X/ 'l/J is monotonic. In case (i) we have (5.70) if'l/J decreases and X increases. In case (ii) (5.70) holds if xtw decreases.

166

5. Hermite-Hadamard's and Jensen-PetroviC's Inequalities

If 'l/J and X are continuous, positive, and strictly monotonic, then D", and Dx are comparable under the conditions in the case (ii) of Theorem 5.52, and the inequality

5.53. Theorem.

(5.71)

holds if X/ 'l/J is decreasing. Jensen (1906) used (5.67) in the proof of the following theorem (see also Mitrinovic, 1970, p. 52 and p. 78): 5.54. Theorem.

numbers and r1 , r;;,! 2= 1. Then

Let a ij (i = 1, ... , n; j = 1, ... , m) be posunie real rm be positive real numbers satisfying r 11 + ... +

••• ,

Similarly, by using generalizations of (5.67) we can give some further generalizations of Theorem 5.54 (see Vasic and Pecaric, 1979a). The following result of Mulholland (1950) presents a generalization of Minkowski's inequality: Let the function f be increasing and convex for x 2= 0 and f(O) = O. Furthermore, let the function F defined by F(t) = logf(e') be convex for all real t. Then

5.55. Theorem.

vf(a + b)::s vf(a)

+ vf(b)

(5.72)

holds for all a = (aI' ... ,an), b = (b , , ... ,bn) such that a, 2= 0 and b, 2= 0 (i = 1, ... , n). Note that Milovanovic and Milovanovic (1978) proved vf(a + b, p)::s vf(a, p)

+ vf(b, p)

(5.73)

under the same conditions, and another inequality for sums was given by Klamkin and Newman (1975): n (

(r

)1I(r+l)

+ 1) ~ a ~

(

2= (s

n

)1/(5+1)

+ 1) ~ 1 a ~

,

where 0=a O::sa 1::S"'::san, a i - a i _ 1::s1 (i=l, ... ,n), r2=1, + 12= 2(r + 1).

s

(5.74) and

5.2. Jensen-Petrovlc's Inequalities

167

A weighted version of (5.74) is given in Meir (1981). A refinement of one of Meir's results is given in Milovanovic and Milovanovic (1986), and an improvement can be found in Pecaric (1989a). In the following we state a generalization of Meir's result given in 1. Milovanovic (1980) and Milovanovic and Milovanovic (1986). 5.56. Theorem. Let f(x) and g(x) be differentiable functions on [0,00) satisfying f(O) = 1'(0) = g(O) = g'(O) = O. Suppose that f'(x) and g'(x) are convex on [0,00), and denote h(x) = g(f(x». Then for any given 0= ao:5 a1 :5 ... :5 an and 0:5 Po:5 P1:5 ... :5 p; satisfying 1 a, - ai-1:52 (Pi

+ Pi-1)

i = 1, ... ,n,

for

(5.75)

we have

h - 1 ( Pih'(a ~ i») : 5 f - 1 ( p;/'(a ~ i»)'

(5.76)

If instead of (5.75) the following condition a, - ai-1 :5 Pi

for

i = 1, ... , n

(5.77)

is satisfied, then we have

) h- 1( n- 1Pi + Pi+1 h'(a i) ) :5f-1 (n-1 Pi + Pi+ 1f'(a i). 2 i=1 i-1 2

2:

2:

(5.78)

Klamkin and Newman (1975) gave a similar integral inequality which was generalized by Pecaric (1985d): 5.57. Theorem. Let g: [a, b] ~ ~ be a nonnegative differentiable function with g(a)=O, let f:[O, 0 0 ) ~ [ 000) , and w:[O, 0 0 ) ~ [ 0 00) , be differentiable increasing functions with f(O) = w(O) = 0, and let p: [a, b ] ~ ~ be a nonnegative integrable function. (a) LetO:5g'(x):5p(x)forallxE[a, b]. Ifwisaconvexfunction, i.e., w' is an increasing function, then b

b

h- 1(f p(x)h'(g(x» dX) :5f-1(f p(x)f'(g(x» a

dx),

(5.79)

a

where h(x) = w(f(x». If w is concave (i.e., if w' is decreasing), then the reverse of the inequality in (5.79) holds.

168

5. Hermite-Hadamard's and Jensen-Petrovic's Inequalities

(b) If g'(X) "2:. p(x) for all x E [a, b], then the reverse of the results in (a) is valid. Furthermore, the equality in (5.79) holds iffg(x) = f ~p(x) dx.

Proof.

(a) Since f'(g(x))g'(x) ::sp(x)f'(g(x)), we have x

f(g(x))::s J f'(g(x))p(x) dx. a

We also have h'(g(x)) = w'(f(g(x)))f'(g(x)). Then, using the fact that w' is an increasing function, we obtain x

p (x)h (g(x)) ::s p (x)1' (g(x))w I

I

(J p (x)1' (g(x)) dx)

(5.80)

a

and b

b

J p(x)h'(g(x)) dx::s w(J p(x)f'(g(x)) dX).

(5.81)

a

a

We then obtain the inequality in (5.79) from (5.81). If w is concave, then the reverse of the inequalities in (5.80) and (5.81) are valid; thus the reverse of the inequality in (5.79) holds. The proof of (b) is similar. 0 Forf(x)=x n+ 1 and w(x)=x(m+l)/(n+l) (XE[O,OO)), it follows from Theorem 5.57 that 5.58. Corollary. Let g: [a, b ] ~ ~ be a nonnegative differentiable function such that g(a)=O and O::sg'(x)::sp(x) for all xE[a,b], where p : [a, b] ~ ~ is a nonnegative integrable function. If m "2:. n "2:. 0, then b

b

«m + 1)) J p(x)(g(x))mdx)lI(m+l)::s «n a

If g'(x) "2:. p(x) for all x holds.

+ 1) J p(x)(g(x)t dx)lI(n+l). a

E

(5.82)

[a, b], then the reverse of the inequality in (5.82)

For p(x) = M for all x Newman (1975).

E

[a, b], we obtain a result of Klamkin and

5.2. Jensen-Petrovii's Inequalities

169

5.59. Remark. The following result similar to (5.48) is also valid (Delange, 1947): Suppose that f(x) and g(x) are real-valued and continuous functions on [a, b]. Letf"be continuous on [a, b] withf":50 and let g/ be continuous on the same interval. Furthermore, let g(a)=f(a), g(b)=f(b), and g ( x ) ~ f ( x f) or a<x 0 on [a, b], then b

J h(g'(x» a

b

dx

~ J h(f'(x»

dx,

a

and equality holds ifff(x) is identical to g(x) on [a, b].

0

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Chapter 6

Popoviciu's, Burkill's, and Steffensen's Inequalities

6.1. Inequalities of Popoviciu and Burkill The following result was proved by Popoviciu (1965) (see also Mitrinovid, 1970, p. 174):

6.1. Theorem. Let n 2 3 and k be positive integers such that 2 5 k 5 n - 1. Let f be a continuous function on an interval I. Then f si convex iff

holdr for all x , , . . . , x,

E I.

We also observe the following result:

6.2. Theorem. Let f :[a, b]+ R be a convex function. Then for all x , y , z E [a, b ] and all p , q, r > 0 we have

Proof(PeEari6 (1986a)). The inequality in (6.2) can be easily proved by applying Theorem 1.7; i.e., for (6.2) to hold for every continuous convex 171

172

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

function on [a, b] (and hence for every convex function on [a, b D, it suffices to show that it holds for the functions

II(x)=4+p,

for

a:=:;x:=:;b

and Mx)=lx-cl

for

a:=:;x:=:;b,

where c E [a, b] is arbitrary but fixed. Obviously II satisfies (6.2); thus we need only show that 12 also satisfies (6.2). For real numbers x, y, z, c and positive real numbers p, q, and r the inequality (p

+ q + r) Px + qy + rz - c I -

I

p+q+r

(q

+ r) Iqy+rz - c I q+r

qy - (r + p) rz + Px - c I - (p + q) Ipx+ - cI r+p p+q I

+ pix -

c] + q

Iy -

c] + r [z - c] ~

°

is equivalent to

Ip(x - c) + q(y - c) + r(z - c)I-lq(y - c) + r(z - c)I-lr(z - c) + p(x - c)I-lp(x -c) + q(y - c)1 + Ip(x - c)1 + Iq(y - c)1 + r(z - c)1

~ 0,

which is just Hlawka's inequality (Mitrinovic, 1970, p. 171)

[u + v + wi - lu + vl- Iv + w] - Iw + u] + lui + Ivl + Iwl ~0, where u, v, ware arbitrary real vectors in a pre-Hilbert (unitary) space withu=p(x-c), v=q(y-c), andw=r(z-c). 0

6.3. Remarks. (a) Theorem 6.2 was proved by Burkill (1974) under the assumption that I is twice differentiable. In Vasic and Stankovic (1976) and Baston (1976) this assumption was removed, and both proofs use Fuch's generalization of a majorization theorem. Another proof of Theorem 6.2 can be found in Lupas (1982). We note that the differentiability condition in Burkill's result can be directly eliminated by using the fact that it is possible to approximate uniformly a continuous convex function by convex polynomials. Using the fact that the convexity of Ion [a, b] implies its continuity on (a, b) and l ( a ) ~ I ( a + ) I , ( b ) ~ I(b-), we can easily prove the validity of (6.2) for an arbitrary convex function. (b) The previous proof of Theorem 6.2 is similar to Popoviciu's proof of inequality (6.1). 0

6.1. Inequalities of Popoviciu and Burkill

173

As in Vasic and Stankovic (1976), let us denote by (Cn,d the inequality

l S i t < ~ < i k s (n ~ P i ) f ( ~ P i h/ ) ( ~ P i i )) ~ G=~ ) G = ~ ~pJ(xi) + (t/i)f((tl PiXi)/

(tl Pi)))'

where Pi>O, xiE[a,b] (i=1, ... ,n), and f : [ a , b ] ~ 1 Ris a convex function. (In fact, C3 ,z) is just inequality (6.2).) Then the implication (C 3 ,z) ::} (Cn,k) for

2 ~ k ~ n - 1 ,

n2::3

holds, and it was proved in Vasic and Stankovic (1976) (see also Pecaric, 1986a).

6.4. Remarks. (a) In the previous discussion, we require that Pi> 0 (i = 1, ... , n) for defining (Cn,k)' This condition can be replaced by a weaker one, namely, that the real numbers Pi (i = 1, ... ,n) are nonnegative such that Pit + ...

+ Pi, > 0 for 1 ~ i. < ... < ik

~ n.

This is equivalent to the condition that no more than k - 1 of the p;'s can be zero. In fact, without loss of generality we can assume that if the function f is continuous and convex, then the inequality for (C n •k ) follows under weaker conditions as we let Pi ~0+. (b) Vasic and Stankovic (1976) also proved the reverse implication (Cn,d ::} (C 3 . z) using Popoviciu's inequality. This implication can be proved as follows: let Pn = 0 (Pn ~0) in (C n.k), then Pn-l = O. Continuing D this process to P4 = 0, we finally obtain (C 3 ,z) (Pecaric, 1986a). A similar result is: 6.5. Theorem. Let the function f:(O, a ] ~ 1 Rbe such that f(x)/x is convex of order m -1, and let Xi E (0, a] (i = 1, ... , m), I:::1 Xi E (0, a]. Then the inequality

- ... + (_1)m-1

L f(Xi) 2:: 0 cn

(6.3)

174

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

holds, where

Proof. We give the proof for m = 2 and m = 3 only. For m = 2, (6.3) becomes f(XI + X2) 2::f(XI) + f(x2), which is obviously true (see Theorem 5.26). For m = 3, (6.3) becomes f(x + y

+ z) - f(x + y) - f(y + z) - f(z + x) + f(x) + f(y) + f(z) 2:: O. (6.4)

By the substitutions

f(x)- f(x)/x,

Xl=x +y +z,

X2=X+y,

and

X3=X for

x, y, z >0

and the inequality in (1.5) we have _ , - f ( - , - x _ + - - , - y ~ + ~ z_ - - , -_ ) f(x + y) + f(x) >0' z(y+z)(x+y+z) zy(x+y) xy(y+z)- ,

i.e., X

~

( y) + z(x +y

+z)

Z

f (x + y + z ) - f (x + y ) + ( )f(x) "2::0. x +Y Y+ z

If y and z are interchanged, we have

u

x y )f(x) "2::0. f (x + y + z ) - f (x + z ) + ( ( y+z ) (x+y+z ) x+z y+z Thus by adding these inequalities we have

x x x - - - f ( x+ Y + z) - - - f ( x + y) - - - f ( x + z) x+y+z x+y x+z

+ f(x) "2::0.

Similarly, we have

--,--y-f(x + y x+y+z

+ z) - -y-f(x + y) - -y-f(y + z) + f(y) 2:: 0, x+y

y+z

z z z - - - f ( x+ Y + z) - - - f ( y + z) - - - f ( x + z) x+y+z y+z x+z By adding the last three inequalities we then obtain (6.4).

+ f(z) "2::0. D

6.1. Inequalities of Popoviciu and Burkill

175

6.6. Remark. Popoviciu (1946) poved the inequality in (6.3) for the case in w h i c h f : [ O , a ] ~ is1 Rconvex, f(O) =0, andf(m-l) exists. Vasic (1968c) proved the same inequality for an arbitrary 3-convex function f without the existence of f(2) and then used this result to prove a more general result. A different proof of Vasic's result was given in Pecaric (1980e). Keekic (1970) showed that the differentiability condition in Popoviciu's result can be removed; he gave a proof for m = 4 only and concluded that the same procedure can be extended to any m. Lackovic (1975) noted that his proof remains correct if f is continuous on [0, a), f(O) = 0, and f(x)/x is a (n -I)-convex function on [0, a). In fact, if f(O) = 0 then clearly (6.3) becomes an equality for some Xi = O. Thus if f is defined on [0, a] and f(O) = 0, then the result can be extended immediately. However, the proof we give here remains correct even if f is not defined at X = O. Thus our result is more general than the result obtained in Lackovic (1975) (see Pecaric, 1986a). 0

A generalization of (6.3) that is similar to (Cn,k) was given by Keckic (1970). Additional generalizations of results of this type can be found in Vasic and Adamovic (1969) and Keckic (1969) and Pecaric (1986a). In the following we state the results due to Pecaric. Let D be a commutative additive semigroup, and let the nonempty set E c D satisfy the condition n

a, E E for i = 1, ... , nand

m

L a, E E::} L ai E E j

i=1

j ~ 1

Further, let G be a commutative additive group which is totally ordered (i.e., G possesses a totally ordering relation ::s satisfying the condition a, b, c E G and a < b ::} a + c < b + c). We start with the following simple result which will be used later. 6.7. Theorem. For any given function f: E (C n ) denote the condition

~G

and n

= 2, 3, ...

let

n

n

2:

2, a,

E

E for i = 1, ... , n,

and

L

a,

E

E.

i=1

Then (a) the implication (C z)::} (C n ) is valid; (b) for n > 2 the implication

176

6. Popoviciu's, Burkill's, and Stelleusen's Inequalities

(Cn ) ::? (C z) is valid if the neutral element 0 of D exists, 0 E E, and f(O) = o.

Proof. (a) can be easily proved by induction, and the proof of (b) is obvious. 0 6.8. Remark. A function satisfying the condition (C z) is usually called subadditive. 0 The following result is a minor modification of a theorem in Vasic and Adamovic (1969):

6.9. Theorem. For any given function f: E denote the condition

~G

and 2:::; k < n let (C n •k )

where a, E E for 1:::; i:::; n, I : 7 ~ 1a, E E. Then (a) the implication (C3,z) ::? (Cn,k) is valid. (b) For (n, k) (3,2) the implication (Cn. k ) ::? (C 3 •Z) is valid if D has the neutral element 0,0 E E, and f(O) = o.

*

Proof. The proof is similar to the proof in Vasic and Adamovic (1969). 0 6.10. Remark. Under the assumption 0 E E, we can obtain the result of Vasic and Adamovic (1969) by applying Theorem 6.2 to the function f(x) - f(O). 0 6.11. Remark. A function f satisfying condition (C 3 ,z) is said to be H-positive (Hlawka-positive; see Burkill, 1974 and Baston, 1976). 0 The preceding theorems are generalized by the following theorem which simultaneously generalizes the analogous result in Keckic (1969).

6.1. Inequalities of Popoviciu and Burkill

6.12. Theorem. For any given function f : E let (Cm,n,k) denote the condition

2J 2: a; (k) j=1 k

(

J

)

S

~G

and 3 S m

S

177

k

+1S

n

(n -m+ 1) 2: f (m-z) 2: a., k- m

+2

_ (k - m 1

j~1

(m"-2)

+ 2)(n -

m

k- m

J

2)

+ +3

k - 2 ) ( n - 2) +(_1)m-1 ( m-3 k-1 for a, E E (1 sis n),

L7=1 a, E E,

2: (m"-3)

f ( ~ 3aii) + ... j=1

(n-m+1) (n

~ f ( a i ) +k -m+l/

)

~ a i

where for all

k <no

Then (a) the implication (Cm,m,m-I) =? (Cm,n,k) is valid; (b) the implication (Cm,n,k) =? (Cm,m,m-I) is valid if D has the neutral element 0, 0 E E and f(O) =0. '

6.13. Remarks. (a) Under the assumption 0 E E, we can obtain the result of Keckic (1969) by applying Theorem 6.5 to the function f(x) - f(O). (b) We shall say that a function f is superadditioe of nth order if it satisfies the condition (Cn,n,n-I)' For n = 2, this property reduces to ordinary superadditivity, and, for n = 3, to H-positivity. 0 For a given function f : E ~G (E and G are as defined above) and for a given sequence {a;}iEK (defined on the set J( of all natural numbers or on some sufficiently large set M of J(), we define

<1>(/) = <1>(/, a, f) =

f ( ~a i)

-

~f(a;)

for

a,

E

E, i

E

I,

and

2: a, E E iel

to be a function on the power set of M. Then we have the following 6.14. Theorem. (a) Let I and J be disjoint nonempty sets of natural numbers, and let a, E E (i E I U J), and LiElUJ a, E E. If the function f

178

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

satisfies condition (C 2), then the inequality cI>(I U J) ~ cI>(I) + cI>(J) holds; i.e., the function cI>, in a limited sense, is subadditve. (b) Let I, J, and K be disjoint nonempty sets of natural numbers, and let a, E E (i E I U J U K) and ~ i E I U ] Ua,K E E. If the function f satisfies condition (C 3,z), then the inequality cI>(I U J U K) - cI>(J U K) - cI>(K U I) - cI>(I U J)

+ cI>(I) + cI>(J) + cI>(K)

~0

holds; i.e., the function cI> is H -positiue, also in a limited sense. (c) More generally, let a, E E (i E UZ=l Ik = A), ~ i E a, A E E. If the function f satisfies the condition (Cn,n,n-I)' then l

L cI>Ci Iii) ~ L c I > ( ~ I2ii) - L cI>(Y Iii) + ... ( n ~ l )

( n ~ 2 )

/=1

( n ~ 3 )

/=1

/=1

L cI>(Ii) + cI>(i I} _m

+ (-lr- l

, ~ l

which means that the function limited sense.

cI> is superadditive of nth order,

again in a

6.15. Remark. In the following we shall confine our attention to the subadditive case because properties of positive or superadditive functions of index set can be preserved by pasing to the limit. 0 Proof of Theorem 6.14. Letting a2) ~ f ( a l )+ f(az), we obtain

a l ~~ i E l a a i nd

a z ~ ~ i E ] a iin

f(al

+

and

cI>(I UJ) = f(

L

at) -

iEIU]

L

tEIU]

f(a;)

~ f ( ~ai) - ~ f ( a ; +) f ( ~ai) - ~ f ( a t ) =

cI>(I) + cI>(J).

This establishes the result in (a). The proof of (b) and (c) is similar. 0

6.1. Inequalities of Popoviciu and Burkill

179

6.16. Corollary. Let Ik = {l, ... , k}, (k = 1, ... ,n), a; E E (i = 1, ... ,n), and ~ 7 = a;1 E E. If the function f satisfies condition (C 2) , then the following inequalities hold: (In)::5 (In)::5

(In-I)::5 . . . ::5 <1>(1 2 ) ::5 0,

min (f(a;

+ a) -

f(a;) - f(aj))

::5 0.

(6.5)

ls.i<j:5n

Proof. For I=I n- 1 and J={n}, (6.1) yields (In)::5<1>(In-I); (6.4) we have

and by

Since the points a I and a2 can be replaced by arbitrary a; and aj, (6.5) follows. 0

6.17. Remarks. (a) Corollary 6.16 is an improvement of Theorem 6.7(a). (b) Let L be a real linear space and U be a convex set in L. A function f: U -IR is said to be convex if for all Xl , X2 E U and a E (0, 1) (6.6) (see Definition 1.1). We show that the well-known Jensen's inequality for convex functions can be obtained easily from (6.6) using Theorem 6.7(a). For this purpose we introduce a semigroup structure on Lx IR+ (=D). The operation "+" is defined by X+Y=(x,p)+(y,q)= (

) PX + qy ,p+q p +q

for

X, YED. (6.7)

We can easily show that this operation is commutative and associative and that X, ... , X n

where E

E

E::9 Xl + ...

+ Xn

E

E,

= U x IR +. We further define the function g: E -IR by g(X) = pf(x) for

X = (x, p)

The condition (Cn) in Theorem 6.7 becomes

E

E.

(6.8)

180

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

which is Jensen's inequality for the convex function f: U - - ~ . We note that condition (C z) is equivalent to inequality (6.6) defining the convexity of f Similarly using Theorem 6.14(a) we can obtain Theorem 3.14 and Corollary 3.15(a) when the p;'s are positive. D Note that Vasic and Stankovic's implication (C 3 ,z) ::} (Cn,k) for convex functions follows directly from Theorem 6.9(a) applied to the case when D = ~ x ~ + , E = [a, b] x ~ +where the operator "+" is defined in (6.7) and the function g: E -- ~ in (6.8). Also note that inequality (6.3) is identical to the condition (Cm,m,m-I)' Thus, by Theorem 6.12, condition (Cm,n,k) is also valid. An interesting application of Theorem 6.14(a) is that the set function considered in Theorem 6.12 is not only subadditive, as shown in Theorem 3.14, but also H-positive (Pecaric, 1986a).

6.18. Remark.

Let a be a given positive n-tuple, k an integer such that

1:5 k s: n, and denote byaik ),

.- .. ,

a ~ ) the

K

= (:) k-tuples formed

from the elements of a. If s, t E ~ , then the mixed mean of order sand t of the positive n-tuple a, when taking k at a time, is

] the power mean of order u of a given n-tuple. Ozeki where M ~ u denotes (1973) considered the special cases A k = M(l, 0; k, a), Bk = M(O, 1; k, a), and C; = M(l, -1; k, a). An inequality for A k was given by Kober (1958) (see also Mitrinovic, 1970, p. 380). Note that the following consequences of (6.1) are generalizations of related results of Ozeki (1973) and Kober (1958):

(n - k)AI + (k - l)An 2: (n - l)Ak,

(n - k)C I + (k - l)Cn 2: (n - l)Ck.

By using Vasic-Stankovic's inequality (the condition (Cn,k))' we can easily give generalizations of these results for arbitrary sand t and arbitrary weights. D

6.2. Stelfensen's Inequality

6.2.

181

Steft'ensen's Inequality

The following result was given by Steffensen (1918) (see also Mitrinovic, 1970, p. 107, and Beckenbach and Bellman, 1961, 1965, p. 48): 6.19. Theorem. Assume that two integrable functions f and g are defined on the interval [a, b] such that f never increases and that O:=; g(t) :=; 1 on [a, b]. Then b

b

a+A

J f(t) dt:=; Jf(t)g(t) dt J f(t) dt, z;

a

b-A

(6.10)

a

where b

A=

Jg(t) dt.

(6.11)

a

Proof. (6.10) follows directly from the following identities (Mitrinovic, 1969, and Mitrinovic, 1970, p. 117); a+A

b

a+A

J f(t) dt - Jf(t)g(t) dt = J (f(t) - f(a + A))(1 - g(t)) dt

a

a

a b

J (f(a + A) - f(t))g(t) dt,

+

(6.12)

a+A b

b

b-A

Jf(t)g(t) dt - J f(t) dt = J (f(t) - f(b - A))g(t)) dt a

b-A

a b

+

J (f(b - A) - f(t))(1 - g(t)) dt. b-A

(6.13)

o 6.20. Remarks. (a) Steffensen (1919) derived the Jensen-Steffensen Inequality (Theorem 2.19) using the second inequality in (6.10) (see also Pecaric, 1979b). Bullen (1970a) derived Steffensen's inequality, Theorem 2.19, using Jensen-Steffensen's Inequality «2.19)). Therefore these inequalities are equivalent.

182

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

(b) The first and second inequalities in (6.10) are equivalent. This can be proved by using the substitution g ( t ) 1~ - g(t) (see Mitrinovic, 1969, and Mitrinovic, 1970, p. 108). (c) Hayashi (1919) generalized inequality (6.10) slightly by imposing the condition: 0:::; g(t):::; A where A is any positive real number (instead of 0:::; g(t):::; 1). However, his result can be easily obtained from (6.10) by using the substitution g ~g / A. (d) Note that Steffensen's inequality cannot be found in Hardy, Littlewood, and P6lya (1934, 1952), and his paper (1918) was not reviewed in Jahrbuch tiber die Fortschritte der Mathematik. However, the paper was cited by G. Szego in his review (see Hayashi, 1919, 1920). (e) Davies (see Mitrinovic, 1970, p. 108) proved Steffensen's inequality by showing that the function a+G(x)

S(x) =

J

x

f(t) dt -

a

Jf(t)g(t) dt, a

where G(x) = S ~ g ( t dt, ) is increasing, Vasic and Pecaric (1984) used that result in the proof of Theorem 3.20 to show that monotonicity of S(x) actually follows from Steffensen's inequality. D In his short note (which contains no references) Apery (1951,1953) proved a different version of Steffensen's inequality. His result states (see also Mitrinovic, 1970, p. 116 and Riekstyn's (1986, p. 23»:

6.21. Theorem.

Let f be a decreasing function on (0,00), and g be a measurable function on [0, 00) such that 0:::; g(x) :::; A (A is a positive real number). Then A

00

J f(x)g(x) dx ss A J f(x) dx, o

0

where

f 00

A=

~

g(x) dx.

o

6.22. Remark. Apery proved Theorem 6.21 by using an identity similar to (6.12). In fact, Mitrinovic (1969) proved Theorem 6.19 by using Apery's idea. D

6.2. Steffensen's Inequality

183

Applying integration by parts, (6.12) becomes a+A

b

a+A

J f(t) dt - J f(t)g(t) dt = - J a

a

a

x

(J (1 - g(t» dt) df(x) a

b

-

J a+,t

b

(6.14)

(J g(t) dt) df(x), x

where A is defined in (6.11). Thus it is obvious that the condition 0:5 g(t):51 (for every t E [a, b]) can be replaced by the weaker condition x

J g(t) dt zzx - a for every x E [a, a + A]

and

a b

Jg(t) dt

~0 for

every x E [a + A, b].

(6.15)

x

However, the conditions in (6.15) are also necessary. Indeed, for f(t) = 1 (t :5x) and f(t) = 0 (t > x for every x E [a, b]) we obtain (6.15) from the second inequality in (6.10). On the other hand, from (6.15) we have b

x

b

J g(t)dt= J g(t)dt- J x

a

g(t)dt~A-(x-a)~O

a

for every x E [a, a

+ A]

(6.16)

and x

b

b

J g(t) dt = J g(t) dt - J g(t) dt s: A:5 X- a for every x E (a + A, b]. a

a

x

(6.17)

Combining (6.15), (6.16), and (6.17) we obtain (6.15)::} (6.18), where x

Jg(t)dt:5X-a a

b

J g ( t ) d t ~ OforeveryxE[a,b].

and

(6.18)

x

Moreover, it is clear that (6.18)::} (6.15). Thus we conclude that (6.15) and (6.18) are equivalent. Consequently, we have proved the following theorem (Milovanovic and Pecaric, 1979):

184

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

6.23. Theorem. Assume that f and g are integrable functions on [a, b], and let A be defined in (6.11). Then the second inequality in (6.10) holds for every decreasing function f iff (6.18) holds. Similarly we can prove: 6.24. Theorem. Let f and g be integrable functions on [a, b]. Then the first inequality in (6.10) holds for every decreasing function f iff x

b

J g(t)dt$b -x and

J

x

a

g ( t ) d t ~ Ofor every X

E

[a, b],

(6.19)

where A is defined in (6.11).

An immediate consequence of Theorems 6.23 and 6.24 is that (Vasic and Pecaric, 1981a): 6.25. Theorem. Let f and g be integrable functions on [a, b]. Then the inequalities in (6.10) hold for every decreasing function f iff x

b

0$ J g(t)dt$b-x and

0$ J g(t)dt$x-a foreveryxE[a,b]. a

x

(6.20)

Some converse results are considered in Pecaric (1982d). 6.26. Theorem. Let f:l ~ ~ ,g : [a, b ] ~ ~ ([a, b] c 1 where 1 is in the interval in ~ ) be integrable functions, a + AE 1 where A is given by (6.11). Then a+A

J f(t) dt a

b

$

f

f(t)g(t) dt

(6.21)

a

holds for every decreasing function f iff x

J g ( t ) d t ~ x - fa orxE[a,a+A] a

b

and

Jg(t)dt$O x

forxE(a+A,b]

(6.22)

6.2. Stelfensen's Inequality

185

and 0 :s: A:s: b - a; or x

f g ( t ) d t ~ x - af orxE[a,b];

(6.23)

a

or b

f g(t)dt:s:O

(6.24)

forxE[a, b].

x

Proof. For f(t) = 1 (t:s: x) and f(t) = 0 (t > x) for all x E I we claim that, from (6.21), (6.22) or (6.23) or (6.24) must be satisfied. To show the other direction, (i) if O:s: A:s: b - a, we can obtain (6.21) from (6.18); (ii) if A> b - a, then a+A

b

b

a+A

f f(t) dt - f f(t)g(t) dt = f f(t)(l- g(t» dt + f f(t) dt a

a

a

b

b

= f (f(t) - f(b »(1 -

sv» dt

a

a+A

+ f (a+A-x)df(x) b

x

b

(J (1 - g(t» dt) df(x)

=- f a

a

a+A

+ f (a+A-x)df(x):s:O; b

186

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

and (iii) if A< 0, then a+.l.

b

J f(t) dt - Jf(t)g(t) dt

a

a

b

J f(t) dt - Jf(t)g(t) dt

= -

a

a+).

a

b

=

Jg(t)(f(a) - g(t)) dt a b

J (x-a-A)df(x)

+

a+.l. b

=-

b

J(J g(t) dt) df(x) a

x a

J (x-a-A)df(x):50.

+

a+.l.

D

Similarly we can prove

6.27. Theorem. Let f: I - 7 functions, b - AE I. Then

~ ,

g: [a, b] - 7

b

~

([a, b] c I) be integrable

b

Jf(t)g(t) dt:5 J f(t) dt a

(6.25)

b-.l.

holds for every decreasing function f iff x

J g(t) dt:50

b

for x

E

[a, b - A]

and

a

Jg(t) dt

?

b- x

x

for x and 0 :5 A :5 b - a; or b

Jg(t)dt?b-x

forxE[a,b];

x

or x

J g(t)dt:50 b

forxE[a,b].

E

(b - A, b]

6.2. Steffensen's Inequality

187

6.28. Theorem. Let g: [a, b] ~ ~ be an integrable function such that there exists aCE [a, b] satisfying g(x) ~1 for x E [a, c] and g(x) ~0 for x E (c, b]. Then (6.21) holds for every decreasing function f: I ~ ~ provided that [a, b] c I and a + A E I. Proof.

Let 0

~A ~b

- a. If c

~a

+ A,

then clearly we have

x

b

J g ( t ) d t ~ x - aforxE[a,c] a

and

J g ( t ) d t ~ O forxE[a+A,b]. x

Suppose that for some Xl E (c, a + A) we have f ~ lget) dt <x, - a. Since f ~ l get) dt ~0, we have f ~get) dt < Xl - a, i.e., a + A < Xl' which is clearly a contradiction. Similarly in the case c > a + A we can prove that (6.13) also holds. Now let A> b - a. Then obviously we have gg(t) dt ~ X - a for X E [a, c] and for some X E (c, b Jwe have x

b

b

Jg(t)dt= Jg(t)dt-J a

a

x

b

g ( t ) d tJ ~ g(t)dt~b-a~x-a; a

i.e., condition (6.23) holds. Similarly, in the case A < 0 we can prove that (6.24) holds. Thus from Theorem 6.19 we obtain Theorem 6.21. D From Theorem 6.20 we can also prove

6.29. Theorem. Let g: [a, b ] ~ ~ be an integrable function such that there exists aCE [a, b] satisfying g(x) ~0 for X E [a, c] and g(x) ~ 1 for X E (c, b]. Then (6.25) holds for every decreasing function f: I ~ ~ provided that [a, b] c I and (b - A) E I. 6.30. Theorem. Let g: [a, b]- ~ be an integrable function such that g ( x ) ~ 1(or g ( x ) ~ O for ) every xE[a,b]. Then for every decreasing function f: I - ~ the reverse inequalities in (6.10) hold provided that a + A, a - AE I. Proof.

This is a simple consequence of Theorems 6.28 and 6.29.

D

6.31. Remark. As noted in Remark 6.20(a), Steffensen (1919) derived the Jensen-Steffensen Inequality using the second inequality in (6.10). In

188

6. Popoviciu's, Burkill's, and Steffensen's Inequalities

the following we show that Theorem 3.3 (the reverse of the JensenSteffensen Inequality) can be obtained from Theorem 6.28; namely, letting Xl;:::' •• ;:::x n and fortE(Xk+l,xd

g(t)=gk=Pk/P n

and

1 ~ k ~ n - 1 ,

where Pk = r . ~ = P 1 i' we show that

Since I can be approximated uniformly on [a, b] by polynomials with nonnegative second derivative, without loss of generality we may assume that ['(x) exists and is increasing, i.e., -['(x) is a decreasing function. Then, from (6.21), we obtain (3.2). 0 By letting get) = AG(t)/ S ~G(t) dt, where A> 0 and S ~G(t) dt > 0, we obtain that (Mitrinovic and Pecaric, 1988c):

IJ b

b

~Jl(t)G(t) dt /

I(t) dt

b-A

a

b

a+A

JG(t) dt

~ I J I(t) dt

a

(6.26)

a

holds iff b

b

J

J

x

a

O ~ A G ( t ) d t ~ ( b - x )G(t)dt x

and

b

J

J

a

a

O ~ A G ( t ) d t ~ ( x - a )G(t)dt

(6.27)

hold for every x E [a, b]; and that the second inequality in (6.26) is valid iff x

J

A G(t) dt a

b

~ (x -

a)

JG(t) dt a

b

and

JG(t) dt;::: 0

(6.28)

x

hold for every x E [a, b]. Of course, using these results we can give extensions of the results related to Bellman's generalization of Steffensen's inequality (Pecaric, 1982b, 1984f). The following theorem (Godunova, Levin, and Cebaevskaja, 1967) is also a consequence of the result stated above:

6.2. Steffensen's Inequality

189

6.32. Theorem.

Let f(x) be a nonnegative decreasing function on [a, b], and <jJ( u) be an increasing convex function for u 2:: 0 with <jJ(0) = O. If g(x) is a nonnegative increasing function on [a, b] such that there exists a nonnegative function gl(X) defined by the equation gl(X)<jJ(g(X)/gl(X)) = 1,

(6.29)

and that J ~gl(t) dt ~ 1, then the following inequality is valid: b

a+).

b

~ iJ

<jJ(J f(t)g(t) dt / J g(t) dt) a

a

(6.30)

<jJ(f(t)) dt,

a

where A = < j J ( J ~g (t) dt).

For <jJ(u)

= uP (p > 1),

Theorem 6.32 becomes

6.33. Theorem. f

Let f(x) be a nonnegative decreasing function on [a, b], Lp(a, b), and let g(t) be nonnegative and increasing on [a, b] such that J ~gq(t) dt ~ 1, where p > 1 and q = p/(p - 1). Then E

b

(J f(t)g(t) dt a

r

a+).

~

(6.31)

J fP(t) ft a

holds, where A = ( J ~g(t) dtY'.

Proof of Theorem 6.32.

Using Jensen's inequality for convex functions and the second inequality in (6.26) we have b

<jJ

(J f(t)g(t) dt / a

b

b

~J

J g(t) dt) a

b

g(t)<jJ(f(t)) dt / J g(t) dt

a

i

~

a

a+).

J

(6.32)

<jJ(f(t)) dt

a

provided that x

b

<jJ(J g(t) dt) J g(t) dt a

a

b

~ (x -

a) J g(t) dt a

b

and

J g(t) dt

2:: 0

(6.33)

x

hold for every x E [a, b]. The second condition in (6.33) is obviously satisfied. On the other hand, the increasing convex function <jJ with

190

6. Popoviciu's, Burkill's, and Stelfensen's Inequalities

<j>(0) = 0 is starshaped, thus <j>(ax) ::;a <j> (x) holds Consequently by (6.29) and Jensen's inequality we have

for

0 < a ::; 1.

x

b

<j>(J get) dt) J get) dt a

a b

b

x

b

I

= <j>(J gl(t) dt(J get) dt) (J gl(t) dt)) J get) dt a

a

a

b

a

b

x

b

::; (J gl(t) dt)<j>(J gl(t)(g(t)jgl(t)) dtl J gl(t) dt) J get) dt a

a

a

b

a

x

::; J gl(t)<j>(g(t)jgl(t)) dt J get) dt a

a b

=J a

x

x

1 . dt J get) dt

= (b - a) J g(t) dt.

a

a

Since g is an increasing functiori, we have (see, for example, Mitrinovic, 1970, p. 9) b

x

_1_ J get) dt b-a

~ _ 1 J_ get) dt,

a

x-a

a

i.e. x

b

(b-a) J g(t)dt::;(x-a) J g(t)dt. a

a

Hence the first condition in (6.33) is also satisfied.

0

We note that (a) (6.32) is an interpolation inequality of (6.30). Also note that the condition in (6.29) can be replaced by (6.29') or, more generally, by b

J gl(X)<j>(g(X)jgl(X)) dx s: b - a;

(6.29")

a

and (b) the proof given above can be found in Mitrinovic and Pecaric (1988b). The same is true for Remarks 6.34(a).

6.2. Steffensen's Inequality

191

Inequality (6.31) was given by Bellman (1959), and has been stated in Beckenbach and Bellman (1961, 1965, p. 41) and Mitrinovic (1970, p. 111). However, this result is incorrect as stated, as noted by Godunova, Levin, and Cebaevskaja (1967). The results given above represent a more general and correct version of that in Bellman. Another corrected version of Bellman's inequality can be found in Bergh (1973):

6.34. Theorem. Let f and g be positive functions on (0, (0), f decreasing and g measurable. If for some p 2: 1, f E LP + L and g E Lq n L 1 such that IIfllL" = 1 and IIgllo = t, where lip + 1/q = 1, then 00

tP

00

r p

J f(x)g(x) dx :521/q(J (f(x»)P dx o

0

(6.34)

holds, and the constant 2 1/q is the best possible.

°

A similar inequality for < P :5 1 can be found in Godunova and Levin (1968). In fact, that result is a consequence of a more general result given below: Let K(x, t) (x EX, t E [0, a]) be a given nonnegative kernel. We say that the function 1>: [0, a ] ~ ~ belongs to the class U(K) if it can be expressed in the form ep(t) = Ix K(x, t) do(x), where o(x) is an increasing function on [0, a] with f x do(x) = 1. Let f be a positive, increasing, and strictly convex function such that f' (x )/f"(x) is concave, and let t

h(t)

=J

t

dh(s),

get)

=J

h(O) = g(O) = 0,

dg(t),

h(a)=1,

o

o

where get) and h(t) are increasing functions on [0, a]. Then a

a

(J f( ep(t») dh(t)

J ep(t) dg(t) :5 f- 1 o

0

holds for every function ep in the class U(K) iff a

J

a

(J f(K(x, t)) dh(t)

K(x, t) dg(t):5 f- 1

o

holds for every x E X.

0

(6.35)

192

6. Popoviciu's, Burkill's, and Stell'ensen's Inequalities

The following generalization of Steffensen's inequality is given in Pecaric (1982f):

6.35. Theorem. Let h be a positive integrable function on [a, b] and f be an integrable function such that f(x)/h(x) is increasing on [a, b]. If g is a g(x):s for every x E [a, b], real-valued integrable function such that then (6.21) holds, i.e.,

O:s

b

f

1

a+A

~

f(x)g(x) dx

a

f

(6.36)

f(x) dx

a

where A is the solution of the equation a+A

f

b

hex) dx =

a

f

h(t)g(t) dt.

a

If f(x)/h(x) is a decreasing function, then the reverse of the inequality in (6.36) holds.

Proof. a+A

a+A

b

J f(t) dt - Jf(t)g(t) dt = f a

a

b

f

g(t)(1 - g(t»f(t) dt -

a

f(t)g(t) dt

a+).

a+A

: s ~ ~ : ~: ~ f

h(t)(1- g(t» dt

a b

-f

f(t)g(t) dt

a+A b

a+A

= ~ ~ : : ~(f~ h(t)g(t)dt- f a

h(t)g(t)dt)

a

b

- J f(t)g(t) dt a+A

f ( b

= and the proof is complete.

g

a+A

D

t )h (t )( f (a + A) h(a + A)

f(t») dt:S 0 h(t) ,

6.2. Steffensen's Inequality

193

Applying Theorem 1.43(a) we obtain, from Theorem 6.34: 6.36. Theorem. Let g be an integrable function such that O:s g(x) :s 1 for every x E [a, b]. (a) If the function f is convex of order n with t
f b

A= (n

lin

(t - at-lg(t) dt)

.

(6.37)

a

(b) Iff is a nonnegative and concave function of order n with f(k)(a) = 1, ... , n - 2), then the reverse of the inequality in (6.36) holds.

=0

(k

6.37. Remarks. (a) Theorem 6.36(a) is proved in Milovanovic and Pecaric (1979), and a similar generalization of the first inequality in (6.10) is also given there. (b) By using Theorems 6.35 and 6.36 we can obtain Theorems 2.68(b) 0 and 2.69, respectively (see Pecaric, 1982f). Some additional generalizations of results in Milovanovic and Pecaric (1979) are given in Fink (1982b). Let f E Mo where Mo is the class of nonnegative and increasing functions. Then f(x) = H dv(t) for some nonnegative Borel measure v, If f(O) > 0, then it includes an atom at 0 and, to facilitate the arithmetic we introduce the notation x , = max{x, O}. Also x':- denotes (x. )" except that 00 will be interpreted as being 1. Thus the indicator function of [t, (0) is (x - t ) ~ . The above formula for f E M o may be written as

f 1

f(x) =

(x -

t ) ~ dv(t).

o

The following class of functions which we consider is larger than that containing such f(x). Let M; denote the class of functions f with the representation

f 1

f(x) =

(x -

t ) ~ dv(t)

for x

E

[0, 1]

o

for some v which is a nonnegative regular Borel measure. Note that k need not be an integer, although the case of great interest is when it is an integer. In particular, M, is the class of increasing convex functions with a value zero at O. More generally, if f E c(n+l)(o, 1) with f(i)(O) = 0 for i = 0, ... , n - 1, f(n) 2:: 0, and I":" 2:: 0 on [0,1], then f E M n .

194

6. Popoviciu's, Burkill's, and Stetrensen's Inequalities

6.38. Theorem. Let J.. be a regular Borel measure such that f6ldJ..(x)1 < 00, and let dx denote Lebesgue measure, then f6f(x) dJ..(x) 2: fof(x) dx holds for all f E Mo iff

f 1

dJ..(x) 2: 0 foreverytE[0, 1]

t

and 1

a

~Or:!}!:l {t +

f

dJ..(x)}.

t

Therefore 1

a = Or:!}!:l {t -

f

dJ..(x)}

t

is the best possible choice.

6.39. Theorem. Let J.. be a (signed) regular Borel measure such that HIdJ..(x)1 < 00. Then f6f(x) dA(X)2: fof(x) dx holds for all f E M; iff

f 1

(x -

t ) ~ dJ..(x) 2: 0

for every t E [0, 1]

o

and (6.38) Therefore the best possible choice is that a equals the right-hand side of (6.38).

6.40. Remark. Some generalizations of Steffensen's inequality for functions of several variables are given in Pecaric (1980d), Fink (1982b), and 0 Pecaric and Janie (1989). Pecaric (1989b) proved the following theorem:

6.41. Theorem. Let G: [a, b] ~ ~ be an increasing and differentiable function and f: I ~ ~ be a decreasing function (I is an interval in ~ such

6.2. Stelfensen's Inequality

that a, b, G(a), G(b)

E

195

I). (a) If G(x) ?x, then

b

G(b)

f f(x)G'(x) dx?

f

a

f(x) dx.

(6.39)

G(a)

(b) If G(x) =5 x, then the reverse of the inequality in (6.39) is valid. Proof.

By letting G(x) = z we have b

b

G(b)

f f(x)G'(x) dx = f f(x) dG(x) = a

a

f

f(G- 1(z» dz.

G(a)

If G(z) > z, then G-1(z) < z and f(G-1(z» ?f(z). Thus we have Glb)

G(b)

f(G-1(z» dz > G(a)

f

f(z) dz;

G(a)

i.e., (6.39) holds. If G(z) =5 z then, of course, we obtain the reverse of the inequality. 0

6.42. Remarks. (a) For a = 0 and G(O) = 0, we obtain a result in Ostrowski (1970, pp. 83, 161,263) from Theorem 6.41(b). , G ( b ) ~ o o we , obtain the result in Volkov (b) For a=O, b ~ o o and (1969). This result is a generalization of the following inequality of Gauss (see, for example, Mitrinovic, 1970, p. 300): Letfbe never increasing for x> 0 then, for any A> 0, 00

2

1.. f A

00

f ( X ) d x = 5 x~ 2f(x)dx. f 0

Indeed, this inequality follows if we also let G (x) = 4x 3 /271.. 2 + A for 1..>0. (c) Theorem 6.19 is, in fact, a consequence of Theorem 6.41. To see this, let G(x) = a + f ~ g(t) dt in Theorem 6.41 where g satisfies the conditions of Theorem 6.19, then G(x) =5x holds and we obtain the second inequality in 6.10. To obtain the first inequality we let G(x) = b - f ~ g ( t d) t in Theorem 6.41. (d) For another generalization of Gauss' inequality, see Petschke 0 (1989).

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CebySev-Gruss’ , Favard’s, Berwald’s, Gauss-Winckler’s, and Related Inequalities

Chapter 7

7.1.

Cebysev-Griiss Inequality

A classic result due to CebySev (1882,1883) is stated in the following theorem. 7.1. Theorem. Let f, g : [ai b ]+= R and p : [a, b ]+= R + be integrable functions. I f f and g are monotonic in the same direction, then b

6

a

a

b

a

b

a

provided that the integrals exist. I f f and g are monotonic in opposite directions, then the reverse of the inequality in (7.1) is valid. In both cases, equality in (7.1) holds iff either g or f is constant almost everywhere. A discrete analogue is given by 7.2. Theorem. Let a and b be two n-tuples of real numbers monotonic in the same direction, and p be a positive n-tuple. Then

If a and b are monotonic in opposite directions, then the reverse of the inequality in (7.2) holds. In either case equality holds iff either a , = . . = a, o r b , = b,.

-

=

a

CebySev’s inequality can be generalized for m (>2) functions (or n-tuples).

197

198

7. Related Inequalities

7.3. Theorem. Let t., ... .t: (m > 2) be nonnegative real-valued functions and p a positive function on [a, b]. If t., ... .I; are monotonic in the same direction, then b

(fp(x) dx a

r-

1

il

b

b

f p(x) rlfi(X) dx 2= (f p(x)f;.(x) dX). a

(7.3)

a

If It, ... ,fm are positive on [a, b], then the equality in (7.3) holds iff at least m - 1 of the functions f1' ... .I; are constant almost everywhere. 7.4. Theorem. Let aj (j = 1, ... , m; m > 2) be nonnegative n-tuples which are monotonic in the same direction, and p be a positive n-tuple. Then

(7.4) If all n-tuples are positive, then the equality in (7.4) holds iffat least m - 1 n-tuples among a1' ... , am have identical components.

7.5. Remarks. (a) The history of Cebysev's inequality and the question concerning its priority are considered in the expository paper by Mitrinovic and Vasic (1974). It was noted that for the special case in which Pi = a, (i = 1, ... ,n), inequality (7.2) was obtained earlier by Laplace (1749-1827), and that inequality (7.1) with p(x) == 1 was obtained by Winckler (1866). (b) Cebysev's papers were published in 1882 and 1883, and inequality (7.3) with p(x) == 1 was considered by Andreief (1883a). (c) There exist several results which show that Cebysev inequalities are valid under weaker conditions: (i) The condition that the functions be monotonic can be replaced by the condition that they be similarly ordered. The same is valid for sequences. In this case Theorem 7.1 is a simple consequence of the following identity:

f b

a

f b

p(x) dx

f b

p(x)f(x)g(x)dx -

a

a b

b

= ~ J a

f b

p(x)f(x) dx p(x)g(x) dx a

Jp(x)p(y)(f(x)-f(y»(g(x)-g(y»dxdy. (7.5)

a

7.1. Cebysev-Griiss Inequality

Note that the functions f: I ordered if

~IR

and g: I

~IR

199

are said to be similarly

(f(x)-f(y))(g(x)-g(y))2:0 forevery

x. v e I

holds, and they are said to be oppositely ordered if the reverse inequality holds. A similar definition applies to sequences. Of course, the generalization of the identity in (7.5) for functions with several variables is also valid (see Berljard, Nazarov, and Svidskii, 1967, or Mitrinovic and Vasic, 1974). Thus similar generalizations of Theorems 7.1-7.4 are valid (for similarly ordered functions and sequences). The first such result is given in Hardy, Littlewood, and P6lya (1934, 1952, p. 168). (ii) The condition that the functions be monotonic can be replaced by monotonic in mean. Such a result for two functions (sequences) was given by Biernacki (1951), and for m (2:2) functions (sequences) by Burkill and Mirsky (1975). A simple proof and some interpolations were given in Vasic and Pecaric (1982d), and additional generalizations for functions with increasing increments were given in Pecaric (1984g) (see Theorems 2.30, 3.12, 3.17 and Remark 3.19(b)). (iii) Steffensen (1920) noted that Theorem 7.1 is valid when f is an increasing function on [a, b] and g satisfies the condition x

P ~ XJ ) p(t)g(t) dt:5 P ~ b J ) p(t)g(t) dt, a

x

b

where

P(x) =

Jpet) dt. a

a

(7.6) Pecaric (1980b) (see also Vasic, Stankovic, and Pecaric, 1985b) noted that instead of pet) > 0, we need only that P(x) > 0 and P(b) > O. Steffensen and Pecaric also gave corresponding discrete analogous results. Steffensen's result contains a result of Biernacki (1951), and Pecaric's results is a generalization of Popoviciu (1959b) (see also (iv)). (iv) The condition that pet) > 0 (Pi> 0) can be replaced by

0:5 P(x):5P(b)

for

a:5x:5b.

(7.7)

The same condition also appears in the Jensen-Steffensen Inequality; thus Cebysev's inequality and the Jensen-Steffensen's inequality are related. If the functions have increasing increments, then both (7.1) and (7.2) follow from Jensen-Steffensen's inequality. For the general case, a proof for (7.2) can be found in Popoviciu (1959b).

200

7. Related Inequalities

(v) The conditions that the functions (and sequences) be positive in Theorems 7.3 and 7.4 were weakened by Ahlswede and Daykin (1979). That is, if p(x) == 1 and Pi == 1, then Theorems 7.3 and 7.4 are valid for increasing functions and increasing sequences aj which satisfy

t

b

t ( 0 ) + b ~ a J t ( X ) d x : 2 for :0

j=1, ... ,m

(7.8)

a

and 1

n

ajl + - - L aji:2: 0 for n -1 i=2

j

= 1, ... , m.

(7.9)

(vi) Another modification of the conditions for Cebysev's inequality was given by Levin and Steckin (see, for example, Karlin and Studden, 1966, pp. 414-415). The following result is a generalization of their result and can be proved similarly: Let v: [0, 1] ~ ~ be an increasing function such that v(x) = -v(1- x), and let f: [0, 1 ) ~ ~ be an integrable function with respect to v such that the following two conditions hold:

f ( X ) i s i n C r e a S i n g f O r X E [ oand ,~l

f(x)=f(1-x)

for

xE[0,1].

Then for every continuous convex function <j> we have 1

1

1

1

J dv(x) J f(x) <j>(x) dv(x):::; J f(x) dv(x) J <j>(x) dv(x). o

0

0

0

As a special case we have the inequality of Leven and Steckin: 1

1

1

J f(x) <j> (x) dx:::; J f(x) dx J <j>(x) dx, 0 0 0

For a more general result, see Clausing (1980). (d) For other similar results, see the noted paper by Mitrinovic and Vasie (1974) and Section 8.l. (e) Theorems 7.1-7.4 have been generalized for monotonic functions of several variables. The case p(x) == 1 was generalized by Vietoris (1974). Pecaric, Janie, and Beesack (1982) showed that an analogous generalization is valid for the case when p(x) = P\(Xl) ... Pk(xd, and for similarly defined sequences of weights. In fact, they gave a related result for Stieltjes' integral. Additional generalizations are given in Pecaric and

71..

Cebykv-Griiss Inequality

201

Mesihovik (1988), where a similar generalization of (b) is also given. Some applications in number theory are given in Rutkowski (1989). 0 In the rest of this chapter we shall use the following notation

a

a

A simple consequence of (7.10) is the following result: Let p be a nonnegative function. If 2 0 for all x i , . . ,x,

I.L(xi)ln Igi(xi)ln

E

[a, b ] ,

then (7.11) Simple consequences of (7.11) include the inequalities of CebySev, Cauchy, and other inequalities. Of course, Gram’s inequality

II

b

a

p(x).L(xl&(x)dx

1

20

(7.12)

is also an obvious consequence of (7.11). Identity (7.10) for Stieltjes’ integral is given in Chokhate (1929). A generalization of (7.10) for functions of several variables is also valid (the

202

7. Related Inequalities

case p ( x ) = 1 is given in Ogura, 1920). Results analogous to (7.11) and (7.12) are also valid in this case. An interpretation of (7.12) for the case p ( x ) = 1 is given in Ogura (1920), and using the idea in his proof we can obtain the weighted version of his result: Let q51(x), & ( x ) , . . . , q5n(x), . . . be a system of normalized orthogonal functions with respect to p, i.e.,

a

If we denote by a i ( i ) the following

I

b

ai(fi) =

p(x)fi(x)&(x) dx for j = 1, . . . ,p ,

i = 1, 2, . . . ,

a

then we have 2

h

where C denotes the summation with respect to m , , . . . , mP satisfying 1P m , < m2 < * . < mP 5 k ; k z p being any fixed positive integer. Of course, similar results can be formulated for Stieltjes’ integral. Discrete versions of (7.10) are also valid. For example, we have

If Ne let f i= x i , x ; = y i , g $ = z j , g$ = u j , then, after simple algebraic manipulations, we obtain the following identity in Seitz (1936/37):

I

i=l j=1

;=I j = 1

As an immediate consequence of the above identity we can formulate the following (Seitz, 1936/37): If, for all positive integers i, j , r, s such that

7.1.

l l i < j s n and l l r < s < r n , we have 2 0 and

IYi

CebySev-Griiss Inequality

;: : 1

203

:;120,

then the inequality

holds. This inequality is also a generalization of CebySev's and Cauchy's inequalities (see, for example, MitrinoviC and VasiE, 1974). Finally, we give an interesting consequence of (7.11) for generalized convex functions. Let f : (a, b ) 4 R be a convex function with respect to an ECT-system of functions {ui}: (see (7.4)) (in symbols, we write f E C(bio, u l , . . . , u,)). The following result is a simple consequence of (7.11): Let f E C ( u o ,u l , . . . , u,) and g E C(vo,' u l , . . . , vn), and let p be a nonnegative integrable function. Then

2

a

l a

0.

(7.13)

a

+

If f, g : (a, )+ R are two (n 1)-convex functions, i.e., if f, g E C(1, t, . . . , t"), then (7.13) becomes (for p ( x ) = 1)

...

a

...

...

...

a

a

204

7. Related Inequalities

For n = 1, we obtain that for all convex functions inequality is valid: b b b f f(x)g(x) dx - b

~ _a f

a

f and g the following

f(x) dx f g(x) dx

a

a

7.6. Remarks. (a) Inequality (7.14) was proved by Lupas (1972a). Moreover, in the case when S: (x - (a + b )/2)g(x) dx = 0, this result was proved by Atkinson (1971) with the additional condition that the second derivatives of f and g exist. (b) In Pecaric (1989d), inequality (7.14) is generalized for (2,2)convex f u n c t i o n f : P ~ (/= 1 R [a, b]): b b b

~a f

f f(x, x) dx - b a

a

;:::

12

f f(x, y) dx dy a

(b - a)3

fbfb(x_a+b)(y_a+b)f(X y)dxdy. 2 2'

(7.15)

a a

(c) Some other results related to inequality (7.14) are given in Lupas D (1972a), Vasic and Lackovic (1979), and Pecaric (1983d). Fan (1953) considered the inequality b b

b

f f K(x,y)f(x)g(y)dxdy:5Bf f(x)g(x)dx a a

(7.16)

a

for nonnegative and decreasing functions f and g; there exist many generalizatons of this result (see Pecaric, 1980d, Pecaric and Crstici, 1981b, and Pecaric, 1984d, 1987). For example, the following theorem holds:

7.7. Theorem. Let p : P ~IR and q:I ~IR (/ = [a, b]) be two integrable functions. Then for every (1, I)-convex function f: P ~IR the inequality b b b K(f,p,q)=

f a

a(x)f(x,x)dx-

ff a

a

p(x,y)f(x,y)dxdy;:::O (7.17)

7.1. Cebysev-Griiss Inequality

holds ifffor every x, y Pea, y) = Q(y),

E

205

[a, b] we have P(x, a)

= Q(x),

P(x, y):s Q(max(x, y»,

(7.18) where Q(x)

=

f ~ q(t) dt and P(x, y) = f ~

g pes, t) dt ds.

Interpolation inequalities of Cebysev's inequality are given in Vasic and Djordjevic (1973), Vasic and Pecaric (1981b, 1982c), and Mesihovic (1985). In the following we give the result in Pecaric (1987). 7.8. Theorem. Let Pij (ij = 1, ... , n) be nonnegative real numbers such that Pij = Pji (i, j = 1, ... , n). If (7.19) holds, then

(7.20) where n

Cn(a, p)

= 2:

i=l

n

n

n

2: Pijajj - 2: j=l 2: Pijaij' j=l i ~ l

7.9. Remark. Theorems 7.3 and 7.4 follow from Theorems 7.1 and 7.2 by induction. Note that by applying Theorem 6.7 we can obtain many results that are usually proved by induction. The same is true for Cebysev's inequality. Indeed, the set of all positive and decreasing (or increasing) functions X defined on [a, b], with the multiplication of functions as an interior operation, is a semigroup D. Let the set E c D be nonempty and have the property that Xl' X 2 E E implies Xl + X 2 E E, and let us define the function f : E - ~ by b

f(X) =

b

JX(t) d{t(t)/ Jduit), a

a

{t being a nonnegative measure such that f ~dll(t)

'*

0 and such that the integral in the numerator exists for every X E E. For the operation "+" in ~+, let us take the ordinary multiplication of positive numbers; then this operation has neutral element 1. Let us denote b

r: = }] [f Xi du./ a

b

b

f J/[f du

a

a

b

Xl' .. x; du./

f J. du

a

206

7. Related Inequalities

Since the function X(t) = 1 (a ~t E, we have (by Theorem 6.7) T2

~ b)

is a neutral element and belongs to -s 1, and Corollary 6.15 yields

~T;

~1

(7.21) and b

b

b

b

~ l s ~ ~ s (f n Xi du f x, dll) / (f du f XiX dll) ~ 1.

t;

(7.22)

j

a

a

a

a

o

G. Gruss (1935) proved the following converse of Cebysev's inequality (conjectured by H. Griiss-see footnote in G. Gruss, 1935):

7.10. Theorem.

Let f, g be two integrable functions defined on I

= [a, b]

and satisfying

(7.23) for every x

E

(a, b), where c l' C2> d 1, d 2 are fixed real numbers. Then

(7.24) where b

T(f, g) = b

b

~ a f f(x)g(x) dx -

(b

b

~ a)2 f f(x) dx f g(x) dx,

a

a

a

and the constant ~ in (7.24) is the best possible.

A discrete analogue of (7.24) is given in Biernacki, Pidek, and Ryll-Nardzewski (1950):

7.11. Theorem. d l ~ b i ~ d 2 f

Let a and b be two n-tuples such that lo ~r i ~ n Then .

Cl

~a,

~ C2

and

(7.25) where [x] denote the largest integer

~ xand

7.1. Cebysev-Griiss Inequality

207

The following result is related to Griiss' inequality:

7.il. Theorem. Let f, g be absolutely continuous on I = [a, b], and let be bounded on I. Then 1 IT(f, g)l:::; -2 (b - a)2 sup II'(x)l- sup Ig'(x)l, (7.26) 1 XEI XEI

1', g'

where equality in (7.26) holds iff I' and g' are constant. Moreover, if are continuous, then

1', g'

1 T(f, g) = 12 (b - a f l ' ( ~ ) g ' ( 1 / ) for some ~ , 1/ E I.

(7.27)

7.13. Remarks. (a) Inequality (7.26) was proved by Cebysev (1882), and identity (7.27) can be found in Ostrowski (1970). (b) Several generalizations of Theorems 7.10-7.12 are given in Pecaric (1984d, 1987). For example, the following result is valid: (i)

Let a = (ai, ... , an) and b = (b l , . . . , bn) be real n-tuples monotonic in the same direction, and let p = (PI, .. _, Pn) be a real n-tuple such that k

0:::; Pk

:::;

P;

for

k

= 1, ... , n -

1

and

Pk

= L. Pi'

(7.28)

i=I

If lilakl

~ m and lilbkl ak+1 - akJ then

~r

T(a, b; p)

for k = 1, ... , n -1 where Sa; = ~ mrT(e, e;

p)

~ 0,

(7.29)

where

and e = (0,1, ... ,n -1). If a and b are monotonic in opposite directions, then

T(a, b; p):::; mrT(e, e; p):::; 0,

(7.30)

where e= (n -1, ... ,1,0). (ii) Let a and b be real n-tuples monotonic in the same direction such that lilakl ~ m, lilbkl ~ r for k = 1, ... , n - 1, and let p be a real n-tuple such that either 0< P; :::; Pk

for

k: = 1, ... , n - 1

(7.31)

208

7. Related Inequalities

or

0:5 P; :5 Fk

for

k = 2, ... ,n

and

h

= P; - Pk -

I •

(7.32) Then the reverse of the inequalities in (7.29) is valid. If a and b are monotonic in opposite directions, then the reverse of the inequalities in (7.30) holds. l c and I ~ b k :l 5 d (iii) Let a and b be two real n-tuples such that I ~ a k :5 for k = 1, ... , n - 1, and let p be a real n-tuple such that (7.28) is satisfied. Then IT(a, b; p)l:5 cdT(e, e; p) holds. If (7.31) or (7.32) holds instead, then IT(a,b;p)l:5cdT(e,e;p). These results are generalizations of results in Lupas (1981) and Mitrinovic (1970, p. 341). In fact, for m = r = 0 (7.28) is a generalization of Cebysev's inequality, as noted in Remark 7.5(c.iv). Pecaric also proved some more general results. For a (l,l)-convex sequence {aiJ ((1, 1)concave sequence {aij}) he obtained the inequality

The continuous analogue of (7.33) is that (iv) Let F be a (l,l)-convex function, and let f and g be two real-valued functions such that F(f(x), g(y)) is integrable over [z and a :5f(x):5 A, b :5g(x):5 B for all x E [a, b], where a, A, b, B are fixed real numbers. Then b

b

b

I b ~ a f F ( f ( X ) , g ( X ) ) d x - ( b ~ af ) F(f(X),g(y))dXdyj zf a

a

a

1 :5 4: (F(A, B) - F(A, b) - F(a, B)

+ F(a, b)).

The following Pecaric's result generalizes (7.27): Assume that p, q satisfy the conditions in Theorem 7.7, and let K(f, p, q) be defined as in (7.17). We have (1) If f: P ~ ~ has continuous partial derivatives t., fz, and fZI on P, then

Ki], p, q)

= fZI(;, 1/)K((x - a)(y - a), p, q) for all

;,1/ E [a, b];

7.1. Cebysev-Griiss Inequality

209

and (2) if f, g : P ~IR have continuous partial derivatives f1' fZ1' gl' gz, and gZl with gZl 0 on P, then

*"

f Z l ~ ~1J ~ K(g, p, q) gZl ,1J

K(f, p, q) =

for all

fz,

~ , 1 JE [a, b].

(c) Similar converses of Theorems 7.3 and 7.4 are given in Pecaric (1980g) and Roghi (1971-72). Pecaric's result states: Let

(J p(t) dt) m-1 JDh(t) dt -/1 (J jj(t)p(t) dt). b

D m(f1, ... ,fm; p)

=

b

a

b

a

a

If jj(x) are monotonic functions on [a, b] for j = 1, ... ,m, and if p is a nonnegative function on [a, b], then IDm(f1,'" ,fm;P)1 ::5(m _1)m-m/(m-1) b

x

(J p(t) dtf jU (ljj(b)1 + Ijj(b) - jj(a)l). a

Furthermore, the constant is the best possible. (d) In the proof of Theorem 7.10 given in Mitrinovic (1970, pp. 70-71), the following inequality is given: T(f, gf::5 T(f, f)T(g, g).

(7.34)

Note that a more general result is also valid: Let b

[f, g]

= IJ p(x)t(x)gj(x) dxl a

n

be the determinant given in the identity (7.10). Then [f, gf::5 [f, n[g, g]. The result is due to Davis (see Beckenbach and Bellman, 1961, 1965, p. 61), where the inequality with p(x) =' 1 is given, and was first proved by Everitt (1957). 0 Ostrowski (1970) gave the following result:

7.14. Theorem.

Let f be a bounded measurable function on I such that ::5f(x)::5 Cz for x E I, and assume that g'(x) exists and is bounded on I. Then

Cl

1 IT(f, g)1 ::5 -8 (b - a )(c z - c 1) sup Ig'(x)l· xel

Furthermore, the constant A in (7.35) is the best possible.

(7.35)

210

7. Related Inequalities

The following two theorems are proved in Lupas (1973) and are refinements of the results in Ostrowski (1970). For notational convenience let us define b

Ilfllz =

(b ~ a J If(x)I dX) Z

liZ

.

a

7.15. Theorem. Let f, g be locally absolutely continuous on I = (a, b) with f', g' E Lz(I)· Then (b - a)Z IT(f,g)l:5

n

z 11f'llz·llg'llz.

(7.36)

Furthermore, the constant 1/nz in (7.36) is the best possible.

7.16. Theorem. Let f be locally absolutely continuous on I with f' E Lz(I) and let g be bounded and measurable on I with d, :5g(x):5 d z on I. Then 1 IT(f, g)l:5 2n (d z - d 1)

,

Ilf liz·

(7.37)

7.17. Remark. The weighted versions of (7.36) and (7.37) are given in Milovanovic and Milovanovic (1979). D Of course, the Gruss Inequality can be improved if we consider smaller classes of functions. A function f defined on (a, b) is said to be monotonic of order p if it is a convex (concave) function of order 1, ... ,p. If f is monotonic of order p in (a, b) for every p = 1, 2, ... , it is said to be absolutely montonic in (a, b). If f is absolutely monotonic in (a, b), then it has derivatives of all orders and f(kl(X) 2':

°

or

f(k l(X):5

°

for all

k

=

1, 2, . ..

and x

E

(a, b).

A function f is said to be completely monotonic on (a, b) if f( -x)

absolutely monotonic in (-b, -a) or equivalently, and it satisfies f'(x):50,

f"(x) 2': 0,

f"'(x) :50,

...

f"(x) :5 0,

f"'(x) 2': 0, ...

or

IS

7.1. Cebysev-Griiss Inequality

for x in (a, b). Griiss (1935) proved that if monotonic functions, and if (7.23) holds, then

211

f and g are absolutely

4 IT(f, g)l:5 45 (cz - cl)(d z - d 1 ) ,

(7.38)

and the constant 15 is the best possible. Pecaric (1983e) noted that (7.38) is also valid if f and g are completely monotonic functions. If f is absolutely monotonic and g is completely monotonic, then (7.39) where the constant -b is also the best possible. Pecaric also gave a result for three functions, and a related result was given by Hardy (1936) (see also Mitrinovic, 1970, p. 72). Landau (1935) proved that (7.38) holds if f and g are monotonic of order 4. Needless to say, this is a nontrivial improvement of Griiss' result. For functions of order k = 2, 3, Landau proved that 1 ' IT(f, g)1 :5 9(cz - Ct)(d z - d 1 )

for

k = 2,

(7.40)

9 IT(f,g)I:5100(cZ-Cl)(dz-dl) for

k=3.

(7.41)

Note that using inequality (7.34) we can obtain a series of results when

f and g belong to different classes of functions. For example, if f is monotonic of order 2 and g is monotonic of order 3, then (7.42) Some other bounds for IT(f, g)1 can be obtained by using the Berwald inequality (see Section 7.2). As a special case, we have

where f is a positive and concave function on (a, b). This inequality is equivalent to the inequalities b

T(f, f):5

~ (b ~ a J(f(x))Z dx) z a

b

and

T(f, f):5

~ J (f(x))Z dx.

212

7. Related Inequalities

Thus, using (7.34) we obtain b

IT(f, g)l:5

b

~ IIfllzllgllz:5 ~ (b ~ a Jf(x) dx )(b ~a Jg(x) dx ), a

a

(7.43) where f and g are positive and concave functions on (a, b). This is an interpolation inequality for an inequality given in Mitrinovic (1970, p. 73) (see also Gruss, 1935, and Franck and Pick, 1915). Other bounds can be obtained from theorems which are proved in Franck and Pick (1915) and Blaschke and Pick (1916). For other related results, see Fempl (1965) and Knopp (1935). Griiss' inequality provides bounds for the difference in T(f, g). An analogous result for the ratio 1

R(f, g)

=

1

1

(J f(x) dx Jg(x) dx ) / (J f(x)g(x) dX) 0 0 0

was obtained by Karamata (1948). He proved that if integrable functions on [0, 1] and if O
and

O
for

f

and g are

0:5X:51,

then K

-z :5 R(f, g):5 K zwith · y(ib+VAB K = Viii YAh >- 1. aB+

Ab

Lupas (1978) gave an analogous result for positive linear functionals.

7.2.

Favard's, Berwald's, Gauss-Winckler's, and Related Inequalities

Favard (1933) proved the following result:

7.18. Theorem. Let f(x) be a nonnegative continuous concave function on [a, b], not identically zero, and let ep(y) be a convex function on [0, 2f], where b

- b -a 1 f=

J f(x)dx. a

(7.44)

-r

Then

;1

7.2. Related Inequalities

213

b

<jJ(y) dy

~b ~ a f

o

<jJ(f(x)) dx.

(7.45)

a

7.19. Remarks. (a) In Karlin and Studden (1966, p. 412) a more general inequality is given: 2J-c

21 ~ 2c f

b

<jJ(y) dy

~b ~ a f

<jJ(f(x)) dx,

(7.46)

a

c

where c satisfies 0 < c s; fmin (where fmin is the minimum of f(x)) and <jJ is convex on [c, 21- c]. The reverse inequality is also given in Karlin and Studden (1966, pp. 412-13): If f is a continuous nonnegative convex function on [a, b], d > fmin, and <jJ is convex on [d, 21- d], then 2J-d

b

2 1 ~ 2 d f < j J ( Y ) d y s ; b ~<jJ(f(x))dx. af d

(7.47)

a

(b) In Popoviciu (1944, p.' 35), Favard's inequality is given in the form (with a = 0 and b = 1) 1

1

(7.48)

f <jJ(f(x)) dx s; f <jJ(21x) dx. o

0

This fact suggests that Theorem 7.18 is a consequence of majorization theory (see Section 12.1) in an integral form. 0 Pecaric (1984b) used (7.48) in the proof of the following generalizations of Favard's inequality:

7.20. Theorem. Let x(t) = (Xl(t), ... ,Xk(t)), where the components Xj(t) are nonnegative concave functions defined on [a, b] such that b

i = b

~a

f

Xj(t) dt

for

j = 1, ... , k.

(7.49)

a

r.

If f: I ~ ~ (I = [0, p 2i s; p, and x(t) E I for t function with increasing increments, then

f

E

[a, b]) is a continuous

2i

1 2i

o

f(y,···, y) dy ~ b _1 af(x(t)) dt.

(7.50)

214

7. Related Inequalities

If the components xlt) are nonnegative convex functions with xj(a) = the reverse of the inequality in (7.50) holds.

o(j = 1, ... , k), then

An important generalization of Favard's inequality is given by Berwald (1947):

7.21. Theorem.

Let f(x) be a nonnegative, continuous concave function, not identically zero on [a, b], and X(t) be a continuous and strictly monotonic function on [0, Yo], where Yo is sufficiently large. If i is the unique positive root of the equation z

~

f

b

X(y) dy

=b

~a

o

then for every function we have

f

(7.51)

X(f(x» dx,

a

1>: [0, Yo]

~IR

which is convex with respect to X,

~o J 1 > ( Y ) d Y 2 : b1>(f(x»dx. ~af

(7.52)

a

7.22. Remark. A generalization of Berwald's inequality for concave functions of several variables is given in Borell (1973b). 0 The following related results (Theorems 7.23 and 7.24) were given by Popoviciu (1939):

7.23. Theorem.

Let 1>:[a, b ] ~ 1 R be convex and f:[O, 1 ] ~ 1 Rbe continuous, increasing, and convex such that a ~f(x) ~ b for x E [0, 1]. Then 1

f

1>(f(x» dx ~

b

+ a - 2c b_ a

f b

2(c - a) 1>(a) + (b _ a)2

o

1

1>(x) dx,

C

=

f

f(x) dx.

o

a

(7.53) If 1> is strictly convex, then the equality in (7.53) holds iff xf(x)=a+(b-a)

A+ Ix - AI

2(1-A)

,

where

A= b +a -2c. b- a

(7.54)

7.2. Related Inequalities

215

7.24. Theorem. Let ¢: [a, b] ~ ~ be continuous and convex, and let f: [0, 1] ~ ~ be convex of order 1, ... , n + 1 such that a :s; f(x) :s; b for x E [0, 1]. Then 1

1

J¢(f(x)) dx:s; J ¢ ( ~ ( A x)), dx o

0

tor

J'

ja

+b

U - 1)a

+b

- - <: C <: -'-'-----'--j +1 - j ,

2:s; j:s; n,

(7.55)

and 1

J¢(f(x))dx:S;V(c)

for

o

na +b a:S; c :s;-n+1 '

(7.56)

where ja +

b) + (U - l)aj + b-t)).

~ ( t , x ) = a + j U((+t1 )j+1

X X

1- 1

(7.57) and b

¢(x)dx _ b + na - (n + 1)x () (n+1)(x-a)f V(x) b-a ¢ a + n(b_a)(n+l)/n (x_a)(n-l)/n' a

(7.58) If ¢ is strictly convex, then equality in (7.55) holds iff f(x) =

(7.59)

~ ( c ,x);

and equality in (7.57) holds if f(x)=a+(b-a)

X (

A+ Ix - AI)n

2(1-A)

,

where

A= b

+ na - (n + 1)c. b-a (7.60)

All of the results stated above may be regarded as converses of Jensen's inequality. Thus Popoviciu used Theorem 7.23 to obtain a bound for c'" - ¢(c",) where c'" = f6 ¢(f(x)) dx. More general results are given by Pecaric (1986b). For example, Pecaric proved 7.25. Theorem.

(a) Let ¢ and f be defined as in Theorem 7.23, and let

g : J2 ~ ~ be increasing in its first component, where J is an interval such

216

7. Related Inequalities

that ¢(x) E J for all x

E

[a, b]. Then b

g(cep, ¢(c» s:

max

xE[a.(a+b)/2]

b + a - 2x g( b ¢(a) a

-

2(x-a)f ) + (b )2 ¢(x) dx, ¢(x) . - a

a

(7.61) (b) Under the same conditions as in (a) except that g is decreasing in its first component, we have g(cep, ¢(c» ~

. nun

(b +b a - 2x ¢(a) + 2(x - a) f ) g (b )2 ¢(x) dx, ¢(x) . b

xE[a.(a+b)/2]

a

-

- a

a

(7.61') Proof.

(a) By (7.53) and the increasing property of g(', y), we have b

b + a - 2c 2(c-a)f ) g(cep,¢(c»s:g ( b-a ¢(a)+(b-af ¢(x)dx;¢(c) -

s:

max

xE[a.(a+b)/2]

g(

b

+a b

-

a

f

)

b

2x

a

2(x - a) ¢(a) + (b )2 - a

¢(x) dx; ¢(x) ,

a

since c E [a, (a + b)/2]. (b) Similar. D If g(x, y) =x - y in Theorem 7.25, then from (7.61) we obtain the noted Popoviciu's result. Also, if we let X(y) = y' and ¢(y) = yS for O
7.26. Theorem. Iff is a nonnegative concave function on [a, b], then for o< r < s we have b

[ ~ ~ ~f a

11

(f(x»S- . Ss:

b

[ ~ ~ ~f

11

(f(x», - . '.

(7.62)

a

7.27. Remark. For s;::: p ;::: 1 and r = 1, we obtain the inequality of Favard, which is a consequence of (7.45). D

7.2. Related Inequalities

217

Thunsdorff (1932) proved the following similar result: 7.28. Theorem. Iff is a nonnegative, convex function with f(a) ifO
= 0,

and

7.29. Remark. Theorems 7.26 and 7.28 provoked a strong interest among mathematicians, and thus there exist a number of results related to these inequalities. A new proof of Theorem 7.28 is given in Nishiura and Schnitzer (1971). The weighted version of Theorem 7.26 is given in Pecaric (1983c) (which also contains a related result for concave sequences). The weighted version of Theorem 7.28, i.e., the related generalization for k-convex functions, is given in Stankovic (1975, 1976). Analogues of Theorem 7.26 for convex sequences are proved in Rahmail (1978) and Vasic and Milovanovic (1977), and a generalization for k-convex sequences can be found in Milovanovic and Milovanovic (1983,1985). Although this problem has a long history, it is not well-known that the first result of this type was given by Gauss (1821). Let F: IR ~[0, 1] be a probability distribution function, a E IR, and let

f 00

v, = vr. a =

Ix - air dF(x) for

r ;:::: 0

(7.63)

denote the rth absolute moment of F about a. Let us consider the probability functions P(x)

= Pr(IX - al ;::::x),

Q(x) = 1- P(x)

= Pr(IX -

al <x).

If F is continuous, then we have

Q(x) = F(a +x) - F(a -x),

x ;::::0,

from which it follows that

f 00

v, =

x r dQ(x).

(7.64)

o

Note that Q(O) = 0, Q(x) = 1, and Q is increasing on [0,00). Inequalities involving the absolute moments have a long and rich history dating (at least) back to Laplace (1810) and Gauss (1821). For example, one classic result is (7.65)

218

7. Related Inequalities

If Q' is continuous and decreasing on (0,00), then we have the

Gauss-Winckler inequality (7.66) which is an improvement of (7.65). Winckler (1866) obtained (7.66) using an argument which was later found invalid, and the first correct proof is due to Faber (1922). A special case of (7.66) with n = 2 and r = 4 is (7.67) which was stated without proof by Gauss (1821, Art. 10); a proof of (7.67) can be found in Krafft (1937). For other proofs of (7.66), see Bernstein and Krafft (1914), and Fujiwara (1926). 0 We also note that Theorems 7.26 and 7.28, as well as the GaussWinckler Inequality, are related to a well-known result of Marshall, Olkin, and Proschan (1967) for the monotonicity of ratio of means, and their result was proved by using the theory of majorization. Note that this result was previously proved in Izumi, Kobayashi, and Takahashi (1934) and later given in Sunouchi (1938). A simple proof of their result with weights is given by Vasic and Milovanovic (1977), and it can also be proved using a generalization of a majorization theorem (Pecaric, 1984b). Moreover, by using an idea in Vasic and Milovanovic's paper a more general result can be obtained:

7.30. Theorem. For k = 1, ... ,r let Uk: [ak> b k]- ~ be increasing du;(x;), A = {x = (Xl' ... ,Xr ) :ak $,Xk $, bi; 1 $, functions, du(x) = k $, r}, and let f, g : X - ~ be positive real-valued functions. Then for rEI (I is an interval in IR) there exists a function F(r) defined by

m=l

F(r)

= (f

r

(f(x)Ydu(x)/

A

F(O) = exp(

f (g(x)YdU(X)f

for

r *0,

A

f 10g(f(x)1g(x)) du(x)/ f dU(X)). A

A

Furthermore, when g and fig are similarly ordered on A, then F(r) is increasing on ~ ; when g and fig are oppositely ordered on A, then F(r) is decreasing on ~ .

7.2. Related Inequalities

219

Proof. The proof given here depends on Cebysev's inequality and the well-known theorem concerning the power means H(r) defined for positive functions h given by (f p(x)(h(x))' du(x)/ f p(x) dU(X))lIr H(r)

=

A

{

for

,,#0,

for

r=O.

A

exp(f p(x)log h(x) du(x)/ f p(x) dU(x)) A

A

The function H is increasing and continuous on ~ (we assume that I = ~ ) . Now for convenience we rewrite Cebysev's inequality in the form (f p(x) du(x)) (f p(x)F(x)G(x) dU(X)) A

A

s;

(f p(x)F(x) du(x) )(f p(x)G(x) du(x)) , A

(7.68)

A

provided that F, G are oppositely ordered on A, with the inequality reversed if F, G are similarly ordered. For s s; r (s 0, r 0), take p(x) = (f(x))" F(x) = (g(x)lf(x))" and G(x) = (g(x))s-r in (7.68) to obtain

*" *"

(f (f(x))' du(x))(f (g(x))S dU(X)) A

A

2:

(f (g(x))' du(x))(f (f(x)Y(g(x))s-rdU(X)), A

(7.69)

A

«:

provided that (g If)" are similarly ordered, or the inequality in are oppositely ordered. If (7.69) reversed, say (7.69'), when (gln r , r > 0, it follows from (7.69) that

«:

{f (f(x)Y(g(x)y-rdu(x)/ f (g(x)Ydu(x)}lIr A

A

-s

{f (f(x))' du(x)/ A

f (g(x))' dU(X)}lIr

for

s s; r

(7.70)

A

holds when g and fig are similarly ordered. Similarly, if r < 0, then (7.70) follows from (7.69') when fig and II g are oppositely ordered, i.e.,

220

7. Related Inequalities

when g and we have

f /g are similarly ordered. By the increasing property of

H,

(f p(x)(h(x»S du(x)/ f p(x) du(x)fS A

A

::; (f p(x)(h(x»' du(x) / f p(x) du(x)f ' A

for s::; r (s, r:#: 0).

A

(7.71)

In (7.71) we then let p = gS and h = fg- 1 to obtain

{f (f(x»S du(x)/ f (g(x»S du(x)} A

lis

A

::; {f (f(x)Y(g(x)Y-' du(x)/ f (g(x)Y du(x» A

A

r'

for

s s: r.

(7.72) By (7.70) and (7.72) we have F(s)::; F(r) for s s; r (s, r:#: 0) whenever g and f /g are similarly ordered. Suppose now that s = 0 < r so that by the power mean inequality exp(f p(x)log hex) du(x)/ A

f

p(x) du(x»)

A

::; (f p(x)(h(x»' du(x)/ f p(x) dU(x)f' A

holds. By letting p == 1 and h exp(f log(f(x)/g(x» du(x)/ A

(7.73)

A

= f /g in (7.73) we obtain

f

dU(X»)

A

-s

(f (f(x)Y(g(x»-' du(x)/ f du(x)f'· A

A

When g and f /g are similarly ordered, so are f".g r and gr. Hence by Cebysev's inequality with m = 2 we have -r

f (f(x)Y(g(x»-' du(x) J(g(x»' du(x)::; Jdu(x) J(f(x»' du(x). A

A

A

A

(7.74)

7.2. Related Inequalities

221

Combining with the preceding inequality yields F(O) ~ F(r). Now, in the case r < 0, the inequality in (7.73) is reversed (and this reverse inequality is denoted by (7.73'», and again we take p = 1, h = fig. When g andf Ig are similarly ordered, so also are gr andj'g-r, thus (7.74) still holds, and, from r
7.32. Theorem.

if the function f(Q(x»Ix is decreasing, then the reverse inequality in (7.75) is valid.

Proof. Since x andf(Q(x»lx are similarly ordered, by Theoem 7.30 the function a

F(r)

a

U

= (!f(Q(x»'dQ(x)/!xrdQ(X») o

is increasing. Thus holds. D

r

for

r>O

0

for

n ~ r

we have

F(r):2=F(n),

and

(7.75)

222

7. Related Inequalities

7.33. Examples. (a) For f(t) == 1, we have (7.65). (b) For f(t) == t, i.e., in the case when Q(t)/t is increasing, we have (7.66). If Q is a starshaped function, i.e., if Q(t)/t is decreasing, we have the reverse of the inequality in (7.66). (c) If f(t) = t llk (k > 0), i.e., if Q(x)/xk is decreasing, then

(r+k k v, )lIr - (n +k k ) lin <:

Vn

for

n::::;r;

(7.76)

if Q(x)/xk is increasing, then the reverse of the inequality in (7.76) 0 holds. We note that for the rth absolute moment given in (7.63) (or (7.64)), it is known that the function fer) = log v, is convex (see Theorem 13.8 and (14.13)). Furthermore, in the case when Q' is continuous and decreasing on (0,00), the function j.fr) = log((r + l)vr ) is also convex. Beesack (1984) implicitly proved the following generalization of these results: 7.34. Theorem. If Q is a probability distribution function with Q(x) = 0 for x::::; 0, Q(O) = 0, Q(oo) = 1, and if (_I)k-lQ(k) is positive, continuous, and decreasing on (0, 00) for k = 1, 2, ... , N, then

is a convex function of r for k = 1, ... , N. In the following we give some corollaries which follow from this result and some known results concerning convex functions (see Mitrinovic and Pecaric, 1986): 7.35. Corollary. (7.78)

7.36. Remark. Corollary 7.35 is the corrected version of a result Beesack (1984). o

In

7.2. Related Inequalities

7.37. Corollary.

223

For m:5 n z; r we have (7.79)

7.38. Remark. Corollary 7.37 is a generalization of the well-known Lyapunov Inequality. 0

7.39. Corollary.

7.40. Remark. (1927). 0

For m :5 nand r :5 s we have

Corollary 7.39 is a generalization of Izumi's inequality

Similarly, we can give a generalization of Narumi's inequality (1927), i.e., for I defined in (7.77) wehave

q-t t-p s-t t-r --A(p)+--A(q) 2::.-lk(r) +-lk(S) q-p q-p s-r s-r for

p

< r < s < q,

p:5 t s; q.

We also note that in their papers Favard (1933) and Berwald (1947) considered similar inequalities with several positive concave functions. Results of this type are extensively treated in the mathematical literature. In this connection we consider the following expression b

H(fl' ...

b

.t.: g) = [(f g(x) Jljj(X) dX)/

(f g(x) dX)]/

a

a

II (f b

a

b

g(X)jj(XYidx/

f

g(x) dx

f

Pi ,

a

where g, t., ... .I; are nonnegative integrable functions, and Pl , ... , P« positive real numbers. In the case g(x) == 1, we write Htf.; .L«: g) as H(ft, ,1m); and for Pi = ... = P« we write H(fl, .I«: g) as T(ft, ,In). In the literature H is well-known as Holder's ratio, and T as Cebysev's ratio.

224

7. Related Inequalities

7.41. Theorem. (a) Let j;:[a, b ] ~ 1 (Ri = 1, ... , n) concave functions. Then the following results are valid: (i) For Pi

~ 1 (i

= 1, ...

, n) we have

~(n ~ 1)! ( [ ~ } ) ( [ n 1; }) [II (Pk + 1)lI

Pk

H(fl' ... ,fn) (ii) For 0 < Pi

be nonnegative

~ 1 (i

= 1, ...

, n) we have 1

•

(7.81)

n

H(fl' ... , f n ) ~ -IT- (Pk+1)lIPk. n +1 k ~ l

(7.82)

(b) Let j ; : [ a , b ] ~ 1 be R nonnegative convex functions with j;(a)= o(i = 1, ... ,n). Then, for 0
7.42. Remarks. (a) Inequality (7.81) was proved by Godunova and Levin (1972), and it is an improvement and a generalization of results in Nehari(1968) (see also Bellman, 1956). Inequality (7.82) is proved in Pecaric (1983c) and Rahmail (1972a), and inequality (7.83) in Vasic and Pecaric (1980b). (7.81) and (7.82) are generalizations of results in Barnes (1969). The weighted version of (7.82) and (7.83) are given in Pecaric (1983c) and Vasic and Pecaric (1980b), where in the latter a related generalization for k-convex functions is also given. The weighted version of (7.81) for n = 2 is given in Pecaric (1983c). (b) For PI

= ... = Pn = 1,

(7.82) yields Favard's inequality (1933):

T(fl' .. '

2n , f n ) ~ - ,

n+1

(7.84)

when t.. .. . .I: are nonnegative concave functions, and (7.83) yields Anderson's inequality (1958) (see also Mitrinovic, 1970, p. 306):

T(fl' .. '

2n , f n ) ~ -

n

+1

(7.85)

when fl> ... .I; are nonnegative convex functions. The weighted version of (7.84) is given in Vasic and Pecaric (1981c, 1982a), and the weighted version of (7.85) in Vasic (1972). For a generalization of (7.85) (i.e., of a result in Vasic, 1972), see Stankovic (1974, 1976) (for the case of k -convex functions).

7.2. Related Inequalities

225

(c) For discrete versions of previous results (convex, concave, and k-convex sequences) and some similar results see Barnes (1970,1984), Pecaric (1982e, 1982f, 1983c, 1984b), Vasic and Djordjevic (1973), Vasic and Pecaric (1980b, 1981c, 1982a), Milovanovic and Milovanovic (1982a, 1982b, 1983, 1985), Rahmail (1978), Wang (1978a), Stankovic and Milovanovic (1980), Vasic, Stankovic, and Pecaric (1985a), Borell (1973b), and Beesack and Pecaric (1985b). 0 It should be noted that inequalities (7.82) and (7.83) and their weighted versions are simple consequences of the following result (Pecaric, 1982f and Beesack and Pecaric, 1985b):

Let g be a nonnegative integrable function on [a, b] and be integrable positive functions on [a, b] such that they are all similarly ordered. Furthermore, let O
t.. hi (i = 1, ... , n)

H(ft, ... ,f;. ; g)

2:.

Hilt«, ... , hn ; g).

(7.86)

If they are oppositely ordered, then the reverse ofthe inequality in (7.86) is valid. A discrete analogue of Theorem 7.43 is that 7.44. Theorem. Let p, aj, b j (j = 1, ... , m) be nonnegative n-tuples such that aj , bj (j = 1, ... , m) are all similarly ordered, bji > 0, and 0< qi :s 1 (i = 1, ... , n). If b, and a)bj (= (ajr/b jl, ... , ajn/b jn)) are also similarly ordered, then

H(aj, ... ,am; p)

2:.

H(br, ... ,bm ; p),

where

If b, and aJbj are oppositely ordered, then the reverse of the inequality holds. 7.45. Remark. For the special case in which hl(x) = ... = hn(x) == 1 and b i = ... = b., = (1, ... ,1), both Theorems 7.43 and 7.44 yield Cebysev's inequality. In fact, Godunova and Levin (1972) proved (7.81)

226

7. Related Inequalities

as a special case of a more general result obtained by using the Bellman (1956) method. An additional generalization of his result is given in Mitrinovic and Pecaric (1987b). 0 Next we consider the nonnegative kernel K(x, y) (a::; x::; b, a::; y ::; b) such that, for some P (given as Pi in Theorem 7.46), we have

f b

0<

K(x, yfdx <00.

a

We shall say that the function u belongs to the class U(K) if it admits the presentation

f b

u(x) =

K(x, t)v(t) dt,

a

where v is an integrable nonnegative function (v(t) *- 0).

7.46. Theorem.

Let Ui(X) E U(Ki), i nonnegative integrable function. Let

= 1, ... ,n,

and g: [a,

b ] ~IR

be a

b

f g(X>C~l

ui(x») dx

T(Ul(X), ... , un(x);g(x»=

a

b

.

i ~ l (f g(x )Ufi(X) dX)

lip;

a

(a) If Pi 2= 1 (i

= 1, ... , n),

then

T(Ul(X), ... , un(x); g(x»

2=

C(K),

(7.87)

where C(K) = inf T(K1(x, Yl), ... , Kn(x, Yn); g(x».

(7.88)

Yi

(b) If Pi::; 1 (o#oO)(i (7.88) holds, where

= 1, ... ,n), then the reverse of the equality in

C(K) = sup T(K1(x, Yl), ... , Kn(x, Yn); g(x». Yi

Furthermore, the constants (7.88) and (7.89) are the best possible.

(7.89)

7.2. Related Inequalities

Let P; > 1 (i

Proof.

227

= 1, ... ,n). Then

b

b

b

1 Y;)V;(Y;»)dY1 ... dy; dx Jg(x > C ~ u;(x») 1 dx J ... J g(x > C ~ K;(x, a

a

a

b

b

b

;91 (J g(X)(U;(X)Y'idX) ;9 (J g(x)(u;(x)Y'i- J K;(x, y;)v;(y;) dy, dx) 1

1

a

a

a

It! 12 2= C(K)13/14

=

(n (J

b

= C(K)/

Pi, g(X)(Ui(X)Y'idX) l-lI

a

where b

11 =

b

J .. -J T(K

1(x,

a

01 (J n

X

lIPi

b

Vi(Yi»)dYI, ... , dYn,

nJ

b

J

(Vi(Yi) g(x)Ki(x, Yi)(U;(X)Y'i-1 dx) dy.,

a

b

13 =

) ( K ; ~ Y;) , Y'i dX)

g(x

a

b

12 =

Yl), ... , Kn(x, Yn);g(X»

a

b

(n

a b

11

J ... J (J g(x)(K;(x, Y;)Y'idX) PiVi(Yi)dYI... dYn, a

a

a

and n 14=]1

Jb Vi(Y;)(Jb g(X)(K;(X,y;)Y'idx)lIPi(Jbg(X)(U;(x)Y'idx)(pi-1)/pi dy., a

a

a

Therefore (7.87) is valid. For Pi < 1 (i = 1, ... , n), the reverse of the inequalities is valid (with C(K) given in (7.89». A similar proof applies 0 for Pi = 1 or by letting Pi - 1 in the above proof. 7.47. Remarks. (a) Inequality (7.87) for p(x) == 1 is proved in Godunova and Levin (1972). (b) Let o for a:Sx:Sy K 1=K(x,y)= {1 for

y

<x :sb.

228

7. Related Inequalities

Then x

U(X) =

Jv(y) dy

E

U(K1),

o

where U(K 1 ) is the class of nonnegative increasing functions with u(O) = O. The last condition can be easily eliminated, and in this case Godunova and Levin (1972) obtained the Cebysev inequality for n nonnegative increasing functions. Note that we can obtain the weighted version of this result from Theorem 7.46 by a similar argument. When the two functions are monotonic in opposite directions, the reverse inequality can also be so obtained. (c) Theorem 7.46 can also be applied to obtain other important results. For example, some results in Barnes (1969) are simple consequences of Theorem 7.46. (d) Some related results are given in Atanackovic (1981) and Barnes (1982, 1984). 0

Chapter 8

8.1.

Hardy's, Hilbert's, Opial's, Young's, Nanson's, and Related Inequalities

Hardy's, Hilbert's, Opial's, and Related Inequalities

In this section we give some useful inequalities for convex functions which imply well-known results as special cases. First, we observe the following theorem due to Boas (1970b):

8.1. Theorem. Let ep be a continuous convex function. Let f be measurable and nonnegative, A be increasing and bounded, and L = A(oo) - A(O). Then

f 00

f 00

x- 1ep{ L -1

o

f 00

~

f(ux) dA(u)} dx

0

x- 1ep(f(x» dx,

(8.1)

0

If ep is a continuous concave function, then the reverse of the inequality in (8.1) holds. Proof.

If

ep is convex,

then by Jensen's inequality we have

00

ep{ L -1

00

Jf(ux) dA(U)} ~ L Jep(f(ux» dA(u), -1

o

0

229

230

8. Special Related Inequalities

i.e., we have 00

00

JX-I{ L Jf(ux) dA(U)}dx

~ L -1 JJx-

-1

o

0

0

= L -1

dA(U)dx

1(f(ux»

dx dA(U)

0

JJxo

=L- 1

1(f(ux»

0

JJto

1(f(t»dtdA(U)

(t=ux)

0

00

=

Jt-

1(f(t»

dt.

0

o

8.2. Remark. A special case of Theorem 8.1 is the well-known Hardy Inequality. Indeed, let (x) =x P , a> 0, f3 > 0, and a - 1U '" for A1(U) = { 1 afor

O ~ u ~ l , /1, 2':

(thus L 1 = a-I),

1,

o

O ~ U~ 1,

for for

Az
U2':1.

Then (8.1) becomes x

00

J xor

1-cxp(J

o

00

~ «» J x-1(f(x))P dx,

t"'-lf(t) dtf dx

0

00

J

00

(Jt-fl-y(t) dtf dx

x- 1+ flP

o

(8.2)

0 00

~

Jx-

f3-P

x

1(f(x))P

dx

(8.3)

0

for either p > 1 or p < O. If 0 < P < 1, then the reverse of the inequalities in (8.2) and (8.3) is valid. If we let f(t) = t 1-"'g(t) and a = (k - l)/p in (8.2) andf(t) = t 1 + flg(t) and f3 = (1- k)/p in (8.3), we have x

00

Jx-k(J g(t) dtf dx o

1fJ 00

~ (k ~

0

xp-k(g(x))p dx

(8.4)

xp-k(g(x))p dx

(8.5)

0

when either k> 1, P > 1 or k < 1, P < 0; and 00

00

Jx-k(f g(t) dtr dx ~ (1 o

x

oc

~ kr f 0

8.1. Hardy's, Hilbert's, Opial's and Related Inequalities

231

when either kl or k>l, pl, the reverse of the inequality in (8.4) holds; and for 0 < P < 1, k < 1, we have

f

x

00

f 00

X-k(fget) dty dx >:

(1

~ kY

xp-k(g(x))P dx.

0 0 0

(8.6)

o

Godunova (1965) (see, for example, Bullen, Mitrinovic, and Vasic, 1987, pp. 271-73) proved a result by using a similar method:

8.3. Theorem. Let F: ~+ ~ ~ be continuous and strictly increasing with limx->o+ F(x) = 0 or -00, and F- 1 be convex. Let {ad and {bk} be two positive real sequences such that for every n e: 1, w(n) = { W r ) } k ~ isl a positive n-tuple satisfying n

2:

w ~ n )= 1

and

k=l

2:

w ~ n ) b n :C 5

for

k e: 1.

n=k

If F',. is a quasi-arithmetic mean (for definition, see Bullen, Mitrinovic, and Vasic, 1987, p. 215), then 00

2:

n=l

00

bnF',.({ad,w(n»):5C

2: an;

(8.7)

n ~ l

if in addition 1 n C = lim bk> n-+ oo n k=l

2:

then the constant in (8.7) is the best possible.

8.4. Remark. For F(x) = !ogx, w ~ n =) s.ln, and b; = 1/(k + 1), we have 1

2: - n=l n + 1 00

Since e- 1 < (n!)lInl(n

(

n

n!

IT ak k=l

) lin

<

2: an' n=l 00

+ 1), this implies Carleman's inequality 00

2:

k=l

00

Gk({ak})<e

2: ak>

k=l

where G; ( {ak}) = Va 1 • • • an' Of course, many other examples can be found in Godunova (1965) and Bullen, Mitrinovic, and Vasic (1987, pp. 272-73). Godunova also gave many integral analogues of the results. 0

232

8. Special Related Inequalities

In her proof Godunova used the well-known inequality for quasiarithmetic means. Since this result is a simple consequence of Jensen's inequality, by using Jensen's inequality we can have more general results. This fact is noted in Vasic and Pecaric (1982c), where the following results are given: 8.5. Theorem. Let f: 1-->; ~ be a convex function, Xi E I (i = 1, 2, ... ), {cd be a positive sequence, and for every n ~ 1 let qn = ( q ~ , ... , q ~ ) be a positive n-tuple such that ~ 1 : ~ 1ql: = 1 (n ~ 1). If ~ cnql:::=; d k

for

k ~ 1,

(8.8)

n ~ k

then (8.9)

If f is a concave function and the reverse of the inequality in (8.8) holds, then the reverse of the inequality in (8.9) holds.

Proof.

By Jensen's inequality we have

Thus

co

cc

= ~ f(xd ~ cnql:::=; k ~ l

n ~ k

~

~ dd(xd·

0

k=l

8.6. Theorem. If in Theorem 8.5 we replace I by a convex set U in ~ m and the points Xi (i = 1, 2, ... ) by points in U, then the conclusion remains valid.

Proof.

The proof is similar to the one-dimensional case.

0

,

8.1. Hardy's, Hilbert's, Opial's and Related Inequalities

Let us consider the two quasi-arithmetic means (see Mitrinovic, and Vasic, 1987, pp. 215-82) given by Kn({ak}, p)

=

233

Bullen,

K - l ( ~ n ~ 1PkK(ak)),

Ln({bd, p) = L

- l ( ~ n ~ lPkL(bk)),

Then the following result holds:

8.7. Coronary. Let f : [k 1 , k 2 ] x [ e1 , e2] ~ ~+ be a real-valued function, {ad, {bd, and {cd be positive sequences, and assume that, for every l = 1. n 2: 1, qn = ( q ~ , ... , q ~ ) is a positive n-tuple such that ~ Z = qZ (a) If H(s, t) then

=

f(K-1(s), L -l(t)) is a convex function and (8.8) holds,

00

00

2: cnf(Kn({ak}, qn), Ln({bd, qn)) L dnf(a n, b n). $

n ~ l

(8.10)

n ~ l

(b) If H(s, t) is concave and the reverse of the inequality in (8.8) holds, then the inequality in (8.10) is reversed. Proof. This follows from Theorem 8.6 by letting m = 2, f(s, t) = H(s, t), s, = K(aJ, and t, = L(bJ for i = 1, 2, . . . . 0 8.8. Remark. In the previous result we assume that all sums are finite. Of course, we can use other generalizations of Jensen's inequality (e.g., a generalization of Theorem 8.1) and related inequalities to obtain similar results. For example, in Vasic and Pecaric (1982c) the Jensen-Petrovic Inequality is used, and Imoru (1977) contains a generalized Hardy's 0 inequality.

Similarly, Godunova (1967b) proved the following result:

8.9. Theorem. Let K(t) 2: 0 be defined on ~ = {t = (tl , ... , tn): 0 < t, < 00, i = 1, ... , n} with f v, K(t) d ~= 1, and let V" and Vy be defined similarly. Let (u) be a nonnegative convex function for u 2: 0 and f be such that f(y) 2: 0 for y E Vy , f $ 0, and (f(x))/(x) ... x n) is integrable

234

on

8. Special Related Inequalities

v,..

Then

By using this result Godunova obtained many general inequalities, which include (i) Hardy's and Knopp's inequalities (Hardy, Littlewood, and P6lya, 1934, 1952, p. 250): x

00

00

J e x p ( J~ logf(t) dt) dx < e Jf(x) dx, o

0

0

which follows from Theorem 8.9 by letting n = 1, (jJ(u) = e" and K(t) = {01

for for

0 ~ t ~ 1, t> 1;

(ii) Hilbert's inequality:

by letting n = 1, (jJ(u) = u", and K(t) = [sin(.n/p)/.n][r and (iii) Hardy-Littlewood-P6Iya inequality:

1(1 m ~ ~ ~ ~

y} dy

f

dx <

(p ~If

1

(f(x)Ydx

llP

for

/ (l

+ t)];

p> 1,

0 0 0

by letting n

= 1,

(jJ(u)

= u", and K(t) = (p

-1)/(p2t llp max{l, t}).

In the following we give some results from Mitrinovic and Pecaric (1988c). We say that a function u : [a, b ] ~ ~ belongs to the class U( v, K) if it admits the representation

f b

u(x) =

K(x, t)v(t) dt,

(8.12)

8.1. Hardy's, Hilbert's, Opial's and Related Ineqnalities

235

where v is a continuous function and K is an arbitrary nonnegative kernel such that v(x) > 0 implies u(x) > 0 for every x E [a, b]. We also assume that all integrals under consideration exist and are finite. First we prove the following theorem.: 8.10. Theorem. Let u, E U(v" K) (i = 1, 2), where vz(t) > 0 for every t E [a, b], r(t) ~0 for every t E [a, b], and ¢(u) is convex and increasing for u ~ O. Then

(8.13) holds, where b

() J r(t)K(t, x) d s (x ) - Vz x ( t. Uz

a

Proof.

(8.14)

t)

Using Jensen's inequality for the convex function ¢ we have

b Ja r(x)¢

(I

1 Jb , v 1(t) uz(x) K(x, t)vz(t) vz(t) dt

I)

a b

b

::; J r(x)¢(J K(x, t)vz(t) Iv 1(t) I dt) dx Uz(x) vz(t) a

a

b

b

= f ¢(jV1(t) !)vz(t)(f r(x)K(x, t) dX) dt vz(t) Uz(x) a

a

b

=fS(t)¢(I~:~:~I)dt.

0

a

8.11. Remark. If for a> 0 we let K(x, t)

= { ~ X- t)a-l/f(a)

for for

t ::; x

t >x,

(8.15)

then v is the derivative of order a of u in the sense of RiemannLiouville, and from Theorem 8.1 we obtain Theorem 8.3 from Godunova and Levin (1969) (see also Rozanova, 1976a). 0

236

8. Special Related Inequalities

Note that Theorem 8.1 can be generalized for convex functions of several variables. For example, the following result is valid:

8.12. Theorem. Let u, E U(vi , K) (i = 1, 2, 3), where vz(x) >0 and r(x) 2: 0 for every x E [a, b], and 4>(u, v) is convex and increasing for u, v 2: O. Then b

b

J r ( x ) 4 > ( I ~ : i ; ~ I ,1 ~ : i ; ~ I ) d XJ~ S ( X ) 4 > ( I ~ : i ; ~ I 1, ~ : i ; ~ I ) d X ' (8.16)

a

a

where s(x) is given in (8.14).

Let U1(v, K) denote the class of all functions u E U(v, k) such that K(x, t) = 0 for t > x. Note that if U E U1(x, K), then we have b

u(x) =

JK(x, t)v(t) dt.

(8.12')

a

Let u, E U1(Vi , K), vz(x) > 0, and r(x) 2: 0 for every x E [a, b]. Further, let 4>(u) and f(u) be convex and increasing for u 2: 0 and f(O) = O. Iffis a differentiable function and max K(x, t) = M, then

8.13. Theorem.

b

M

.

J vz(x)4>(I ~ : ~ !~ )f'(u z(X)4>(I~ : i ; ~I)) dx

a

b

~ f ( MJ v z ( t ) 4 > ( I ~ : i : ~ I ) d t .(8.17) a

Proof. Since f' is an increasing function, by using (8.12') for the function Ul and the well-known inequality for the absolute value of a function, we have

b

~M

x

)4>( Ivz(x) v1(x) 1)f'(uz(X)4>(J K(x, t)vz(t) Iv1(t) I dt)) dx. uz(x) vz(t)

J vz(x a

a

8.1. Hardy's, Hilbert's, Opial's and Related Inequalities

237

Now using Jensen's inequality for the convex function ep and the condition K(x, t) :s: M, we have x

b

V z C X ) e p ( I ~ ~ i ; ~ l ) fK(x, ' ( f t ) V z C t ) e p ( I ~ : i dt) : ~ 1 )dx

I:s:M f a

a x

b

:s: f

M V 2 ( X ) e p ( I ~ ~ i ; ~ I ) fM' V( fz C t ) e p ( I ~ : ~ : dt) ~ I ) dx

a

a b

= f(M f V2(t)ep( I

~ : ~ : ~1) dt.

0

a

8.14. Remark. Note that for Riemann-Liouville's derivative we have For a=l, we have M=l, and we obtain Theorem 1 in Rozanova (1972a). Therefore, Theorem 8.13 is a further generalization of Theorem 2 in Godunova and Levin (1967). 0 M=(b-a)a-1jr(a).

In the following we give another generalization of Theorem 2 in Godunova and Levin (1967). 8.15. Theorem. Let ep: [0, (0) ~ ~ be a differentiable function such that for q> 1 the function ep(x1/q ) is convex and ep(O) = O. Let u E U1(v, K) where ( f ~(K(x, t)Y' dt) lIP :S: M, P -1 + = 1. Then

«'

b

b

f lu(x)1

1- qep'(!u(x)l)

Iv(xW dx:s:

1/

~ q
(8.18)

a

a

If the function
Proof.

Using Holder's inequality we have x

lu(x)l:S: f K(x, t) Iv(t)1 dt s; a

x

1/X

(f (K(x, t)Y' dt) (f Iv(tW dt) P

a

a

1/

q

238

8. Special Related Inequalities

To prove (8.18), let z(x) = f ~ Iv(tW dt, so that z'(x) = Iv(xW and lu(x)l:5 M(Z(X»l/q. Further, from the convexity of fj>(x llq) it follows that the function x llqfj>'(x) is increasing. Thus, we have b

q f lu(x)11- fj>'(lu(x)l) Iv(xWdx a b

:5 f M 1-q(z(X»lIq-lfj>'(M(z(X»lIq)Z'(X) dx a b

q) q) = :q f fj>'(M(Z(X»lI d(M(z(x»lI a

o a

8.16. Remark. Similar results can be given for the class of functions U2(v, K) in which K(x, t) = 0 for t <x. If u E U2(v, K), then we have b

u(x) = f K(x, t)v(t) dt.

(8.12")

o

x

8.17. Corollary. Let fj>, q, and p be defined as in Theorem 8.15. If uCn-1)EAc[a,b) and either uCk)(a)=O for k=0,1, ... ,n-1 or uCk)(b) = 0 for k = 0, 1, ... , n - 1, then b

f lu(x)1

b 1-qfj>'(lu(x)1)

11

luCn)(xWdX:5 :q fj>( M(f luCn)(tW dt) ' )

a

a

where M = (b - at- lIq/«n -I)! (np - p + l)lI p ) . For some related results, see Rozanova (1972a, 1976a, 1976b). The following theorem is due to Rozanova (1972b). 8.18. Theorem. Let y(x) be an absolutely continuous function on [0, a), y(O) = 0; r(x) be increasing, and r(O) = O. Let fj>(w) and F(w) be convex

8.2. Young's Inequality

239

and increasing functions for w > 0, F(O) = 0, Q(w) be a convex and increasing function, and 'ljJ(w) be an increasing function with 'ljJ(0) = 0. If F'(z(x»z'(x )'ljJ(11 z '(x)') ~(F(z(a »1z(a »'ljJ' (x 1z(a»,

(8.19)

where z(x) = H r'(t)¢(Iy'(t)llr'(t» dt, then

f a

F'(r(x)¢ )(1 y(x)llr(x»G(r' (x )¢(I y' (x )llr'(x» dx

o a

~ H ( r'(x)¢(ly'(x)llr'(x» J dX),

(8.20)

o

where G(w) = wQ('ljJ(l/w», H(w) = F(w)Q('ljJ(alw». Further, the equality in (8.20) holds iff y(x) =Ax, rex) = Bx, and 'ljJ(w) = CF(aw), where A, B, C and a are constants.

If in (8.20) we let y(x) = f(x), f'(x) > 0, f(O) = 0, ¢(w) = w, F(w) = 1P(w) = w 2, and Q(w) = VI + w, we obtain the following result (see Rozanova, 1972b): Let a and b be given positive real numbers, and let f be a real-valued function such that f(O) = 0, f(a) = b, f(x) ?::. 0, and f(x )/1' (x) ~x on the interval [0, a]. Then

8.19. Remark.

f(a)

= b,

f a

2

f(x)(l + (f'(X»2)112 dx

~b(a 2 + b 2)1I2,

(8.21)

o

and equality holds iff f(x) = (bla)x. Inequality (8.21) is given in P61ya (1947) with a stronger condition that f"(x)?::.O instead o f f ( x ) / f ' ( x ) ~ x . 0 Rozanova (1972b) also gave some other examples which give generalizations of Opial's inequality.

8.2.

Young's Inequality

The following result is known as Young's inequality: 8.20. Theorem. Let f be a real-valued, continuous, and strictly increasing function on an interval 1= [0, c] (c > 0) such that f(O) = 0, and let

240

8. Special Related Inequalities

a

b

ab::::; ff(X)dX+ f g(y)dy forall o

a,b::::;c,

(8.22)

0

= f(a).

and equality holds iff b

This inequality was proved by Young (1912) with the additional condition that f be differentiable, and another proof is given in McShane (1947, pp. 131-32). Proofs of Theorem 8.20 under the present conditions are given in Diaz and Metcalf (1970), Bullen (1970b), and Nieto (1974). For a geometric interpretation of the result in Theorem 8.20, consider Figures 8.1 and 8.2 given below. The area of the curvilinear triangle OAP is given by SU(x) dx, and the area of the curvilinear triangle ORB is given by S ~g(x) dx. Thus the inequality in (8.22) is justified. On the other hand, Young's inequality is related to an integral representation of convex functions (see Theorem 1.2(a)). Namely, let f: [0, 00) - [0, 00) be a continuous and increasing function such that f(O)=O and f(x)_oo as x_ oo. Then exists and has the same properties as f. Further, if we let

r:

f

f

o

o

y

x

F(x)

= f(s) ds and F*(y) =

f-1(t) dt,

y

B(O,b)

x A(a,O)

C(X,O)

Figure 8.1.

8.2. Young’s Inequality

241

Y

A

Figure 8.2.

then F and F* are both convex functions on [ o , ~ ) Thus . the following results are valid (Roberts and Varberg, 1973, pp. 29-30):

+ +

(i) xy 5 F ( x ) F * ( y ) for all x 2 0 and y 2 0 (Young’s inequality), (ii) xy = F ( x ) F * ( y ) iffy = f ( x ) = F ’ ( x ) , (iii) (F*)’ = (F’)-’, (iv) (F*)* = F, and ( 9 F * ( Y ) = SUPX20 ( X Y -f(x>). Note that property (v) is used for defining a conjugate function: If f :I+ R is a convex function defined on an interval I, then f * :I*+ R denotes the conjugate function given by f * ( y ) = supxEI(xy - f ( x ) ) with domain I*= { y E (w :f*(y) < m}. Some properties of conjugate functions are given in Roberts and Varberg (1973, pp. 28-36). In a similar fashion we can define a conjugate function corresponding to a convex function f of several variables (see PSeniEnyi, 1980, p. 64):

f*(Y)=sup ((x, Y> -f(x)). This function is also convex with the property f ( x ) = ( f * ) * ( x ) . In this case Young’s inequality is also valid, i.e., we have

f(x) +f*(Y)

2

(x, Y>.

A function f is called an N-function (Krasnosel’skii and Ruticii, 1958 and 1961) if it admits to the representation M ( u ) = J g ’ p ( t )dt where the

242

8. Special Related Inequalities

function p is continuous from the right for t ~ 0, increasing, positive for t > 0, and such that p(O) = 0, limHoop(t) = 00. The class of such p functions will be denoted by '!fl. Let p E '!fl. For a function q defined by q(s) = supp(t)"Ss t for s ~ 0, we say that it is the right-inverse function of p. It can be easily verified that q has the same properties as p. We now introduce a concept of complementary N-functions: Let M be an N-function and p E '!fl be its right-derivative. Then the N-function v, N(v) of the form N(v) = fb q(s) ds, where q is the right-inverse function of p, is called the complementary N-function to function M. M and N are mutually complementary N-functions. For these functions Young's inequality is also valid, i.e., the following results are valid (Krasnoselskii and Rutickii, 1958 and 1961): (i) If M and N are mutually complementary N-functions, then for every u, v E ~ Young's inequality is valid, i.e.,

uv ::s;M(u) + N(v).

(8.23)

(ii) Equality in (8.23) holds iff either

v=p(u)

or

u=q(v)

for

(8.24)

u , v ~ O .

Consequently we have

up(u)

= M(u) + N(p(u»

and

vq(v) = M(q(v» + N(v) for

u, v

~ O.

(8.25)

(iii) If for a given N-function M the inequality in (8.23) holds for all u, v ~0, then the function N is the complementary N-function of the function M. Now let f be an increasing function on an interval I, let (1' = inf{f(x):x E I}, 13 = sup{f(x):x E I}, and let J = «(1', 13) (or [(1', 13), «(1', 13], or [(1', 13] if the values of (1', 13 are attainable). A function g with domain J is called a pseudo-inverse of f if for each Y E J we have

XL(Y)== sup{x :f(x) < y} ::s; g(y) ::s; inf{x:f(x) > y} == XR(Y)' Cunningham and Grossman (1971) proved an extension of Young's inequality for the case a > O. For general a the following result holds: Let f be an increasing function on an interval I containing the points x = and x = a (where a> 0 or a < 0). Let g be a pseudo-inverse of f with domain J. If f(O) = 0 and b e J, then (8.22) holds, and equality holds iff f(a_)::s; b ::S;f(a+).

°

8.2. Young's Inequality

243

Boas and Marcus (1974b) proved the following theorem, which is equivalent to the above result: 8.21. Theorem. If f is an increasing function on an interval I containing the points x = c and x = d, and g is a pseudo-inverse off, then d

f ~ )

cf(c) + ff(x)dx-=:=dt+

f g(y)dy,

e

(8.26)

t

d

t

df(d)+ f g(y)dy-=:=ct+ f f(x)dx fed)

(8.27)

e

for all t in the domain of g. Furthermore, equality in (8.26) holds iff t is between f(d-) and f(d+), and equality in (8.27) holds iff t is between f(c-) and f(c+). If f is decreasing instead of increasing, then the inequalities in (8.26) and (8.27) are reversed. In that case the conditions for the equalities to hold remain unchanged. Proof. Suppose f is increasing. Let f(c) = A, and define F by F(x) == f(x + c) - A. Then F is increasing in I} = {x:x + c E I} with F(O) = O. Furthermore, letting I denote the domain of g, the function G defined by G(y) = g(y + A) - c for y E I} = {y :y + A E I} is seen to be a pseudo-inverse of F. Thus, for arbitrary d E I and t E I we have (d - c) E I} and (t - A) E I}. By the result of Cunningham and Grossman (1971), it follows that d-e

t-A

(d - c)(t - A)::; f

F(x) dx + f

o

o

G(y) dy,

which implies d

dt - cA::; f f(x) dx e

t

+ f g(y) dy. A

Since A = f(c), this is equivalent to (8.26). Moreover, the equality holds iff F«d-c)_)::;t-A::;F«d-c)+) holds; that is, iff f(L)::;t::;f(d+) holds. The inequality in (8.27) is equivalent to that in (8.26), because interchanging c and d in (8.26) yields (8.27), and vice versa. Finally, the proof for the case in which f is decreasing follows in the same fashion by

244

8. Special Related Inequalities

a similar application of a result of Cunningham and Grossman 0 (1971).

8.22. Remark. A result which is analogous to Theorem 8.21 can be 0 found in Milicevic (1975). Merkle (1974) noted that the following converse of Young's inequality is valid: 8.23. Theorem. Let f be a continuous and strictly increasing function on an interval I which contains x = 0 such that f(O) = 0, and let g = f-l. Then b

a

Jf(x) dx + J g(y) dy s: max{af(a), bf(b)} o

holds for every a

(8.28)

0

E

I and b Ef(I).

8.24. Remark. Similar results can be obtained for other forms of Young's inequality. For example, from (8.25) we obtain M(u) + N(v):5 up(u)

for

p(u)

~v

and M(u) + N(v):5 vq(v) for

Thus for every u, v

~0

p(u):5 v

i.e.,

u

z;

q(v).

we have

M(u) + N(v):5 max{up(u), vq(v)}.

o

The following result is given in Beesack, Mitrinovic, and Vasic (1980): 8.25. Theorem. Let f be continuous and strictly increasing on an interval I containing x = 0 such that f(O) = O. Let g be a continuous function with domain J = f(I) such that (8.22) holds for a E I and b E J, where equality l • holds for b = f(a). Then g =

r

Proof.

For an arbitrary but fixed a

f

E

a

ep(b) =

o

I let

f b

f(x)dx+

0

g(y)dy-ab.

8.2. Young's Inequality

245

Then <jJ(f(a» =0, f(b)?O for all bEl, and <jJ'(b)=g(b)-a exists for all s «: It follows that <jJ'(f(a» =0; thus g(f(a))=a for all aEI. Suppose that g(b) =1= f-\b) for some b e J. Let a = f-\b) so that b = f(a); then a = g(f(a» = g(b) =1= f-l(f(a» = a, a contradiction. Hence g(b)=f-l(b) for all s «: D 8.26. Corollary. Let f satisfy the conditions of Theorem 8.23, and g be a continuous function with domain 1 = f (I) such that g( y) ~f - \y) for all y E I. If (8.22) holds, then g = f-l.

8.27. Remark. In Takahashi (1932) it is assumed that g is continuous and strictly increasing with g(O) = 0 and g-l(X) ?f(x). This result is somewhat weaker than Corollary 8.26. Similarly, a result of Bullen (1970b) is weaker than Corollary 8.26. A similar result for N-functions is given in result (iii) of Krasnoselskii and Rutickii (1958, 1961). D 8.28. Theorem. conditions:

Let T: P - P be an operator satisfying the following

x y ~ T ( p ) ( x ) + T ( q ) for ( y )

p,qEP,X?O,

and

y?O;

xy= T(p)(x) + T(q)(y) if y=p(x),

(8.29) (8.30)

where pEP and q is its right -inverse function. Then

f x

T(p)(x) =

p(t)dt for

pEP

and

x?O.

(8.31)

o

8.29. Remark. Theorem 8.28 is proved in Lackovic (1974b). It is a minor generalization of a result in Hsu (1972). D There exist generalizations of Young's inequality which involve several functions. The following is given in Beesack, Mitrinovic, and Vasic (1980): 8.30. Theorem. Let f be continuous and strictly increasing on an interval I containing the points x = a and x = b, and let g be an increasing function on I. Then

f b

t g ( b ) - f ( a ) g ( a ) ~f(x)dg(x) +

f t

f(a)

g(f-l(y»dy fort Ef(I),

(8.32)

246

8. Special Related Inequalities

= f(b)

and equality holds iff either t

or g is a constant between band

r\t). Cooper (1927), Takahashi (1932), and Oppenheim (1927) also give generalizations of Young's inequality which involve several functions. The following result is due to Oppenheim: 8.31. Theorem. Let t.. ... ,fn be continuous, nonnegative, and strictly increasing on 1= [0, 00). If at least one of them takes the value zero at x = 0, then 'k

D1 fk(tk)

~ktl

f (I] i*k

o

for tk E I (k

= 1, ...

(8.33)

t(x») dA(x)

, n). Moreover, the equality holds iff t 1 = ...

°

Proof. Without loss of generality we may assume that Define the functions Fk (1 ~ k ~ nJ by Fk(x) = A(x) for Fk(x) = fk(tk) for x 2= tk. Then

t1 f (ll

t1 f (ll

4

~ •••

~x

~ tk

~tn' and

~

Fj(x) ) dFk(x) =

f

~

=

°

Since ~ Fj(x) ~ t ( x )and differences), we also have

°

o

Fj(x») dFk(x)

n

d

n

n

(J] Fj(x») = J] Fj(tn) = Jl t(ti)'

~ ~ F k ( X )~ ~ f k ( X h ) old

for all x, j, k (and all

f ( I T F j ( x » ) d F k ( Xf) ~ 4

o

~t1

°

= tn'

4

l*k

0

(ITt(x»)dfk(x) for l*k

1 ~ k ~ n ,(8.34)

and the inequality (8.33) follows. Furthermore, if t 1 = ... = t.: then clearly the equality in (8.34) holds, hence also that in (8.33). On the other hand, if ~ tk < tn for some k, then

°

IT Fj(x) = Fk(x) IT

i*n

i*n,k

Fj(x) < fk(X)

IT

i*n,k

t(x) =

IT t(x)

i*n

holds for tk < x ~ tn ; thus strict inequality must hold in (8.34) for k Consequently, strict inequality also holds in (8.33). D

= n.

8.3. Nanson's Inequality

8.3.

247

Nanson's Inequality

The following result is due to Nanson (1904):

8.32. Theorem.

If the real sequence { a k n : ~isl convex, then

al

+ a3 + ... + aZn+l:> az + a4 + ... + a Zn n+1 n

(8.35)

with equality iff {ak} is an arithmetic sequence.

Proof.

Since {ad is convex, we have ak - 2ak+l

+

a k + Z ~ Ofor

k

= 1, 2, ...

,2n-1.

(8.36)

By virtue of this fact, we conclude that k(n - k

+ 1)(azk-l -

2aZk + aZk+l) ~ 0

for

k

= 1, ... , n

and k(n - k)(aZk - 2aZk+l + aZk+Z) ~ O.

By adding these inequalities we obtain (8.35). Furthermore, equality in (8.35) holds iff equality in (8.36) holds for every k, which occurs iff {ak} is an arithmetic sequence. 0

8.33. Remarks. (a) Another proof of (8.35) is given in Adamovic and Pecaric (1989). (b) Steinig (1981) noted that the following extension and interpolation of (8.35) is equivalent to (8.35): Zn+l

2:

k=l

1 n 1 Zn+l 1 n ( - I ) k + l a k ~ - a Z k + l ~ - 2 - a k ~ - a Zk' n + 1 k=O n + 1 k=l n k=l

2:

2:

2:

(c) Let the sequence {ak} satisfy the conditions m s 6?an s M

for

n ~ 1.

(8.37)

Then the sequences {Cn}n;"l and {dn}n;"l given by en =a n -m(nZj2) and d; = M(n z/2 ) are convex. Thus the following results are valid (see Andrica, Rasa, and Toader, 1984): 2n+l 1 n In 2n+l - - m ::s-aZk+l - aZk ::s--M 6 n +1 k ~ O n k~l 6

2:

2:

248

8. Special Related Inequalities

and n(2n + 1) 6

Z ~ l()k+l

m ~L.

k=l

-1

1 ~ n(2n + 1) ak---1 L. a Z k + l ~ 6 M. n + k=O

(d) Stankovic (1976) gave the following generalization of (8.35): (n - 2p )(al + a3 + ... + aZn+l) + (2p - n - l)(az + a4 + ... + azn)

+ 2p(a l + aZn+l) - p(az + azn)

~

° for all

p ~ 0.

For p = 0, it reduces to Nanson's inequality. Note that this inequality yields the special case neal + a3 + ... + aZn+l) - (n + l)(az + a4 + ... + azn) ~ 2p(a3

+ ... + a Zn- l) - p(az + 2a4 + ... + 2azn- z + azn),

which is weaker than Nanson's inequality. This is so because it follows from

which in turn follows by adding the inequalities 2aZk+l

~ aZk

+ aZk+Z for k = 1, ... , n -

1.

(e) Adamovic and Pecaric (1989) proved the following generalization of Nanson's inequality: 1 n-m+l 1 n+l 2 2 aZk ~ - - 1 aZk-l for m e n - m + k =m n + k= 1

L

L

j{

and

2m ~ n.

However, this inequality is also weaker than (8.35) due to the inequality 1

n - 2m + 2

n-m+l

L k=m

1 n «» ~ -L aZk' n k=l

(8.38)

which will be proved as a consequence of Theorem 8.34 below. (f) Lackovic (1975) proved that the inequality

is valid for every convex sequence {ad iff the sequence {pd is given by Pk = constant (k = 1, ... , 2n + 1). 0

8.3. Nanson's Inequality

249

Adamovic and Pecaric (1989) proved:

8.34. Theorem. Let {akH be a convex sequence and In = {1, ... , n}, and let I, J, and M be nonempty subsets of In such that I and J are non-overlapping. Let I, J, and M have cardinal numbers a, f3, and y, respectively, and denote v = 2: i,

w

=

ieJ

2: i. ieM

If u

v

u

v

-<- and -<m
(8.39)

then the inequality yv 2: a, =5 ll'V -

iEM

f3w - yu 2: a, + ll'W 2: a, f3u iEI av - f3u iEJ

(8.40)

holds. If in addition to (8.39) the condition

w

u+v

y

a+f3

(8.41)

is also satisfied, then

-1 2: a, = 51- - 2: a

y iEM

+ f3 iEIUJ

a,

(8.42)

holds. If M and I U J are also non-overlapping, then the inequality 1 1 1

-y 2: a, =5 a + f3 + Y 2: iEM

iEIU}UM

a, = 5 - - 2: a, a + f3 iEIU}

(8.42')

holds. Furthermore, each of the inequalities (8.40), (8.42), and (8.42') becomes an equality iff {ak} is an arithmetic sequence on the set S = {min(I UJ), min(I UJ) + 1, ... , max(I UJ)}. Note that without the condition

u a

v

- =5 m =5 -

f3

for every n EM,

(8.39')

the conditions u] a < vi f3 and (8.41) do not imply the inequality in (8.42).

250

8. Special Related Inequalities

8.35. Corollary. Let {akH be a convex sequence. Then for m e N and 2m :::5 n - 1 the inequalities 1 n-m 1 n 1 (m ai:::5 a, :::5 a, + n - 2m i ~ m + 1 n i=1 2m i=1

2:

2:

2:

n) 2: a, i ~ n - m + l

(8.43)

hold, and both inequalities become equalities iff {ak} is an arithmetic sequence.

8.36. Remark. If we replace m - 1 by m and aZk by ak in the first inequality in (8.43), we obtain (8.38). 0 Next we observe the following definition: 8.37. Definition. A real sequence { a (p, q > 0) if Lpq(a n) ~0 for n ~ 1, where Lpq(a n) = an+z - (p

d ~is

said to be p, q-convex

+ q)an+l + pqa n·

8.38. Remark. In Milovanovic, Pecaric, and Toader (1985) it is shown that the theory of p, q-convex sequences plays an important role in the ~ by sequence { w n } given p n _qn W = P - 1 for p*"q n { np'"? for p = q. For example, the following result is proved: The sequence {an} satisfies the relation n = 1,2, ... (8.44) iff an = UWn + VWn+l' (8.45) where u and v are arbitrary real numbers.

0

Milovanovic, Pecaric, and Toader (1986) also proved the following generalization of Nanson's inequality: 8.39. Theorem.

If the real sequence {ak}f'+l is p, q-convex, then

(pqta 1 + (pqt-1a3

+ ... + aZn+l

...:..o.-..o...:...----=-_..:.::-::...:.-_--=--

(pqt-1az

+ (pqt- Za 4 + ... + aZn

~ ~ ~ - - - " - - - - - - ' ' ' - - ' ~ - - - ' - - - - - - - . . . = : . . :

(8.46) and equality holds iff {ad satisfies (8.44).

8.3. Nanson's Inequality

251

8.40. Remarks. (a) For s.; = Wl + ... + Wn we have Lpisn) = 1; thus if the real sequence {ad satisfies m

:s; L p q :s; M

for

n = 1, 2, ... ,

then the sequences {b n} and {en} given by

are p, q-convex. A generalization of the first inequality in Remark 8.33(c) can be given by using this fact (see Milovanovic, Pecaric, and Toader, 1986). (b) As shown in Mitrinovic (1970, pp. 205-6), Nanson's inequality for k- 1 ak =X gives an inequality of J. W. Wilson. However, the results given in this section can also be applied to yield results in Mitrinovic (1970, p. 198, (3.24» and in Mitrinovic (1965, p. 139, 2.3.1.4 and 2.3.1.5). All of these results are further generalized in Adamovic and Pecaric (1989), where the following result is proved: Let p and q be real numbers such that p >q.

(i) If either (1) q >

°

°

or (2) p > > q and p + q < 0, then

p

q

a + -1

p +q p- q

-P- -q> - - for O
°

-

a

(8.47)

(ii) If either (1)' p < or (2)' p > 0 > q and p + q > 0, then the reverse inequality in (8.47) holds. 0 Besides the generalizations of Wilson's inequality and other related results, this result represents an improvement of a result of D. Z. Djokovic (Mitrinovic, 1965, pp. 162-63, 2.3.2.8, or Mitrinovic, 1970, p. 276, 3.6.26). It also improves the inequality in Mitrinovic (1970, p. 279, 3.6.31). Furthermore, some other examples are given in Adamovic and Pecaric (1989).

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General Linear Inequalities for Convex Sequences and Functions

Chapter 9

9.1.

Inequalities for m-Couvex Sequeuces and Functions

The following theorem is given in Pecaric (1981e): 9.1. Theorem. Let p = (Pl' ... ,Pn) be a real n-tuple, where n > m. Then the inequality n

2: p.a, 2:: 0

(9.1)

i=1

holds for every m-conuex sequence

{ a i } i~ ff

n

2: (i -

i=1

1)kp i = 0 for

k

= 0, 1, ... , m -

1

(9.2)

+ 1, ... , n,

(9.3)

and n

2: (i -

k

+m -

1)(m-1)Pi 2:: 0 for

k =m

i ~ 1

where

P) = j(j -

1) ... (j - k

+ 1), /0) = 1.

Proof. The sequences a, = (i - 1)k and a, = -(i - 1)k (1 -<:; i -<:; n) are m-convex for every k = 0, 1, ... , m - 1; thus (9.1) becomes (9.2), and the condition (9.2) is necessary. On the other hand, the sequence

o

a i = {(i - k

for

+m -

1)(m-1) for

1 -<:; i -<:; k - 1 k -<:; i -<:; n

is convex for every k = m + 1, ... ,n. Thus by (9.1), condition (9.3) is also necessary. 253

254

9. General Linear Inequalities

That the conditions are also sufficient follows directly from the identity rn-I

n

1

1

n

~p.a, = ~ ok! /j.ka l ~(i -1yk)Pi + (m -1)! X

k ~ ~ + 1 ( ~(i - k + m _1)(rn-I)Pi)/j.rn ak_rn;

(9.4)

this identity represents a generalization of the well-known Abel Identity given by (9.5) which can be easily proved by mathematical induction. 9.2. Remarks.

o

(a) It is clear that conditions (9.2) can be replaced by n

2.: (i -

1)(k)Pi =

i=1

°

for

k = 0, 1, ... , m - 1.

(9.2')

(b) In Theorem 9.1 the result for the case in which m = 1 and Pi is replaced by Pi - qi can be found in Marshall and Olkin (1979, p. 445). The corresponding integral analogue is also given in Marshall and Olkin (1979, p. 444) and Mitrinovic (1970, pp. 114-15), and it was pointed out that this result is due to Steffensen. (c) For m = 2, the result in Theorem 9.1 is given in Karlin and Studden (1966, p. 421). Moreover, Karlin and Studden (1966, p. 422) also gives the following interesting special case: Let p be an n-tuple i 0, (ii) I : ? ~ Iipi = 0; and let { a ; } ~be satisfying the conditions (i) I : ? ~ I P = such that

then for every convex sequence { a i } ~t he inequality in (9.1) is valid. (d) The identity in (9.4) can be expressed as (see Milovanovic and Pecaric, 1986): n

rn-I

.=1

k

2.: p.a, = 2.: X

1 n-k 1 k' Vkan_k (n - i)kpi + ( -1)' ~ O ' . ~ 1 m.

: ~ ~ ( ~(k -

2.: i

+ m _1)(rn-l)Pi)V rna k'

(9.6)

9.1. Inequalities for m-Convex Sequences and Functions

255

Thus the following result is a modification of Theorem 9.1: Let p be a real n-tuple (n > m). Then the inequality n

2 : P i a i ~ O i=l

holds for every V-convex sequence of order m iff n

2: (n - i)kpi = 0

for

k = 0, 1, ... , m - 1

i=l

and k

2: (k - i + m _1)(m-I)Pi ~0

for

k

= 1, ... , n - m.

i=l

(e) In Pecaric, Toader, and Milovanovic (1986) it was shown that, by using (9.4), Taylor's formula for sequences can be given by

(f) The following identities are consequences of (9.4) and (9.6): (9.8)

(9.9)

For Pi =

(:)i we can obtain (from (9.9)) the following identity given in

Drazin (1955):

256

9. General Linear Inequalities

By using this identity, Drazin proved the following inequalities (see also Mitrinovic, 1970, p. 348): (i) If (-1

r-

s

~n-sa s

inequality, then (ii) If

2: 0 (s = 0, 1, ... , n) with at least one strict

I:7=o

~ n - s a 2: s 0

(s

inequality, then

C)iai > 0 for y > -1.

= 0, 1, ...

,n)

with

I:7=o ( ~ ) i a i > 0 for y <

at

least

one

strict

-1.

We can obtain some similar results by using (9.8) for Pi

=

C)yn-i, i.e.,

(g) Let us denote by Km(n) the class of all m-convex sequences, i.e.,

and similarly we denote

Then Theorem 9.1 yields the following result (Toader, Pecaric, and Milovanovic, 1985): p belongs to K:'(n) iff P i = v m q i = ( - I ) m ~ for m q i = 1, ... , n, where

=o for qi{2:0 for

i = 1, ... ,m and i =m + 1, ... , n.

i =n

+ 1, ... , n + m,

Of course, this result is related to the well-known Minkowski-Farkas' lemma (see, for example, Tchernikov, 1968). D

In Pecaric, Mesihovic, Milovanovic, and Stojanovic (1986) it was noted that identities (9.4) and (9.6) can be used for proving the following generalization of Theorem 9.1:

i

9.1. Inequalities for m-Convex Sequences and Functions

9.3. Theorem.

257

Let {Pi}]' be a real sequence (n > m 2: 1). Then

(i) Inequality (9.1) holds for every sequence {aJ 7' that is convex of orders j,j + 1, ... , m (j E {I, ... , m}) iff n

2: (i -

i=l

l)(klp i = 0 for

k

= 0, 1, ... ,j -

1,

(9.10)

n

2: (i -

l)(klp i 2: 0 for

k

= j, j + 1, ... , m - 1,

(9.11)

i ~ l

and n

2: (i -

k

i=k

+ m _1)(m-1 lp i 2:0

where a(kl = a(a - 1) ... (a - k

for

+ 1),

k= m

+ 1, ...

,n,

(9.12)

a(Ol = 1.

(ii) Inequality (9.1) holds for every sequence {aJ7' that is V-convex of orders j, j + 1, ... , m ti E {I, ... , m}) iff n

2: (n -

i=l

i)(klp i = 0 for

k

= 0, 1, ...

,j - 1,

(9.13)

n

2: (n -

i=l

i)(klp i 2: 0 for

k

= j, j + 1, ... , m - 1,

(9.14)

n

2: (k -

i

+m -

1)(m-1lp i 2: 0 for

k

= 1, ... , n - m.

(9.15)

i ~ l

The following definition, given in Toader (1985b), concerns starshaped sequences: 9.4. Definition. A sequence {a i}7' is said to be starshaped of order m if the sequence {(ai+l - a 1)/i}7' is convex of order m-1. The following theorem is proved in Pecaric and Agovic (1988) (see also Pecaric, Toader, and Milovanovic, 1986):

258

9. General Linear Inequalities

9.5. Theorem.

Inequality (9.1) is valid for every sequence starshaped of order j, ... , m iff the sequence { P i } satisfies ~

2)Pi = 0

n

n

(.

~ k(i -1) ~ ~2

LPi=O; ;=1

2)Pi

n

i ~ (i -1) ( .~ ~2

L (i i=k n

1)

2= 0

(i + m -_ k - 2)Pi m

k

for

2

2= 0

is

k = 2, ... ,j;

for

= j + 1, ... for

{ a i } ~t hat

k=m

,m;

and

+ 1, ... , n.

*

Let P 0 be a real number. We define the operator Lp by (see Lackovic and Kocic, 1979b): Lp(a n) = an+l - pan

for n E.N,

where .N is the set of all positive integers. For a given sequence {aJ ~ , we say that it is p-monotone or that it belongs to the class Kp , if the inequality Lp(a n) 2= 0 holds for all n E.N. By the well-known Abel Identity, we then have (9.16) Thus from (9.16) we can easily obtain the following theorem (Pecaric, 1981d):

9.6. Theorem.

Let

{ p J b ~e

(a) Inequality (9.1) 'L.?=lpi-lPi = 0 and

an arbitrary real sequence. Then

holds for

every sequence

{ a J ~

in

K p iff

n

L pi-kPi

2= 0

for

k

= 2, ...

, n.

i=k

(b) Inequality (9.3) holds for every sequence a l 2= 0 iff

{ a J ~in

Kp such that

n

L pi-kPi 2= 0

for

k = 1, ... , n.

i=k

9.7. Remark.

Note that

Lpq(ak) = Lp(Lq(ak» = Lq(Lp(ak» = ak+2 - (p

+ q)ak+l + pqak

9.1. Inequalities for m-Convex Sequences and Functions

259

holds for every k = 1, ... , n - 2; and that the inequality Lpq(ak) 2: 0 for k = 1, ... , n - 2 defines p, q-convex sequences (see Definition 8.37). We D shall denote this class by K pq . Note that necessary and sufficient conditions for (9.1) to hold for all sequences in K pq are given in Milovanovic and Kocic (1989). Instead of the inequality in (9.1), Popoviciu (1959b) considered an inequality of a bilinear form, and proved the following result: 9.8. Theorem. E?=r E7=s x ij ·

Let Xjj (1:5 i, j:5 n) be real

numbers and X r •s =

(a) For all sequences a=={aJ7 and b=={b j}7 monotonic in the same direction, n

F(a, b) ==

n

2: 2: Xijaibj 2: 0

(9.17)

i=l j=l

holds iff X r •s X r • l = 0 for

2:

0 for

r = 1, ... ,n

r, s = 2, ... , n, and

X l •s = 0 for

(9.18) s =2, ... , n.

If a and b are monotonic in opposite directions, then the inequality in (9.17) is reversed. (b) For all nonnegative increasing sequences a and b, F(a, b) 2: 0 holds iffX r •s 2: 0 (r, s = 1, ... , n).

9.9. Remark. As noted in Pecaric (1979a) and Kovacec (1983), the condition Xl,s = 0 is missing in Popoviciu (1959b) and Mitrinovic (1970, p. 38). Pecaric's proof uses a generalization of Abel's identity given in (9.5), D and Kovacec's proof uses Theorem 9.1, with m = 1. A similar result of multilinear form is given in Pecaric (1979a), and a new proof of that result is given in Kovacec (1984). This result states: 9.10. Theorem. and

Let Xi, , ... , x im(1 :5 ik :5 n, 1:5 k :5 m) be real numbers Sl

XS''''Sm =

2: ...

i}=l

s,«

~ Xi,"'Xim ' i",=1

260

9. General Linear Inequalities

Then for all nonnegative decreasing n-tuples aj= {aj1"'" m), the inequality

ajn} ( l ~ j ~

n

2:

F(a1' ... , am) =

Xi,'" Xim a1i, ... amim ~

;t,···,i m = l

holds iff X

s ,

• • • Sm

~

°

for

1

~ Sj ~ n,

° (9.19) (9.20)

l ~ j ~ m .

9.11. Remarks. (a) In Pecaric (1979a) integral analogues of Theorems 9.8 and 9.10 are also given. (b) Generalizations of Theorems 9.8 and 9.10 for p-monotone sequences are given in Pecaric (1981d). (c) Necessary and sufficient conditions for the validity of inequality (9.17) for convex sequences a and b are given in Pecaric (1983d). A generalization of this result for p, q-convex sequences is given in Milovanovic, Pecaric, Stojanovic, and Mesihovic (1988). Analogous results for convex and V-convex sequences of order m and n are given in Pecaric, Mesihovic, Milovanovic, and Stojanovic (1986), and for starshaped sequences of higher order in Pecaric and Agovic (1988). 0

The following generalization of Theorem 9.8 (and many other related results) is given in Pecaric, Mesihovic, and Milovanovic (1989):

9.U. Theorem.

Let

Xij, aij

(i = 1, ... , N;j = 1, ... ,M)

be

real

numbers.

(a) The inequality N

F(a) ==

M

2: 2: Xijaij

~

i=l j=l

°

(9.21)

holds for every sequence {aij} that is (n, m )-convex iff N

M

r=l

s=l

N

M

r=l

s=j

N

M

r=i

s=1

2: 2: (r -l)(i)(s -l)(j)x

rs

2: 2: (r -

l)(i)(s - j

=

°

for

i = 0, ... , n -l;j

+ m - 1)(m-1)xrs =

°

= 0, ...

for

i = 0, ... ,n -l;j = m

2: 2: (r -

i

+n -

1)(n-1)(s - l)(j)x rs =

, m -1,

+ 1, ...

,M,

° for

i=n

+ 1, ... , N;j = 0, ... , m -1,

9.1. Inequalities for m·Convex Sequences and Functions

261

and N

M

2: 2: (r -

i

+n

- 1)(n-1)(s - j

+m

- 1)(m-l)xrs

2:

° for

r=i s=j

i= n

+ 1, ...

, N, j = m

+ 1, ...

, M.

(b) The inequality in (9.21) holds for every sequence {a ij } that is convex of order (u, v) (u =0, ... , n; v =0, ... , m) iff N

M

r=1

s=1

N

M

r=1

s=j

N

M

2: 2: (r-1)(i)(s-1)(j)x 2: 2: (r -

1)(i)(s - j

+m

rs2:0

for

i=0, ... ,n-1;j=0, ... ,m-1,

_1)(m-1)xrs

2:

° for

i = 0, ... , n -1;j

2: 2: (r -

i + n - 1)(n-1)(s - 1)(j)xrs

2:

= m + 1, ...

, M,

° for

r=i 5=1

i

= n + 1, ...

, N;j

= 0, ... , m -1,

and N

M

r=i

s=j

2: 2: (r -

i + n - 1)(n-1)(s - j

+m

- 1)(m-l)xrs i=n

2:

° for

+ 1, ... , N, j = m + 1, ... , M.

9.13. Remark. Theorem 9.1 may be formulated more generally as the following: Let S be a vector space of all sequences, X be a topological vector space, and Pc X be a closed, convex cone in X. The following result is given in Toader (1987b) (see also Lackovic and Kocic, 1984): Let A: S ~X be a continuous linear operator. In order that A(x) E ~ for every x E K m it is necessary and sufficient that (i) A( {ed) = for k =0,1, ... , m -1 and (ii) A({ed) E P holds for k 2:m, where {ed is the sequence with components eki =

(m- 1 + 1i - k) m _

and

° (n) = ° for j

n <j. Another generalization of Theorem 9.1 is given in Toader (1988a). 0 Of course, instead of inequality (9.1) we can also consider similar results for convex functions. First we shall give a result of Popoviciu (1940a) (see also Popoviciu, 1944, p. 35) which contains Theorem 9.1 as a special case. Popoviciu proved this result by using an identity which is a generalization of (9.6).

262

9. General Linear Inequalities

9.14. Theorem. Let f : I ~IR be a discrete n-conuex function and p be a real n-tuple (p 1= 0). Let Xi E I (i = 1, ... , m) and Xl < X2 < .. ·Xm. Then m

2: pJ(x;)

(9.22)

~ O

;=1

holds iff m

2: Pi x 7= 0

for

k = 0, 1, ... , n - 1

i=l

and r

2: Pi(Xi -

Xr+l)(Xi - Xr+2) •.. (Xi - x r+ n- l) :::::;

0 for

r

= 1, ... , m - n.

i ~ 1

Popoviciu (1940a, 1940b) also proved the following theorem (see also Popoviciu, 1944, pp. 35-36 and Mitrinovic, 1970, p. 160):

9.15. Theorem. The inequality (9.22) holds for every 1= 0 (i = 1, ... , m) and every n-conuex function f iff

Xl:::::;·· . :::::; Xm ,

Pi

m

2:

P r X ~= 0

holds for

k

= 0, 1, ... , n -

1

r=1

and k

- 2: Pr(x r -

m

t)n-1

=

r=l

holds for t E

(Xb xk+d

2:

p.i»; -

rr-

l

~0

r ~ k + 1

and k

= 1, ... , m - n.

9.16. Remark. For convex functions we have the following result (Pecaric, 1989f): Inequality (9.22) holds for all m-tuples x and p and all convex functions f iff Pm = 0 and m

2: Pi IXi -

xkl

~0

for

k

= 1, ... , m.

0

i ~ 1

9.2.

Some Generalizations and Refinements

The following results are some generalizations of Theorem 9.15 (see Vasic and Lackovic, 1978 and Lackovic and Vasic, 1979).

9.2. Some Generalizations and Refinements

Let the norm of a function f

E

263

C[a, b] be defined in the usual way:

Ilfll = as/sb max If(t)l· For a sequence of function Un}, fn E C[a, b], we say that the sequence of functions converges (uniformly) to the function f E C[ a, b] if limn--->oc IIfn - fll = O. Let us assume that D c ~ , and let S(D) be one of the normed subspaces of the space of all real functions defined on D, where the norm of a function f E S(D) is denoted by Ilflll' We consider operators A of the following form A:C[a, b ] ~ S ( D and ) , say that A is continuous if from the condition Ilfn - fll ~0 (as n ~(0) it follows that IIAfn - Aflll ~0 as n ~00. Also, we write Af?:. 0 if g(t) = Af?:. 0 holds for every tED, where f is a given function in the space C[a, b]. The set of all functions which are convex of order n and continuous on [a, b] (continuous from the right at a and continuous from the left at b) will be denoted by Kn[a, b]. Clearly we have Kn[a, b] c C[a, b]. We shall consider the classes K n [ a, b] for n e: 2, and define for i = 0,1,2, ....

ei(t) = ti for

a

~t

~b.

(9.23)

Furthermore, for t, C E [a, b] we define the function wn+l(t, c) by t - C

Wn+l(t,C)=

(

+ It - CI)n 2

9.17. Theorem. Assume that A: C[a, tinuous operator. Then

f

E

= (t-c)':-.

b ] ~S(D)

(9.24)

is a linear and con-

Kn[a, b]:? Af?:. 0

(9.25)

holds for every function f iff Aei = 0 for

i = 0, 1, ... , n - 1 and

Awn(t, c)?:. 0 for every Proof.

c E [a, b].

(9.26) (9.27)

(a) If: Let us denote m

Fm(x) = Pn(x) + L cjwn(x, Xj)'

(9.28)

j ~ 1

where Pn(x) E fIn (the set of all polynomials of degree at most n), and assume thatf E Kn[a, b]. Then by Theorem (1.44) there exists a sequence

264

9. General Linear Inequalities

the form (9.28) with Cj 2: 0 (j=1, ... ,m) such that the functions wn(t, c) are of the form (9.24) and { F m ( x ) }of ~

lim IlFm(x) - f(x)11 m---+ oo

= o.

(9.29)

By virtue of (9.28) and the linearity of the operator A, it follows that m

AFm(x) = APn(x) + m

2: cjAwn(x, Xj)

j=l

m

= 2: ajAe/x) + 2: cjAwn(x, Xj)' j ~ O

j=l

(9.30)

Since (9.26) and (9.27) are valid where the e/s and wn's are given in (9.23) and (9.24), respectively, by virtue of (9.29) we obtain m

AFm(x)=2:c jAwn(x,xj)2:0 forevery j=l

m=1,2, ....

(9.31)

By using the continuity property of the operator A in (9.30) and (9.31), we find that Af

= A ( ~ ~ Fm(X)) o o = E ~ o Ao Fm(x) 2:0,

which implies (9.25). (b) Only if: Suppose that the implication in (9.25) is valid for an arbitrary function f E Kn[a, b]. By a direct verification it follows that the functions ej and -ej defined in (9.23) are in the class Kn[a, b]. Thus by (9.25) we have Aej 2: 0 and A(-ej) 2: 0, and (9.26) is satisfied. In the same fashion we conclude that wn+1(t, c) E Kn[a, b]. Using (9.25) one more time, we obtain (9.27). D We say that the operator A:C[a, b] x C[a, b ] ~ S ( D is) bilinear if the operator Bu = A(u, v) is linear with respect to u for every function v E C[a, b] and if the operator Cv = A(u, v) is linear with respect to v for every function u E C[a, b]. From Theorem 9.17 we can obtain the following theorem. 9.18. Theorem. Let the operator A: C[a, b] x C[a, b ] ~SeD) be bilinear and continuous. Then for every pair offunctions (t, g),

9.2. Some Generalizations and Refinements

265

is valid iff (i) A(ei,ej)=O for O::5i, j::5n-1, (ii) A(ei,wn(t,C»= A(wn(t, c), ej) = 0 for every c E [a, b] and every i, j = 0, 1, ... , n - 1, and (iii) A(wn(t, Cl), wn(t, C2»"? 0 for every (c 1 , C2) E [a, b] x [a, b].

9.19. Remarks. (a) For n = 2, instead of w2 (t, c) = (t - c)+, we can let W2(t, c) = oc(t) = It - c] (see Vasic and Lackovic, 1978). (b) Vasic and Lackovic's papers were published in 1978, and earlier Bojanic and Roulier (1974) proved the following general result: Let A: C[a, b ] ~X be a continuous linear operator. Then AU) E P holds for every f E Kn[a, b] (n "? 2) iff we have (i) A(p) = 0 for every p E IIn - 1 (the set of all polynomials of degree at most n - 1), and (ii) A(wn(t, c» E P for every c E (a, b), where X and P are defined as in Remark 9.13. (c) Theorems 2.24 and 2.26 are simple consequences of Theorem 9.17. (d) Vasic and Lackovic (1978, 1979) give some majorization-type theorems which follow by replacing A with A - B in Theorem 9.17. (e) In Kocic and Lackovic (1986) and Kocic (1984) the reverse of the implication in (9.25) is considered. For that result we need the concept of one-sided strong local maximum (OSLM) of real-valued functions: A function rjJ E C(/) has a OSLM at the point Xo E 1 (an interval) if there exists an h > 0 such that for every x E (xo - h, Xo + h) £; 1 we have rjJ(x) ::5 rjJ(xo), and rjJ(x) < rjJ(xo) holds at least in one of the intervals (xo - h, xo) or (xo, Xo + h). We denote by C(/) the set of all functions in C(/) that have a OSLM in at least one point in 1. Now we can give a result of Kocic and Lackovic (1986): Let {A;J be a i family of linear operators such that A),: C ( / ) S ~ (D) and e;(t) = t (i = 0, 1). If (i) A),eo = 0 for every AE A, where A denotes the index set which is at least countable, (ii) A),e1 = 0 for AE A, and (iii) for every rjJ E C(I) there exists at least one Ao E A and Yo E D such that AAo(rjJ, Yo) < 0, then

A),"? 0 for every AE A

f

~

E

K(I) for every f

E

C(/),

where K(I) is the set of all convex functions on I. As a special case, they gave a linear criterion of convexity: Let {A),} be a family of continuous ~ (D) where 1 is a finite interval. If the linear operators A),: C ( / ) S previous conditions (i)-(iii) and (iv) A),0c"? 0 for every C E I, AE A are satisfied, then

Ad"? 0 for A E A¢:}f E K(I) holds. The function o; in (iv) can be replaced by W2(t, c).

266

9. General Linear Inequalities

(f) A result similar to Theorem 9.17 for starshaped sequences is given in Milovanovic, Stojanovic, and Kocic (1986). 0 The following theorem is proved in Brunk (1964):

9.20. Theorem. Let I be an interval in ~ k ; X(t) = (XI(t), ... ,Xk(t» be a vector of functions where the X;'s (1:5 i:5 k) are increasing and continuous from the right on [a, b). Let H be continuous from the left and of bounded variation on [a, b), with H(a) =0. If H(b)=O, fra,b)H(u) dX(u) =0, and

f

H ( u ) d X ( u ) ~for[a,t]c[a,b], O

[a,t)

then

f

~0

f(X(t» dH(t)

(9.32)

[a,b) holds for every continuous function f: I ~ ~ with increasing increments, where JH dX = (f H dXI, . . . , f H dXk)· The conclusion also holds when [a, t] is replaced by [a, t). Now let dp, denote a signed measure on (a, b) such that f ~ Idp,1 < 00. Such a measure possesses a decomposition du = dP,1 - dp,zwhere dP,1 and dp,z are finite nonnegative measures on (a, b). We restrict our attention to the measures du such that for each (jJ E C(uo, u l , ••• , un) the integral f ~ (jJ du is well-defined, with infinite values permitted. Specifically, if (jJ+(t) = max{ (jJ(t), O} and (jJ-(t) = (jJ+(t) - (jJ(t), then we can write b

b

b

f (jJ du = f (jJ+ a

dP,1

a

+ f (jJ- dp,z-

b

b

(f (jJ- dP,1 + f (jJ+

a

a

dP,z).

a

That f ~ (jJ du is well-defined means that at lest one of the sums b

f (jJ+ a

b

dP,1

b

f

f

a

a

+ (jJ- du., and

b (jJ- dP,1

+ f 4>+ dp,z a

is finite. The dual cone of C(uo, u l , . . • , un), denoted by C*(Uo, UI , ... ,un), is the set of signed measures du on (a, b) which

9.2. Some Generalizations and Refinements

267

obey the above integrability requirements and satisfy b

JcI>(t)u(dt)

~0

for all cI>

E

quo,

Ul, •.. ,

un)

a

(see Karlin and Studden, 1966, p. 405). In the following we state some characterizations of the cone C*(uo, Ul,"" un) as given in Karlin and Studden (1966, pp. 405-10.). 9.21. Theorem. A signed measure du is contained in the dual cone C*(uo, Ul,"" un) iff b

J

Uj

du = 0 for

j

= 0, 1, ... , n

(9.33)

~0,

a <x
(9.34)

a

and b

Jcl>n(t, xl dp,(t) a

where cl>n is given by (1.84).

9.22. Theorem. A signed measure du is contained in the dual of the cone nj=k quo, Ul, . . . , uj ) , k:5, n - 1, iff b

J

(i)

Uj

du = 0 for

j

= 0, 1, ... , k;

a

f b

(ii)

Uj

dp.

~o

for

j

= k + 1, k + 2, ... , n;

a

and b

(iii)

J cl>n(t, x) dp,(t)

~0

for

a < x < b.

a

For the next result we need the following definition: 9.23. Definition. A signed measure du is said to have k sign changes on (a, b) if there exists a subdivision of (a, b) into disjoint consecutive

268

9. General Linear Inequalities

intervals .lo,. . . ,Jk such that dp is of alternating sign and non-null on Jo, . . . ,Jk. (In the case that d p =f (t)dt for some continuous function f, the number of sign changes of d p is equivalent to the number of ordinary sign changes of the function f.)

9.24. Theorem. (a) If d p satisfies the orthogonality relations (9.33), then d p exhibits at least n + 1 sign changes. (b) Let d p satisfy (9.33). Zf dy possesses exactly n 1 sign changes on (a, b ) , is a nonnegative measure, and is non-null on some interval extending to the endpoint b, then d p E C*(uo,u1, . . . , u,).

+

Proof. (a) Suppose that d p possesses p 5 n sign changes. Then there exists a subdivision J o , . . . ,Jp such that d p is non-null and alternates in sign on .Io,. . . ,J p . Let ti = sup{t: t eJi} for i = 0, . . . ,p - 1, and define

Then the polynomial u(t) satisfies u(t)dp(t)2 0. Furthermore, since the support of d p cannot be confined to the finite set { t o , .. . , t,-,}, it follows that u ( t )dp(t) > 0. However, this inequality is incompatible with the orthogonality properties assumed for dp. Thus we conclude that dp must possess at least n + 1 sign changes. be the subdivision of (a, b ) associated with d p (b) Let .To, . . . , obeying the precepts of Definition 9.23. (Note that d p is a nonnegative measure on J,,+l.) Define t o , . . . , t, by ti =sup{t:t E J , } for i = 0, 1,. . . , n, and let

s:

8(t) =

(9.35)

where @ E C(uo,u l , . . . , u,). Expanding the determinant in (9.35), we see that s(t)can be written in the form

9.2. Some Generalizations and Refinements

269

With the aid of the orthogonality requirements satisfied by du, we obtain b

f

a

f b

Un)

(J(t) d{l(t) = U(u o' to,

, , tn

ep(t) duit].

a

However, it follows from (9.35) that (J(t) d{l(t) is a nonnegative measure throughout (a, b), so that f ~ ep(t) du ~O. Consequently we have du E C*(uo, Ul, . . . , un)' 0

9.25. Remarks. (a) Theorems 9.21, 9.22, and 9.24 contain many linear inequalities for convex functions, as was shown in Karlin and Studden (1966, pp. 410-31). This is a consequence of the fact that every linear continuous functional has an integral representation (see, for example, Kolmogorov and Fomin, 1972, pp. 347-48). (b) In Kocic (1982b) the implication f E quo, Ul) ~ A f " :0 ? is considered where A is a linear continuous operator defined as in Theorem 9.17. . (c) Note that Theorem 9.14 is used in Kovacec (1984) for a generalization of some classic inequalities for a rearrangement of vectors. (d) In Karlin and Studden (1966, p. 411), the Steffensen inequalities are also obtained as a consequence of Theorem 9.21. 0 In the following we give some refinements and converses of the previous results. First, we note that a simple consequence of Abel's identity in (9.5) is the well-known Abel Inequality (see Mitrinovic, 1970, pp.32-33): 9.26. Theorem. al":?· •. ":? an":?

Let {akH be a sequence of real numbers satisfying and let Wk = al + ... + ak (k = 1, ... , n). If

°

m

=

min Wk and l ~ k ~ n

M

=

max Wk, l ~ k $ n

then n

mai s;

L ;=1

wjaj:S Mal'

(9.36)

270

9. General Linear Inequalities

Bromwich (1908, 1955) gives the following generalization of (9.36): 9.27. Theorem. n), define

Given a real sequence {ad1 and an integer v (1::5 v::5 k

Ak

= 2:

a,

for

k

= 1, ... , n,

i=l

H;

= max

H ~

vsksn

= l:",:,;k:-s:v-l max Ak> Ak

and

,

h; =

min

Ak ,

l ~ k = : : : ; v - l

= min

h;

Y=:::;ksn

Ak ,

If {vd 1 is a decreasing sequence ofpositive real numbers, then n

2: a

hy(Vl - vy) + h ~ v y : : 5

ivi::5

Hy(Vl - vy) + H ~ v y . (9.37)

i=l

9.28. Remark. Bromwich (1908, 1955) also gave an integral analogue of the previous result with many applications. 0 Since the previous results (especially Abel's inequality) are related to the second integral mean value theorem, we give the following theorem and discuss some related results: Let t. g be real-valued functions which are defined and bounded on a compact interval I = [a, b], and let f be Stieltjes integrable with respect to g on I, written as f E L(g) on I. Further, let V(f, [a, b]) denote the total variation of f on I and "r the total variation function of f, i.e., vf(a) = 0 and vf(x) = V(f' [a, x]) for x E [a, b]. 9.29. Theorem (Second integral mean value theorem). Let f be increasing (decreasing), g be continuous, and h e L(g) on [a, b]. Then there exists a Z E [a, b] such that z

b

b

Jf(s)h(s)dg(s)=f(a) J h(s) dg(s) +f(b) J h(s)dg(s). a

a

(9.38)

z

If instead f is nonnegative and increasing (decreasing), then there exists a z, E [a, b] (zz E [a, b]) such that

f b

a b

f b

f(s)h(s) dg(s) =f(b)

h(s)dg(s)

~

~

(f f(s)h(s) dg(s) = f(a) Jh(s) dg(s»). a

(9.39)

271

9.2. Some Generalizations and Refinements

Proof. Denote G(x) = f ~ g ( s d) g(s) for x E [a, b]. Then G is continuous on [a, b] and f E L( G). Integrating by parts yields b

b

f f(s) dG(s) =f(b)G(b)- f G(s)df(s). a

a

By the first mean value theorem there exists a b

[a, b] such that

b

f G(s)df(s)=G(z) f df(s) a

Z E

= G(z)(f(b)-f(a».

a

Combining, we have (9.38).

D

Note that if we redefine f(a) = 0 (or f(b) = 0), then the result in (9.39) follows from (9.38). As an immediate consequence of the second mean value theorem we obtain (Karamata, 1949, p. 264 and Boas, 1970a): 9.30. Theorem. Let f be nonnegative and monotonic on [a, b], and assume that h E L(g) and fh E L(g) on [a, b]. (a) Iff is increasing, then b

f(b )inf{fh(s) dg(s): a z: t s:

-l-

b

f f(s)h(s) dg(s) a

t

b

::Sf(b)sup{f h(S)dg(s):a::st::Sb}. t

(9.40) (b) Iff is decreasing, then t

b

f(a)inf{f h(s)dg(s):a::st::Sb}::S f f(s)h(s)dg(s) a

a t

::Sf(a)sup{f h(s) dg(s):a::s t s; b}. a

(9.41)

Moreover, if g is continuous at b (at a), then we may replace f (b) by f(b_) in (9.40) (f(a) by f(a+) in (9.41».

9.31. Remark. In the same fashion we can obtain analogs similar to the D results of Mitrinovic (1970, pp. 301-2, 3.7.35 and 3.7.36).

272

9. General Linear Inequalities

The following result is a modification of a result in Karamata (1949, pp. 77-79) (see also Marik, 1949): 9.32. Theorem. Let f be a function of bounded variation on [a, b) = I, and let g, h be bounded functions such that h E L(g) and fh E L(g) on I. Then b

IJ f(s)h(s) dg(s) I ~ If(b)1 + V(f, I)

x

~ ~ ~IJ h(s) dg(S)I,

a

(9.42)

a b

b

IJ f(s)h(s) dg(s) I ~ If(a)1 + V(f, I)

~ ~ ~iJ h(s) dg(s)\.

(9.43)

a

Beesack (1975) gave the following generalization of a result of Darst and Pollard (1970): 9.33. Theorem. Let f be of bounded variation on [a, b) = I and h, g be bounded functions such that hE L(g) and fh E L(g) on 1. Let m = inf{f(x): a ~ x ~ b}, then b

v

b

Jf(s)h(s) dg(s)

~m Jh(s) dg(s) + V(f, I) a s ~ ~ ~ s J b h(s) dg(s),

a

a

u

(9.44) b

v

b

Jf(s)h(s) dg(s)

"2

a

m

Jh(s) dg(s) + V(f, I) a

a s ~ ~ ~ s Jb h(s) dg(s). u

(9.45)

An interesting related result is given by Marik (1974) (we state the result in the form given by Beesack): 9.34. Theorem.

Let the conditions of Theorem 9.32 be satisfied. Then v

b

If a

f(s)h(s) dg(s) I

~ ~ (V(f, I) + If(a)1 + If(b)1) U ~ ~ ~Jl h(s) dg(s). U

(9.46)

Proof. Let H o ( x ) = f ~ h ( s ) d g ( s ) , u=inf{Ho(x):xEI}, v= sup{Ho(x):x E I}, c = !(u + v), and H = H o - c. (Note that in general the

9.2. Some Generalizations and Refinements

273

values of u, v need not be attained, the same is true for sUpu,vEI g h(s) dg(s)). Then

IIh II g =

1 H(x) = -2

x

x

t

Ih(s) dg(s) _!2 inf Ih(s) dg(s) +!2 Ih(s) dg(s) tel

a

I

a

a

t

1 - -2 sup tEl

h(s) dg(s)

a

x

=2! SUp teI

x

x

(I h(s) dg(s») +!2 inf (I h(s) dg(s») ::;!sup (I h(s) dg(s») 2 t el

t

t e!

t

t

Thus we have

I b

f(s)h(s) dg(s)

=-

a

II

I b

H(s) df(s) + H(b)f(b) - H(a)f(a),

Q;

b

I::; I b

f(s)h(s) dg(s)

a

IH(s)1 dVf(S)

+ ~II hllg (If(b)1 + If(a)I),

a

and (9.46) follows.

0

The following result is a simple modification of an inequality in Karamata (1949, p. 79):

Let h, g, hf,fg E L(A), and denote G(x) = f ~ g ( s d) A(S), H(x) = f ~ h(s) dA(S)(H(x) > 0 for all x E (a, b D. If f is a nonnegative, decreasing function on I such that f ~ h(s)f(s) dA(S)> 0, then

9.35. Theorem.

I I b

inf G(X)::; a < x ~ H(x) b

a

g(s)f(s) dA(S) ::;

b

sup G(x). H(x)

(9.47)

a-c.x zzb

h(s)f(s) dA(s)

a

Proof. Denote the terms on the left-hand side and right-hand side in (9.47) by u and v, respectively. Then

uH(x)::; G(x)::; vH(x) for x

E

[a, b)

274

9. General Linear Inequalities

and b

b

Jf(s)g(s)dA(S)=f(b)G(b) + JG(x)d(-f(s)) a

a b

~ v V ( b ) H ( bJ) H(x)d(-f(s))) + a b

=V

Jf(s)h(s) dA(S). a

Since f ~ f ( s ) h ( s ) d A ( S » Othe , follows. 0

right-hand

inequality

ill

(9.47)

Karamata gives a discrete analogue of (9.47), and an extension of that result is given by Simeunovic (see Mitrinovic, 1970, p. 223). In the following we give an integral analogue of Simeunovic's result (see also Pecaric and Savic, 1984): If the conditions of Theorem 9.35 are satisfied, h(s) > 0 for all s E I, and A is increasing, then b

J

g(s)f(s) dA(S) inf g(x) ~ inf G(x) ~ . : : . , a b - - - - asxsbh(x) a<xsb H(x) h(s)f(s) dA(S)

J

a

~

G(x) sup - a<xsbH(x)

g(x) sup - - . asxsbh(x)

~

(9.48)

We also give the following generalization of Abel's inequality: Let Xij (i = 1, ... , n, j = 1, ... ,m) be real numbers, and let a = {aiJ (i = 1, ... , n, j = 1, ... , m) be a nonnegative, decreasing (1, i)-convex sequence of real numbers. Then

9.36. Theorem.

n

all min Xij -s

m

L L Xijaij i ~ l

holds, where Xij = ~ ~ ~ 1 ~ ~ = X 1 rs '

j=l

~ all max x;

(9.49)

9.2. Some Generalizations and Refinements

275

Proof,

n-1

= anmXnm -

m-1

x.; !:i.a rm - s=l 2: x; !:i.a r=l 2:

1

2

n-1 :5 max Xij( a nm -

2:

r=l

m-1 !:i.a rm 1

n-I

ns +

2:

s=l

!:i.a ns + 2

m-1

2: 2:

r=l s=l

n-1 m-1

x; !:i.1, 1a.,

2: 2:!:i. «:

)

1,1

r=l s=l

= all maxXij. This establishes the second inquality in (9.49). The first inequality can be proved similarly. D

9.37. Remark. Integral analogues of this and of the next two theorems ~ can be proved similarly. In the next result (proved in Pecaric, 1979a), we shall adopt the notation in Theorem 9.10. 9.38. Theorem.

Let XiI' .. Xim (1:5 ik

:5

n, 1:5 k :5 m) be real numbers.

(a) For all nonnegative and decreasing n-tuples a (1:5 j:5 m), we have

all' .. amIminXst···sm:5 F(a

j , ••• ,

a m) :5 all' .. ami maxXSt"'sm' (9.50)

(b) If aj (1 :5 j :5 m) are monotonic n-tuples, then m

IF(a 1 ,

.•• ,

am)1 :5 max IXs!,,,sml

IT (Iajnl + lajn -

aj11)·

(9.51)

j ~ l

The following result is given in Pecaric, Mesihovic, and Milovanovic (1989): 9.39. Theorem. Let xij (i = 1, ... , N, j = 1, ... ,M) and F(a) be defined as in Theorem 9.12 (a), and let en,m = (i -ltU -1)m.

276

9. General Linear Inequalities

(a) If aij (i = 1, t1n , m aij 2: a (i = 1,

, N;j = 1, ,M) are real numbers such that ,N - n, j = 1, , M - m), then F(a)

a

2: - ' - I

n.m.

F(en,m)'

(9.52)

(b) If lt1n , m aij !::; A (i = 1, ... , N - n;j = 1, ... ,M - m), then A IF(a) I ::;F(en,m)' n!m!

(9.53)

Chapter 10 Orderings and Convexity-Preserving Transformations

10.1. Orderings of Convexity: Generalizations and Related Results Partial orderings of notions of convexity and related preservation properties play an important role in the theory of inequalities. In this section we discuss some useful results on this topic. For notational convenience, we shall express a sequence { a , } ~(defined for n = 0, 1, 2, . . .) simply as {a,}. We first observe the following result due to Ozeki (1968): Let { a , } ; be an increasing sequence, and let the sequences { B , } ; and {C,}; be defined by B, = (lln) C;='=,iaj and C, = (lln) Cy=n bi (bo = 1). Then

for n = 2 , 3 , . . . and C , r C , / 2 . Of course, the main results of Ozeki (1965, 1967, 1968, 1969, 1970, 1971, 1972) are for the sequence {A,}: where A, = l / n C;='=, a,. Ozeki (1972) proved that if the sequence {a,}: is k-convex, then {A,}; is also k-convex. His proof depends on the following identity (see also MitrinoviC, LackoviC, and StankoviC, 1979): (n

+ k)AkA, = (n - l)AkA,-l + Akan

for n = 2,3, . . . .

From this identity we can obtain the following inequality for k-convex sequences:

277

278

10. Orderings and Convexity-Preserving Transformations

For the case k = 2, Ozeki first considered this problem in 1965 and gave the following list of implications:

where the inequalities hold for every n E}( (the set of all positive integers). It should be pointed out that the implication ~ 2 l o an g 2= O=> ~ 2 a n2=0 was proved earlier by Montel (1928). 1 where p = On the other hand, let An(a, p) = (1/ Pn) ~ 7 = p.a., (PI, ... ,Pn) is a real n-tuple and P; = ~ 7 = Pi' 1 Then the following identity is valid: A k+ 1(a, p) - Ak(a, p)

=

:;+1

k

~ Pi(ak+1 -

k k+1 i=1

ai),

(10.1)

and it follows that if the sequence { a n } is ~ increasing, then for arbitrary weights Pi > 0 (i E}() the sequence {An(a, p)} is also increasing. This, of course, is a weighted version of Ozeki's result for k = 1.

10.1. Remarks. (a) A related result is the following (see Pecaric, 1980b): Let p be such that 0 < P; < P; (k = 1, ... , n - 1), and let a be an increasing n-tuple. Then (10.2) Integral analogues of these results are also valid. Note that an integral analogue of (10.2) is given by Lovera (1957) and Ozeki (1965) when the weights are positive. The special case in which the weights are 1 was considered by Mott (1963) (see also Mitrinovic, 1970, p. 9). (b) Note that the monotonicity property of the arithmetic mean stated above can be used to prove the monotonicity property of an arbitrary quasi-arithmetic mean (Bullen, Mitrinovic, and Vasic, 1987, p. 215), i.e., we have

where M: (u, v ) ~ ~ (-oo:s u < v:s 00) is a continuous and strictly monotonic function, u :S a, :S v, and Pi 2= 0 (i = 1, ... , n). 0

10.1. Orderings of Convexity

279

We also note that the generalization of Ozeki's result for k > 1 is not possible for arbitrary weights. To see this, let us consider the sequence {An} defined by (10.3) where a = {an} is a real sequence and p = {Pn} is a positive sequence, and observe that: 10.2. Theorem. If a is a k-convex sequence, then the sequence defined in (10.3) is k-convex iff the sequence p is given by

_ (u +n -1) ,

P« - Po

n

(10.4)

where Po and u are positive real numbers.

10.3. Remark. For k = 2, Theorem 10.2 is proved in Vasic, Kecic, Lackovic, and Mitrovic (1972). For general k, an attempt to prove this theorem was made by Lackovic and Simic (1974), and a proof can be 0 found in Toader (1988c). A result of Toader (1983), which is a modification of the result of Bruckner and Ostrow (1962) (see (1.24)) for sequences, deals with an ordering of convexity. In the following we give some further generalizations of that result. Let K be a class of convex sequences, a = {an}, S* be a class of starshaped sequences of a such that the sequence {(an+lao)/(n + I)} is increasing for n 2: O. Let S be a class of superadditive sequences satisfying a n+ m - an - am + ao 2: 0 for every n, m > 0, and let W be a class of weak-superadditive sequences, i.e., for every n we have a n+ 1 - an - al + ao 2: O. We say that the sequence a = {an} has the property "P" in the u-mean if the sequence AU = { A ~ }given in (10.3) and (10.4) has the property "P." We denote by MUK, MUS*, MUS, and MUW the sets of sequences which are convex, starshaped, superadditive, and weak-superadditive in u-mean, respectively. The following result is given in Toader (1983): If 0 < v < u, then we have K eMuK eMvKeS* eMuS* «u-s-, S* eS e W,

MUS* eMUS eMuW,

MVS* eMVSeMVW,

W eMuWeMVW.

Some more general results can also be found in Toader (1986a).

280

10. Orderings and Convexity-Preserving Transformations

Mocanu (1982) considered the weighted mean

Fg(x)

=

g ( ~ )Jg'(t)f(t) dt,

(10.5)

o

where g is a real-valued function such that g' exists. Toader (1986b) proved the following results: If the transformation (10.5) preserves the convexity (or the starshapedness, or the superadditivity), then the function g is of the form

u>o,

ki=O.

(10.6)

Denoting by F; the function in (10.5) with g given in (10.6), let MUK(b), MUS*(b), and MUS(b) denote the classes of functions f E C(b) with the property that the corresponding functions E, belong to K(b), S*(b), and S(b), respectively, where C(b), K(b), S*(b), and S(b) are the classes of continuous, convex, starshaped, and superadditive functions, respectively. It follows that if 0< v < U, then the following implications are true: K(b) c MUK(b) c MVK(b) c S*(b) c MUS*(b) c MVS*(b), S*(b) c S(b),

MUS*(b) c MUS(b),

MVS*(b) c MVS(b).

Note that (10.6) was also obtained by Lackovic (1975). Some similar results are given in Toader (1986a, 1988b). Lackovic (1975) also proved the following result: Let the function f be defined and continuous on [0, b] such that f(O) = 0, and consider the following conditions: (i) f is m-convex on [a, b], (ii) fis m-convex in mean, i.e., the function F(x) = (1/x) gf(t) dt (0 < X :5 b), where F(O) = 0, is m-convex on [0, b], (iii) the functionf(x)/x is convex of order m -1 on (0, b], (iv) fis superadditive of order m on [0, b] (see Remark 6.13(b», (v) f is superadditive of order m in mean, i.e., F is superadditive of order m on [0, b]. Then the following implications are valid: (i) => (ii) => (iii) => (iv) => (v). Partial orderings for sequences that are convex of higher order was also considered by Toader (1985c). He considered the ordering of order

10.1. Orderings of Convexity

281

three, and provided the following definitions: The sequence {an} is said to be: starshaped of order three if the sequence {(a n + ! convex of order two; (ii) superadditive of order three if

(i)

-

ao)/(n + I)} is

for every

m, n, P ;::: 0;

(iii) 2-starshaped of order three if it satisfies the relation: an + 3

-

ao:> a n + 2

n+3

-

at

n+1

f

or

:> 0 n e: .

Let us denote by K 3 , S;, and S ~ *the classes of convex, starshaped, and superadditive and 2-starshaped of order three sequences, respectively. Further, we denote by M UK3, MUS;, M US3, and M U S ~the * classes of sequences {an} with the pro('erty that { A ~ given } in (10.3) and (10.4) is in K3 , S;, S3' and S ~ * , respectively. If < U < v, then the following implications are true:

°

For results of higher order, see Toader (1986a). Finally, we give some generalizations of Theorem 10.2. Let (Pn,i) (i = 0, 1, ... , n; n = 0, 1,2, ... ) be a triangular matrix of real numbers, let A(a) = {An(a)} be the sequence defined as n

An(a) =

L Pn,n-IJh k=O

for

n = 0, 1, 2, ....

(10.7)

Ozeki (1967) obtained necessary conditions for a triangular matrix (Pn,i) to possess the following property: The sequence {An(a)} defined in (10.7) is convex for every convex sequence {an} (see also Mitrinovic, Lackovic, and Stankovic, 1979; Lupas , 1979; and Kotkowski and Waszak, 1978). The following is a generalization of Ozeki's result (see Lupas, 1979).

282

10. Orderings and Convexity-Preserving Transformations

10.4. Theorem.

Let An(a) be defined as in (10.7). Then the implication /:1r an;::=: 0::;> /:1rA n(a);::=: 0

(10.8)

is valid for every sequence {an} iff

u+ 1, j) = 0

/:1rxn

for

= 0, 1, ... , r -1;

j

n

= 0, 1,2, ...

and

(10.9) /:1rXn(r, i + r);::=: 0 for

i = 0, 1, ... , n;

n = 0, 1,2, ... ,

where Xn(m, k) =

for

{~~k

(n - k

+m -

1 - j)

m -1

LJ

J=O

. Pn,J for

n
(10.10)

n;::=:k.

Theorem 10.4 was obtained in Lupas (1979) by using the following interesting identity: /:1 rAn(a)

r-l

n

u-

= 2: /:1 jao/:1rx n + 1, j) + 2: /:1raj/:1rXn(r, j + r). j=O

j=O

(10.11) A generalization of Theorem 10.4 is given in Pecaric (1982g). By using (9.4), i.e., n

m-l

~Wiai = ~ o/:1

kao

(1k!

n

)

n

i ~i(k)wi + k ~ m/:1m ak_m x (

,±

1 (i - k (m - 1). i=k

where i(m) = i(i -1)· .. (i - m /:1 sAn(a)

+ m - l)(m-l)wi ) , (10.12)

+ 1), and (10.7), we have

n+s

= 2: (/:1

SqnU»aj

j=O

for every s,

(10.13)

where qn( ]") = {O Pn.n-j

for for

n<j n ;::=:j.

Thus from (10.12) and (10.13) we obtain n+s

m-l

/:1 SAn(a)

=

2:

k=O

/:1k ao(/:1sXn(k+1,k»+

L

k=m

/:1m ak_m(/:1sXn(m,k».

Choosing s = m = r we then obtain (10.11) from (10.14).

(10.14)

10.1. Orderings of Convexity

283

By using (10.14) and a similar argument we have (Pecaric, 1982g): 10.5. Theorem.

Let An(a) be defined as in (10.7). Then the implication

/1man 2= a:::} /1sAn(a)

2=

°

for

s

E

.N' (the set of all positive integers)

is valid for every sequence {an} iff /1sXn(k + 1, k) =

°

for

k

= 0, 1, ... , m -1;

n

= 0,1,2,

...

and /1sXn(m,k)2=O for

k=m, ... ,n+s;

n

= 0,1,2,

... ,

where Xn(m, k) is given (10.10).

10.6. Remark. Necessary and sufficient conditions for the inequality ~ 7 = wia o i 2= to hold for every m-convex sequence are given in Theorem 9.1; using that result and (10.13) we can also obtain Theorem 10.5. 0

°

The following theorem is also valid: 10.7. Theorem. Let a = {an} be a real sequence, and let An(a) be defined as in (10.7). If l/1manl:5 N for n = 0, 1,2, ... , and /1sXn(k+1,k)=0

for

n = 0,1,2, ... ,

k=0,1, ... ,m-1;

where Xn(m, k) is given in (10.10), then l/1sAn(a) 1 :5 N

n+s

2:

l/1sXn(m, k)l.

k=m

Proof.

This is an immediate consequence of (10.14).

0

10.8. Remarks. (a) Theorem 10.7 is a generalization of Ozeki's result (see Theorem 2 in Mitrinovic, Lackovic, and Stankovic, 1979): If An*( a ) = -1 - LJ~ ak n + 1 k=O

and

1/1k ajl :5M,

then

* 1/1k An(a)1 : 5M --. k

+1

A similar result is also given in Lupas (1983, 1985). (b) Further generalizations of the previous results are given in Milovanovic and Pecaric (1986) and Pecaric, Mesihovic, Milovanovic, and Stojanovic (1986) (see also Mesihovic, 1987).

284

10. Orderings and Convexity-Preserving Transformations

(c) Analogous results for p-monotonic and p, q-convex sequences are proved in Pecaric (1981d) and Milovanovic and Kocic (1989), respectively. (d) A result similar to Theorem 10.4 for continuous linear operators was given by Tzimbalario (1975) (see also the results given in Remarks 9.13 and 9.19(b)), and he gave necessary and sufficient conditions for the implication

f

e Cia«, Ul,""

u n ) ~ A f E q uu 1 o, · , .

· ,

un).

0

Ozeki (1967, Theorem 2) presented a theorem for logarithmically convex sequences, and it is analogous to Theorem 10.4. To describe his result we first introduce some notation: Let the sequences {Pn} and {rn} be strictly positive, and let

For a given sequence {an} let the sequence {an} be defined by an ==

poao + ... + Pnan

for

r;

n == 0, 1, ....

(10.15)

Then we have 10.9. Theorem (Ozeki, 1967). Assume that the sequences {Pn} and {rn} satisfy the conditions Qo == 0,

2Qn > 1 > Qn for qn

(1 - Qn_lQn)2

$

~ Qn

for

(10.16) n == 1, 2,

n == 1, 2,

4qn(1- Qn-l)(l - Qn) for

,

(10.17)

,

(10.18)

n == 1, 2, .. "

(10.19)

If {an} is a positive and logarithmically convex (written as log-convex) sequence, then the sequence {an}, defined in (10.15), is also positive and log-convex; in other words, the implication is valid.

Proof.

Let totl· .. tn == poao + ... + Pnan

for

n == 0, 1, ....

(10.20)

10.1. Orderings of Convexity

285

Then the sequence {tn } is well defined, and we immediately have tn > 0 for all n. Similarly, we have Pnan = totl ... tn-l(tn -1),

so that from p.a; > 0 we have t; > 1. Since by assumption an-lan+l holds, we have tn + l

~

qntn-l(tn - 1)2 1 + ..:c..:.....:..:.......::....:....:.:_----'-tn(tn-l - 1)

~ a ~

(10.21)

Using (10.15), the definition of the sequence {r,}, and (10.21), we then have

= C(tn+l -

tnQn)

~ c(qntn-l(tn - 1)2 + 1 - tnQ n) tn(tn-l - 1)

= CI/(qn, Q, tn-I, tn), where C and C l satisfy C

~0

and

Cl =

C

tn(tn-l - 1)

> 0 for n > 1,

and

(10.22) is a function of tn only when qn' Qn, tn- l are kept fixed. From (10.18) we have qntn-l - Qntn-l + Qn > O. Clearly the discriminant form of fUn) is

If D :s: 0, then it is immediate that 0n-l 0n+l - o ~ ~ 0; thus we consider only the case D > o. Since t.;.«> 1, we see that t n_ I ( I - 4qn(1- Qn» > 1

286

10. Orderings and Convexity-Preserving Transformations

and 1 - 4qn(1- Qn) > O. Thus the condition D > 0 implies 1 . 1 - 4qn(1- Qn)

(10.23)

~ - l >

To show that 0n-lOn+l - o ~2: 0 holds for all n, we proceed by induction. For n = 1, it follows by a direct verification that 000z -

ai = (poao)Z (-Q1t Z+ (1- 2Ql)t + 1- Ql +Pza z), rorz

where t 000z -

poao

= (Plal)/(poao) > O. Since C = (poaof/(rorz) > 0, we have

of = C(ql = C(ql 2:

z) Ql)tZ+ (1- 2Ql)t + (1- Ql) - qltZ+pza poao Ql)t Z+ (1- 2Ql)t + 1-

e. +pz(aoapoaoz; aD)

C«ql - Ql)t Z+ (1 - 2Ql)t + 1 - Ql) == Cfl(t)

(note that the last step is given incorrectly in Ozeki, 1967). The discriminant form of the polynomial ft(t) is Do = (1- 2Ql)Z - 4(1- Ql)(ql - Ql) = 1- 4ql(1- Ql)'

Applying (10.16) and (10.19) for n = 1, we find that Do ::5 0, i.e. ,fl(t) 2: O. Thus we have 000z - ai 2: O. Now let us assume that (10.24) for some n 2: 1 where C 2: 0 was defined earlier. By a direct verification we have f(Qn-ltn-l)

= tn-1«qn + (Qn-1Qn

Qn)Q~-lt~-l

+1-

2qn)Qn-l tn-l

+ (qn -

Qn-l»

= tn-1F(tn-l), and the discriminant form of the quadratic polynomial f(t) is D 1 = (1 - Qn-1Qnf - 4qn(1- Qn)(1 - Qn-l)'

From (10.19) we have D 1 ::5 0, i.e., f(Qn-ltn-l) 2: O.

(10.25)

Thus the assumption D > 0 implies that equation f(t) = 0 (where f is defined in (10.22» has two distinct real roots, a and {J, say. Moreover,

10.1. Orderings of Convexity

287

we have

a+f3 1 Qn-lln-l - -2- = 2« qn - Qn )i.., + Qn ) x (2Qn-l(qn - Q n ) l ~ - +l (2QnQn-l + 1- 2qn)ln-l -1)

= Cdiln-l), where Cz > O. For the quadratic trinomial jjtr) we have

Iz(O) = -1,

(10.26)

and by a direct calculation we find that

Iz(' -

4 q n ~ -1

Qn»)

= C 3(2Qn -

1)(Qn-l(2Qn - 1) - 1 + 4qn(1- Qn»,

(10.27)

where, from (10.17), 2Qn > 1 and C3 > O. From (10.19) we have

4 (l-'Q) ~(1- Qn_1Qn)Z qn n 1- Qn-l Thus from (10.27) we obtain

Iz(' _

4 q n ~ _1 Qn») ~ C

4(Qn_l)z(1-

Qn)Z

~0,

(10.28)

where C 4 > O. Combining (10.23), (10.25), (10.26), and (10.28), we then have (10.29) and from (10.24) it follows that In (10.25) and (10.29) we obtain

~ In-1Qn-l.

Consequently, from

Note that in Theorem 10.9 if we take r., = 1 and Po = PI = ... = p.; = 1 for all n = 0, 1, ... , then the assumptions (10.26)-(10.29) are satisfied and, from the log-convexity property of the sequence {an} the logconvexity property of the sequence {An} follows, where An is given by An = (lin) ~ 7 ~ a., 1

288

10. Orderings and Convexity-Preserving Transformations

Ozeki (1967) also obtained the following result: If a positive sequence {an} is log-convex, then the sequence {an} defined by for

n = 0, 1, ...

(10.30)

is also log-convex. Together with the sequence {an} let us consider another sequence {Sn} given by (10.31) It is clear that {an} is log-convex iff {Sn} is log-convex, and Ozeki's

(1967) proof is based on this idea.

10.2.

Various Results

Let {an} and {bn} be two given sequences. A result which general than (10.31) was given by Davenport and P6lya (1949):

10.10. Theorem.

IS

more

Let the sequence {wn} be defined by

(10.32) where {an} and {bn} are positive and log-convex. Then the sequence {wn}

is also positive and log-convex. Let the sequence {c n } be defined by n

c, = 2:

ak b n -

k1

(10.33)

k=O

which is the convolution of {an} and {bn}. Then (10.33) gives the coefficients of the expansion

and the following result is valid (Kalusa, 1928; Karamata, 1933;

10.2. Various Results

289

Davenport and P6lya, 1949; Jurkat, 1954; Lorentz, 1954; and Menon, 1969):

10.11. Theorem.

If {an} and {b n} are posuuie and logarithmically concave (written as log-concave) sequences, then their convolution {cn} defined in (10.33) is also positive and log-concave.

Note that if {an} and {bn} are positive and log-convex, then their convolution {cn } need not be log-convex. The following result is proved in Vinogradov (1975): Let {an} and {b n} be two nonnegative log-concave sequences such that ao - Aal 2:: 0 and b., - Ab1 2:: 0, where A 2:: O. Let the sequence {cn } be defined by n

c;

= 2: aib n-i i=O

n-l

A 2: ai+1b n-i for

n

2::

1,

i ~ O

Then {c.} is also log-concave. Ozeki (1969) also considered a sequence related to the convolution sequence {cn}; namely, for given {an} and {b n}, let {cn} be given by (10.34) Then the following theorems are valid:

10.12. Theorem. Let {an} and {bn} be positive and convex sequences. If al2::ao and b l2::b o, then the sequence {cn} defined in (10.34) is convex. In other words, if

i = 1, 2, ... ,

for

(10.35)

then ~ 2 C i _ 2:: l 0 for i = 1, 2, ....

The following theorem concerns log-convex sequences.

10.13. Theorem.

If {an} and {bn} satisfy the conditions bi-1b i + 1 2::

bf,

for

i

=

1, 2, ... ,

then the sequence {c.} defined in (10.34) also satisfies the condition Ci-1C i + 1 2::

cf

for

i = 1, 2, ....

(10.36)

290

10. Orderings and Convexity-Preserving Transformations

It is easy to prove that if {an} and {b n} satisfy the conditions of Theorem 10.12, then they are increasing. This fact follows from (10.35) by induction. The proof of Theorem 10.12, as given in Ozeki (1969), is very similar to that of Theorem 10.9. By a direct calculation we can verify that for the sequence {cn } the equality

L\2(n + I)c,

= (n + 1)L\2cn + 2L\cn+ 1

holds. This implies that if {an} and {b n} satisfy the conditions of Theorem 10.12, then not only is {c.} convex, but also their convolution sequence {cn } defined in (10.33) is convex. Some related problems were studied in Jurkat (1954). The proof of Theorem 10.13 given in Ozeki (1969) is also similar to that of Theorem 10.9. Note that the sequence {cn} defined in (10.34) can be written in the form c; = (l/(n + l»cn , where cn is given by (10.33). Upon applying this identity, the inequality in (10.36) becomes (see Mitrinovic, Lackovic, and Stankovic, 1979).

(i + 1)2 -

"(" 1 1

-

-2

+ 2 )Ci-1Ci+1;:::Ci .

f

or

"1 2

1=

,

, ... ,

which is a weaker result than log-convexity. Ozeki (1970) also gave some theorems related to coefficients of functions given in the form of a power series. In particular, the following result was proved. 10.14. Theorem.

For a given real sequence { P n } let ~ the sequence { q n } ~ be defined by the following equality

1+

k ~ lqkx k = (1- ~ 1Pkxk)

-1;

(10.37)

n = 2,3, . . ..

(10.38)

= 1, 2, ....

(10.39)

that is, let n-1

qn = Pn

+

L Pkqn-k

for

k ~ l

If

{ P n } is ~

positive and log-convex, then qn > 0

and

qnqn+2;::: q ~ + l for

n

The proof of this theorem, given in Ozeki (1970), is based on the following theorem, which can be found in the same paper. We note that the fact qn > 0 was already proved in Karamata (1933). Some similar (but more general) results were obtained by Jurkat (1954).

10.2. Various Results

10.15. Theorem. Let D; (n n X n matrix (a i) where

= 1, 2, ... )

aij = -1

denote the determinant of the j = i - 2,

for

291

and a ij = 0 for

j

< i - 2.

If and then (-ltD n 2= O(n = 1, 2, ... ).

The proof of this theorem by induction is given in Ozeki (1970). A result that is similar to Theorem 10.14 is proved in Ozeki (1971):

10.16. Theorem.

Let {a;}, {b;}, and {c.} be sequences of real numbers such that the following conditions are satisfied:

rr

n i=O

(x

~

+ aZi+l) = j-::O

rr n

(n +i 1) b.x

n+l-i

b o = 1,

,

(10.40)

n

(x

+ azi )

i ~ O

If a, > 0 and aiai+Z - af+l

=

L cjx

n

-

1

Co

,

(10.41)

= 1.

i ~ O

2= 0

(i

= 1, 2, ... ),

then b,

2= c,

(i

= 1, ...

, n).

The proof of Theorem 10.16 given in Ozeki involves a partial ordering k be two power series. They defined below: Let ~ ~ ~ aix" o and ~ ~ = bkx o are said to be partially ordered, denoted by ~ ~ ~ aix">» o ~ ~ ~ bix", o if ak 2= b k holds for k = 0, 1, 2, . . .. Ozeki (1968) proved the following results concerning this partial ordering:

10.17. Theorem. Assume that f(x) = ~ ~ ~ O P k X If k . the sequence {pd is positive and log-convex, then f(k-l)(x)f(k+l)(x) (f(k)(x»)Z (k - 1)! (k + 1)! » k ! for

_ k - 1,2, ... ,

(10.42)

where the derivatives f(m) are taken in the usual sense.

The proof of this theorem, as stated in Ozeki (1968), follows immediately by calculating the coefficient of .r" on the left-hand and the right-hand sides of (10.42).

292

10. Orderings and Convexity-Preserving Transformations

Ozeki (1971) also studied some properties of the partial ordering for polynomials (see also Mitrinovic, Lackovic, and Stankovic, 1979). Let (E, A, fl) be a probability space. For p E ~ let Dp,1' be the set of all functions f: E ~ ~ such that (i) f is measurable and nonnegative (or positive if p 2: 0), and (ii) < f fP du < 00 for p#:O and 0< flogfdu < 00 for p = 0. For f E Dp,1' let us define the LP -mean of f with respect to fl by

°

for p#:O, forp=O. If A denotes another probability measure on (E, A), and q

:5,

p, then we

denote the quotient of the respective means by (10.43)

for positive functions whose domains are the convex cone Dq,A n Dp,w Clausing (1983) proved that:

10.18. Theorem. domain

If q

:5,

1 :5, p, then

¢q,p,A,1'

Dq,p,A,1'

=

is quasiconvex in the whole

Dq,p,A,W

This result is a generalization of a result in Clausing (1982) which concerns the special case q = -1 and p = 1. Using that result Clausing (1982) obtained results in Kantorovich (1948) and Wilkins (1955). The following result can be found in Peetre and Persson (1988). Special cases of this result are treated in Pecaric and Beesack (1986), Capocelli and Tanja (1985), Daroczy (1964), Dresher (1953), Persson (1987), and Sharma and Autar (1973a, 1973b).

10.19. Theorem. Let F: ~ ' : - ~ ~+ be an increasing function, and let g: D ~ ~+ (D is an additive Abelian semigroup) be superadditve. (a) If F is convex and f: D ~ ~ , : - , is subadditive, F(O) = 0, then the function H(x) = g(x)F(f(x)/g(x» is subadditve. (b) If F is concave and f: D ~ ~':- is superadditve, then H (x) is superadditive. Let us assume that the function f is defined and continuous on [0,1]. Bernstein's polynomial Bn(x, f) of order n = 0, 1, ... of the function f is

10.2. Various Results

293

defined by Bn(x,f) =

~ O( ~ ) X k ( I - X r - k f ( ~ ) .

It is a well known fact that the sequence {Bn(x, f)} converges to f(x) uniformly as n ~00 under reasonable conditions on f A systematic study of Bernstein's polynomials of convex functions was first made by Popoviciu (1961) (see also Popoviciu, 1944, pp. 43-44). For example, it is known that Bernstein's polynomials of continuous m-convex functions are also m-convex functions, and similar results are valid for functions of several variables. Moreover, the following result is valid:

10.20. Theorem. A continuous function f: [0, 1] every n = 0, 1, ...

~IR

is convex iff for (10.44)

10.21. Remark. Inequality (10.44) was proved by Temple (1954), and the same result was also proved by Arama (1960). In fact, Arama proved the following result: For an arbitrary continuous function f and ;1, ;2, ;3 E[0, 1], we have

Theorem 10.20 was also proved by Moldovan (1962) under a condition 0 weaker than f E C 2[0, 1] (which was given by Kosmak, 1960). The following theorem is related to Theorem 10.20: 10.22. Theorem. A continuous function f: [0, 1] every n = 0, 1, ... f ( x ) ~ B n ( x , forevery f )

~IR

is convex iff for

XE[O, 1].

(10.45)

10.23. Remarks. (a) Inequality (10.45) was proved independently by Arama (1960) and P6lya and Schoenberg (1958), and Theorem 10.22 was also obtained by Moldovan (1962). (b) Inequalities (10.44) and (10.45) imply that the sequence {Bn(x,f)} is decreasing and converges to f from above, and Theorems 10.20 and 10.22 assert that these properties are valid only for convex functions.

294

10. Orderings and Convexity-Preserving Transformations

(c) The convexity property of the sequence {Bn(x, f)} was considered by Popoviciu (1961) and Arama and Ripianu (1961) (see also Mitrinovic, Lackovic, and Stankovic, 1979). (d) Let S denote the class of all star-shaped functions on [0,1]. Lupas (1972a) showed that

f

E

~Bn(x, f) E

S

S.

(e) Theorems 5.2 and 5.3 are similar to Theorems 10.20 and 10.22. 0 The following theorem was proved by Artin (1931) (see also Marshall and Olkin, 1979, p. 452): 10.24. Theorem. Let U be an open convex subset of ~ n , and let <jJ: Ux(a, b ) [~0, (0) satisfy (i) <jJ(x, z) is Borel-measureable in z for each fixed x, and (ii) log <jJ(x, z) is convex in x for every fixed z. If v is a measure on the Borel subsets of (a, b) such that <jJ(x, .) is v-integrable for every x E U, then b

ljJ(x) ==

J<jJ(x, z) dv(z)

(10.46)

a

is a log-convex function on U.

An important example is that ¢(x,z)=(<jJ(z)Y, where <jJ(z) is a positive function, and in this case the function b

A(X)=

J(<jJ(z)Y dv(z)

(10.47)

a

is log-convex. This is a special case of a result given in Remark 4.19(d). 10.25. Remark. Theorem 10.24 is equivalent to Holder's inequality (see Marshall and Olkin, 1979, p. 461). 0 Suppose that for every x E ~n and every convex subset A c ~m there exists an integral of the form fAf(x,y)dy==I(x,A). Then the following theorems are valid (Tomilenko, 1976): 10.26. Theorem. Let f(x, y) be a function of (n + m) variables, where x E ~ nand y E ~ m .Iffis a finite log-concave function on ~ n + and m A and

10.2. Various Results

B are convex subsets in

295

~ m , then

I(AXI+ (1- A)X2' AA + (1 - A)B) ~I(xl, A)'J(X2' B)l-;" (10.48)

where Xl' X2 E

~ n and

0 < A < 1.

10.27. Theorem. If f(x, y) is a finite log-concave function of (n + m) variables, where X E ~nand y E ~"', then y

I(x, y) is log-concave on

=

Jf(x, t) dt

(10.49)

~ n + m .

10.28. Remark. Theorem 10.26 is a generalization of a result of Prekopa (1973). For Prekopa's result and its applications in statistics, see Section 13.5. 0 In Bodin and Zalgaller (1968) (see also Mitrinovic, 1970, p. 309) the following result is given: '

10.29. Theorem. by

Let A be a negative semidefinite quadratic form given all' a22::; 0 and

alla22 - ai2 ~ o.

If D(m) is a parallelogram in the xy-plane, whose center is m, then the function p defined by p(m) =

JJ eA(x.

y

)

dx dy

D(m)

is a log-concave function of m. Abdel-Hameed and Proschan (1976) considered the transformation g(A) =

J rp(A, x)f(x) duix],

(10.50)

where f and rp are nonnegative Borel-measurable functions of nonnegative arguments, !l denotes the Lebesgue measure on [0, (0) or the

296

10. Orderings and Convexity-Preserving Transfonnations

counting measure on {O, 1, ... }, and the integral is assumed to exist. They showed that various geometric properties possessed by fare inherited by g under appropriate assumptions on cp. To describe their results we first observe some definitions and basic properties. The first definition concerns four geometric properties of a function.

10.30. Definition. Let f : [0, 00) (a) (b) (c) (d)

~[0,

00). We say that

° °

f is star-shaped if f(ax) ~ af(x) for every x 2= and ~ a ~ 1, f is superadditive if f(x + y) 2= f(x) + f(y) for every x 2= 0, Y 2= 0, fis root-increasing iffllX(x) is increasing in x >0, f is supermultiplicative if f(x + y) 2= f(x)f(y) for every x

2= 0,

Y 2= 0.

10.31. Theorem. The following elementary relationships among the geometric properties are valid:

~f

Proof.

Easy.

f star shaped

~et(x)

superadditive

~ef(x) supermultiplicative.

root increasing

0

10.32. Definition. Dual geometric properties may be defined by reversing the direction of the inequality in Definition 10.30(a), (b), and (d), and by replacing "increasing" by "decreasing" in (c). The dual geometric properties are called, respectively: (a') antistar shaped, (b') subadditive, (c') root-decreasing, and (d') submultiplicative. The relationships among (a'), (b'), (c'), and (d') in Definition 10.32 are analogous to those among (a), (b), (c), and (d) in Definition 10.30. We next define completely monotonic functions:

10.33. Definition. A nonnegative function f of a nonnegative argument is completely monotonic if it has derivatives of all orders and (-1),,!(k)(X) 2= for x 2= and k = 1, 2, ....

°

°

We shall need to define two more notions in order to state and prove the main results.

10.2. Various Results

297

10.34. Definition. Let (jJ(A, x) be defined on A x X, where A and X are ordered sets. Then (jJ(A, x) is said to be totally positive of order 2(TPz) if (jJ(A, x) ~0 for AE A, x E X, and (jJ(Al , Xl)

I(jJ(Az, Xl)

(jJ(A l, xz) (jJ(Az, xz)

I

~0

Totally positive functions of order 2 and of higher orders play an important role in analysis, statistics, inventory theory, reliability theory, and many other theoretical and applied fields.

10.35. Definition. A function (jJ(A, x) is said to obey the semigroup property if

where Il denotes the Lebesgue measure on [0, co) or the counting measure on {0,1,2, ... }, AE[O,CO) 'or alternatively, AE{0,1,2, ... } and XE [0, co).

10.36. Theorem.

(a) Let (jJ(A, x) be TP z , and assume that

JX(jJ(A, x) dll(X) = aA

for all

A> 0

(10.51)

holds for some a > O. Then f is star-shaped implies that g is star-shaped.

(b) Let (jJ(A, x) satisfy the semigroup property, and f (jJ(A, x) dll(X) == 1. Then f is superadditive implies that g is superadditive. (c) Let (jJ(A, x) satisfy the semigroup property. Then f is supermultiplicative implies that g is supermultiplicative. (d) (jJ(A, x) satisfy the semigroup property iff f is exponential implies that g is exponential. (e) Let (jJ(A, x) be TP z and satisfy the semigroup property. Then (e-1) f is root-increasing and Il is the Lebesgue measure imply that g is root-increasing. (e-2) f is root-increasing, Il is the counting measure, and for all A> O. lim i;Ho

J (jJ(A, x );X dll(X) == 0

imply that g is root-increasing.

for some

;0 E [-co, co)

298

10. Orderings and Convexity-Preserving Transformations

(f) Let @(A, x ) satisfy the semigroup property. Then f is completely monotonic implies that g is completely monotonic.

10.37.

Remark. Examples of kernels

@(A,

x ) satisfying (e-2) are:

(i) the Poisson kernel @(A, x ) = e-A(A"/x!),A 2 0 and x and 0 (ii) the binomial coefficients (t), A, x = 0, 1, . . . .

= 0,

1, . . . , ;

Proof of Theorem 10.36. (a) For each c > 0, g ( A ) - cA =

I

[

,'I

@(A, x ) f ( x ) - - x d p ( x ) .

Since f is star-shaped, then f ( x ) - ( c / a ) x changes sign at most once in 2 0, and if once, from - to By the variation diminishing property o f the TP, function @(A, x ) (Karlin, 1968, p. 21), it follows that g(A) - cA changes sign at most once, and if once, from - to +. Hence g must be star-shaped. (b) Write

+.

x

+ A*)

=

j

@(A1

+ A2

J

.If

(x) 4 4 x )

=I j @ ( L x

-Y)@(A,,Ylf(x)dp(Y)dCl(x) [by the semigroup property]

[since f is superadditive] [since

I

@(A, z ) d p ( z ) = I].

Thus g is superadditive. (c) Using (10.50), the semigroup property, and the supermultiplicativity o f f , we obtain

Thus g is supermultiplicative

10.2. Various Results

299

(d) Let rjJ(A, x) satisfy the semigroup property and f be exponential. Then by the same kind of argument as in (c), we obtain g(A] + Az) = g(A])g(Az) for all A] 2: 0 and Az 2: O. Since rjJ and f are measurable, it follows from Tonelli's theorem that g is measurable (Royden, 1968, p. 270), and thus g(A) must be exponential, as pointed out by Breiman (1968, p.305). Suppose now that (10.50) maps exponential functions into exponential functions. Take f(x) = e :", s 2: O. For each fixed A, consider the measure VA defined for every Borel set A explicitly by the relation vA(A) =

JrjJ(A, x) d!J(X). A

Define the Laplace transform of

VA

by

Then, by the well-known property of the Laplace transform, for every A] 2: 0 and Az 2: 0, we have that gAl(S)gA2(S) is the Laplace transform of the convolution measure vt. VA2; i.e.,

Since g is exponential in A by assumption, we have gAt(S)gA2(S)= gAt+A2(S); i.e.,

By the uniqueness of the Laplace transform (see Theorem 1a of Feller, 1971, p. 432), it follows that

i.e., rjJ satisfies the semigroup property. (e-1) We can assume that for each A, rjJ(A, x) is strictly positive on a set of positive Lebesgue measure; otherwise g(A) would be zero and we would have nothing to prove. Since f is root-increasing, it follows that for each fixed a, O::s: a < 00, f(x) - a" changes sign at most once in x 2: 0, and if once, from - to +.

300

10. Orderings and Convexity-Preserving Transformations

Also, by the previous result (d). f
changes sign at most once in ).. 2": 0, and if once, from - to conclude that g()..) is root-increasing.

+.

We

(e-2) The proof of (e-2) is similar to that of (e-1), with obvious modifications. (f) Result (f) follows from Theorem 12a of Widder (1946, p. 160) and (d) above. D

10.38. Remark. A dual theorem exists in which each geometric property is replaced by its dual property. Thus the geometric properties defined in Definition 10.32 are also preserved under the integral transformation (10.50). D In the following we give some similar results for means (note that a result of this type was already given in Remark 4.19(b». The following result is proved in Hardy, Littlewood, and P6lya (1934, 1952, pp. 85-88) and Bullen, Mitrinovic, and Vasic (1987, p. 253): 10.39. Theorem. Let a = (a l , . . . ,an), p = (PI' ... ,Pn), P; = r.7=lPi, and let f: ~ nX ~ n ~ ~ be a real-valued function. If f has continuous second order derivatives and is strictly increasing and strictly convex, then the quasi-arithmetic mean

is a convex function of a ifff' If" is concave. A simple consequence of this result is the following (Godunova and Cebaevskaja, 1971): 10.40. Theorem. Let f and t; satisfy the conditions in Theorem 10.39, ak > 0 (k = 1, ... , m); let b, be positive n-tuples and Zk (k = 1, ... , m)

10.2. Various Results

301

be real n-tuples. Then the function

is convex and Sex) ~ f ( I : k ~ akAn(xZk> l bd) holds, where An(xzk> b k ) is the weighted arithmetic mean of the n-tuple XZk with weight vector b k . Let a and p be two positive n-tuples, and let f be a continuous and strictly monotonic function that is positive for all positive x and tends to 00 either as x ~0 or as x ~00. We write (10.52) If Pi ~ 1 (i = 1, ... ,n), we write In(a, p) instead. Of course, in the case P; = 1 we have a quasiarithmetic mean, i.e., In(a, p) == fn(a, p).

10.41. Theorem. Let f have continuous second derivatives and be strictly monotonic and convex.

r.r:

(a) If It!' is a convex function, i.e., if is decreasing, then In(a, p) is a convex function of a. (b) A necessary and sufficient condition that lea, p) is a convex function of a is that the function f is of the form (ax + b > 0, c> 1), or

eax + b •

na

10.42. Remark. The result in Theorem 1O.41(a) is proved in Vasic and Pecaric (1979a), and for Pi == 1 we obtain a result in Hardy, Littlewood, and P6lya (1934, 1952, pp. 85-88), where the result in Theorem 1O.41(b) is also proved. 0 In the following we give some similar results with symmetric functions and means. Let a be a positive n-tuple and k an integer satisfying 1 -s k :S: n. Then the kth elementary symmetric function of a is defined by

e ~ k J ( a= ) ~ ) a ~ l . . . a ~ n(= G ) p ~ k l ( a ) ) ,

(10.53)

where S(A) = {J.:Ai=O or 1 and I:7=lAi=k}, e ~ O ] ( a ) = p ~ O l ( a and ) = l , p ~ k l ( a is ) the kth symmetric mean of a. It is important to note that the

302

10. Orderings and Convexity-Preserving Transformations

symmetric functions and means can be generated as the following: (10.54)

or equivalently, (10.55)

10.43. Theorem. (a) If a is a positive n-tuple, then the sequence { p ~ k l } k : is ~ log-concave. (b) If a is a positive and log-convex sequence, then P ~ " . ! . 2 P ~ k l ~ ( p ~ k l l ) 2 for

1:s k :s n - 2.

(10.56)

(c) If r and v are integers such that 1:s r s; n - 1, O:s v:s k - 1, then ( ~ve!;-vl

?

~ ( ~ v e ! ; - v - 1 ) v(e!;-v+l». ~

(10.57)

10.44. Remark. Theorem 1O.43(a) is a old result (see Bullen, Mitrinovic, and Vasic, 1987, pp. 285-88), (b) is proved in Ozeki (1972) as a consequence of Theorem 10.9, and (c) can be found in Mitrinovic (1967) (see also Bullen, Mitrinovic, and Vasic, 1987, pp. 295-96). 0 Let a again be a positive n-tuple, k be an integer such that 1 :s k :s n, and .N* = {O, 1,2, ... }. Then the kth complete symmetric function of a is defined by

c ~ k l ( a =) ~ ) a ~ l ... a ~ n( = (n +: - 1 ) q ~ k l ( a » ) ,

(10.58)

where S(I.) = {I.:AiE.N*, I : 7 ~ l A i = k } c , ~ o J ( a ) = q ~ O l ( a ) and = l , q ~ k J ( a )is the kth complete symmetric mean of a. The complete symmetric functions can be generated similarly, i.e., (10.59)

10.45. Theorem. (a) If a is a positive n-tuple, then the sequence { q ~ k l } is log-convex. (b) If the sequence {ai } is log-convex, then (10.60)

10.2. Various Results

303

10.46. Remarks. (a) In Hardy, Littlewood, and P6lya (1934, 1952, p. 164) it is noted that Theorem 10.45(a) is due to Schur. This inequality has been cited in many papers (see Mitrinovic, Lackovic, and Stankovic, 1979, and the references given therein). Theorem 1O.45(b) was proved by Ozeki (1967) as a consequence of Theorem 10.10. (b) For some further generalizations of symmetric means and of Theorems 1O.43(a) and 1O.44(a), see Bullen, Mitrinovic, and Vasic (1987, 0 pp. 317-30). Ozeki (1973) proved the following result: Let Ak> Bi , C; (1 < k s; n) be defined as in Remark 6.18. Then the following inequalities are valid:

10.47. Theorem.

A'-l

+ A'+l 2: 2A, (n 2: r + 1),

(10.61) (10.62)

C,-l + C,+l

2:

2C,

(n

2:

r

+ 1).

(10.63)

Note that Theorem 10.47 is a simple consequence of Theorem 3.30. McLeod (1958/59) and Bullen and Marcus (1961) proved the following theorem as a generalization of an inequality of Marcus and Lopes (1957):

10.48. Theorem.

Let rand s (1 ::s; r ::s; s ::s; n) be integers.

is a concave function of a. 10.49. Remarks. (a) By assuming that C(a) = ( c ~ ] ( a ) / c ~ - p l ( a ) ) l(1::s; Ip p::S; r) is a convex function of a, McLeod (1958/59) proved his own conjecture for p = r. In Baston (1978) this conjecture was proved for p=r-l.

(b) A similar result for the well-known Dresher Inequality was proved by Godunova (1967a) (see Section 4.3). She proved this result by using Theorem 10.39 and the following result: Let A and B be sets in two linear spaces LA and L B , let x = Sea) and y = T(b) be real-valued functions which transform A and B into R, and Ry , respectively. Assume that x = Sea) is concave (convex) on A and y = T(b) is convex (concave) on

304

10. Orderings and Convexity-Preserving Transformations

B; and that M(x, y) is a concave (convex) function of two real variables,

defined on R;

X

R y, and is increasing in x when y is kept fixed. Then

M(S(a), T(b)) is concave (convex) on AxB. If S(a)=a or T(b)=b, then the monotonicity condition for M(x, y) in x or y is not necessary.

(c) In the same paper Godunova (1967a) proved the following result: ,Xn-I) is a concave (convex) function of n -1 variables for ,Xn-I ~0, then, for Xn > 0,

If f(xI' XI ~0,

is a concave (convex) function of (XI' ... ,xn ) . (d) Of course, Theorem 10.48 and Dresher's inequality are related to the well-known Minkowski Inequality. For many other related results, see Bullen, Mitrinovic, and Vasic (1987, especially pp. 267-268, 309, D 323-324, 330), and Pales (1982). Let LrCx, y) = (x r+ 1 + yr+I)/(x r + yr), then the following result is valid (Alzer, 1988). 10.50. Theorem. Let X and y be two positive real numbers such that X y, then the function L(r) = Lr(x, y) is strictly convex in ~_, strictly concave in ~+, strictly log-convex in (-00, -1/2], and strictly logconcave in [-1/2, +(0).

*"

Proof.

From L"(r)

= (xy)'(log(x/y))2(y -x)(xr -

we have L"(r) > 0 for r fer) = log Lr(x, y). Then r(r)

yr)/(x r + yr)3,

< 0 and L"(r) < 0 for r > O. Similarly, let

= (xy)'(log(x/y)?(y -x)(X2r+ l

_

y2r+I)/«x r + yr)(x r+ 1 + yr+l)f

holds. Thus r(r) > 0 for r < -1/2, and r(r) < 0 for r> -1/2. For arbitrary but fixed x, y (x F,.(x, y) = (r/(r

D

*" y) let us define

+ 1))(xr+ 1 -

F'o(x, y) = (x - y)/(logx -logy),

yr+I)/(x r - yr)

for

r ='= 0, -1,

F_ 1(x, y) =xy(logx -logy)/(x - y).

10.2. Various Results

305

(Note that Lr(x, y) = F,.(x 2, y2)1F,.(x, y).) Then the following results holds (Alzer, 1989b): 10.51. Theorem. The function F(r) = F,.(x, y) (x =1= y) is strictly logconvex in (-00, -1/2] and strictly log-concave in [-1/2, +(0). A similar result is also given in Alzer (1989b): Let

Then we have 10.52. Theorem. Let x, y, u, and v be positive real numbers such that max(x, y)/min(x, y) > max(u, v)/min(u, v). Then the functions SlIlx, y)/SlIr(u, v) is strictly convex for r E IR+.

10.53. Remark. There exist many other results which can be included in this section. See, for example, Lupas and Muller (1967,1970), Ibragimovand Gadziev (1970), Wood (1971), and Ross (1978,1980) (see also Riekstyn's (1986, pp. 22, 25-26). D

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Chapter 11 Convex Functions and Geometric Inequalities

In this chapter we give some results concerning convex functions and geometric inequalities, mainly from Mitrinovic, Pecaric, Tanasescu, and Volenec (1988) and Tanasescu (1989). Note that the recent book of Mitrinovic, Pecaric, and Volenec (1988) contains a chapter with the same title, and many useful applications of convex functions to geometric inequalities are also given there. The theory of majorization is closely related to inequalities based on convex functions; it was treated comprehensively by Marshall and Olkin (1979), and will be reviewed in Chapter 12. Some recent related results concerning majorization and geometric inequalities can be found in Mitrinovic, Pecaric, Tanasescu, and Volenec (1988), and results in Tanasescu (1990) concern the concavity property of some hyperbolic forms. Those results can provide an answer to some geometric problems and their generalizations, and will be treated in this chapter.

11.1.

Old and New Results via Majorization Theory

11.1. 1. A Partial Ordering of Triangles

Let il(s) denote the set of all metrically distinct triangles with given semi-perimeter s > 0 (degeneracy allowed). An irredundant abstract representation of ~ ( s ) is immediate: il(s) = {T = (x, y, z): s

2:

x

2: Y 2: Z 2:

307

0,

X

+Y + z

=

2s}.

308

11. Convex Functions and Geometric Inequalities

Of course, a vector T = (x, y, z) uniquely determines a triangle with (ordered) sides x ~y ~ z, thus for convenience we shall use "T = (x,y,z)" to mean a triangle, where x ~ y ~ are z the lengths of the triangle T.

11.1. Definition. The set of triangles ~ ( s ) is a partially ordered set (denoted by p.o. set) with the partially ordering relation T=(x,y,z)-
iff x:Sx' and

z ze z "; (11.1)

that is, the vector Tis majorized by the vector T' (see Definition 12.1). Clearly, ( ~ ( s ) ,-c) has a "1" and a "0"; that is, letting T 1 = (s, s, 0) and To = ('ls13, 'ls13, 'ls13), we have To -< T <, T 1 for every T E ~ ( s ) . The most important sup-p.o. sets are the chains, i.e., totally (linearly) ordered sub-p.o. sets. (Here we use the language of ordered structures.) As a chain is a special kind of interval, we may abuse the notation a little and write L = [T', Til], which means that T', Til E ~ ( s ) , T' -< Til, are the extreme elements of the chain L. In fact, a chain in ~ ( s ) is a one-free-parameter family of triangles. (a) Apart from To and T 1 , we distinguish the following elements in (see Fig. 11.2):

~ ( s)

Tz = (\12 - 1)(2,2, \12 - l)s,

T3 = (\12 - 1)(2,

\12, W)s,

T= ( 1 , ~ , ~ ) s . 4

(b) Distinguished chains: (i) L 1 = {(x, x, 2s - 2x):x E [2(W -l)s, s]),

(ii) (iii) (iv) (v) (vi)

L z = {(x, x, 2s - 2x):x E [2s13, 2(\12 -l)s]), L 3 = {(x, s -x/2, s -x/2):x E [2s13, 2(\12 -l)s]), L 4 = {(x, S - x12, s - x/2):x E [2(\12 - l)s, s]), L s={(s,sI2+t,sI2-t):tE[0,sI2]}, and L 6 = {(x, S - xl2 + (x Z/4 + sx - SZ)lIZ, S - xl2 - (x z/4 + sx x E [2(W -l)s, s]).

(c) Distinguished sub-p.o. sets: (i) (ii)

~ a ( s ) : properly ~ a ( s )= ~ a ( s )U

acute triangles, L 6 : nonobtuse triangles; To -< T -< T 1 ,

SZ)1!z:

11.1. Old and N e w Results via Majorization Theory

:

Figure 11.1

X

Figure 11.2

309

310

11. Convex Functions and Geometric Inequalities

(iii) L\o(s): properly obtuse triangles, (iv) L\o(s) = L\o(s) U L s U L 6 : nonacute triangles; T 3 < T < T 1 , (v) L 6 = [T3 , Ttl: right (angled) triangles, L 6 \ {T1 } : properly right triangles, (vi) L 1 U t., U L 3 U L 4 : isosceles triangles, L 1 U t., U L 3 U L 4 \ {T4, TI}: nondegenerate isosceles, (vii)L s: degenerate triangles. In the diagram of the p.o. set (L\(s), <) given in Fig. 11.1, we can easily "read" any particular chain and represent it graphically by an oriented path. Apart from the above described chains L 1 , ••• , L6 , of special interest are the level lines, i.e., chains of triangles with maximal side of constant length. Through each T E L\(s) "passes" a unique level line, i.e.,

LT = {(XT,y, z)}, T=(XT,YT,ZT)' In the diagram in Fig. 11.1, L T appears symbolically as a line "parallel" to Ls , and this represenation is suggestive enough and plainly justified. But it is interesting to notice that any level line is actually a straight line parallel to T 1T4 in the geometric model of L\(s) in 1R 3 (see Fig. 11.2). In fact, this model is simply the graphic representation of the vectors T = (x, y, z) E L\(s) on a tridimensional Euclidean space and consists of the right triangle T 1T4To , where T 1 = (s, s, 0), To = (lsI3, ls13, lsI3), and T 4 = (s, s12, sI2). The plane x = XT cuts T 1T4To along a segment parallel to T 1T4 , which is precisely the representation of LT' The ordering on a level line is indicated by the arrow (see Fig. 11.1). On the other hand, L\T1T4T3 is a compact space topologized relative to the standard Euclidean topology of 1R 3 on T 1T4T3 . This makes significant the later remark that, geometrically speaking, the bounds at the extreme points, or boundary points, for continuous functionals on T 1T4T3 are the best. This is so because L 1 , L z, L 3 , L 6 , L 4 , L, actually belong to the topologic closure of L\a(s) and L\o(s), respectively. 11.1.2.

Schur-Convex and Schur-Concave Functionals

The classic definition of Schur-convex (concave) functionals and the classic results also hold for (L\(s), <); e.g., using Schur's characterization theorem (1923), we can easily show that F = (s(s - x)(s - y)(s -

Z»l/Z

(x, y, z are the sides),

11.1. Old and New Results via Majorization Theory

311

and r = Fs- 1 are Schur-concave on the whole set Ll(s), strongly Schurconcave on L 3 U L 4 , and vanish on L«, Thus the maximal points on Llo(s) and Lla(s) are given uniquely by T 3 and To, respectively. But the circumradius R

= xyz/4F = xyz/4(s(s - x )(s - y)(s -

Z))1I2

is Schur-convex only on Lla(S) (the value at T 1 is defined by continuity). It is interesting to note that on Llo(s)\Ls R is neither Schur-convex nor Schur-concave; although it is strictly decreasing on the intersection of each level line within the domain, and strictly increasing on L 4 and L 6 • Indeed, these properties are simple consequences of elementary calculations performed on the real-valued one-variable functions. It follows that R(T) ~R(T3 ) holds on Llo(s)\Ls, i.e., R ~ s ( v ' -2 1)

and R(T)

(11.2)

on the set {T3},

on Llis), i.e.,

~ R ( T o R(T) ) , =:;R(T 1 )

2 R ~ 3 0 s on the set {To}, R =:;s/2

(11.3)

(11.4)

on {T 1 } .

From the previous discussion concerning F and r, we have F(T 1) -s F(T) =:; F(T o)

and

F(T 1 ) =:; F(T) =:; F(T 3 )

r(T 1) =:; r(T) =:; r(To)

and

r(T 1) -s r(T)

=:; r(T3 )

on

Llis),

on Llo(s).

Consequently we have

y'3

r =:; - s 9

(11.5)

on Lla(S) or {To},

r =:; (3 - 2v'2)s

y'3

on Llo(S) or {T3 },

(11.6)

-

F =:; 9 S2 on Lla(S) or {To}, F =:; (3 - 2v'2)S2

on Llo(S) or {T3}'

It is straightforward to see that the functional 4Rrs Schur-concave (in (x, y, z I). Thus we have 2s 2 Rr=:;27

(11.7)

on Llis)

or {To},

(11.8)

= xyz is

strictly

(11.9)

312

11. Convex Functions and Geometric Inequalities

and (the values on L s are defined by continuity): Rr ~ (50 -7)S2 on L\o(s) or {T3 } .

(11.10)

To give another example, consider the functional f(T) = f(x, y, z) = LXy==xy + xz + yz, which is strongly Schur-concave on L\(s). We have f(T1 )

~ f ( T )~ f ( T o )

and

f(T1)

~ f ( T )~ f ( T 3 )

on L\o(s).

Consequently we obtain the following inequalities: S2

S2

4 S2 3

~Lxy ~-

on L\a(s)

~LXy ~ 2(40 - 5)S2

g(x, y, z)

or {TI , To}

(11.11)

on L\o(s) or {T I , T 3 } .

= (LX2)/(LXY)= 4s2/(LxY)-

2,

(11.11) and (11.12) yield Groenman's inequalities: 1 ~ (Lx 2)/(Lxy)::;'2 on L\is) or {To, T ( 8 0 - 4 ) / 7 ~ ( L X 2 ) / ( L xonL\o(s) y ) ~ 2

(11.12)

I },

or {T3,T1} .

(11.13) (11.14)

Suppose that a certain functional f (on L\(s» can be written in the form f(T)=f(x,y,z)= CP(s,x,y, z)=cP (

X

+ 2y + z ,x,y,Z. )

Then, for verifying the Schur's conditions, we need to evaluate (X_ y)( of _ of) = (x _ y)(! ocp + ocp _! ocp _ oCP) ox oy 2 os ox 2 os oy = (x _ y)(ocp _ oCP), ox oy and f is Schur-convex (Schur-concave) iff (x, y, z) and

cP is permutation invariant in

(X_ y )( 04> _ oCP ) ! 2=0 ox oy s=(x+y+z)/2

(~O)

(11.15)

on I R ~ ,where the partial derivatives are taken by treating s as a constant, and we let s = (x + y + z)/2 only afterward (if necessary) to determine the sign in (11.15).

11.1. Old and New Results via Majorization Theory

313

Another simplification arises if the function f can be written in the form

f(x, y, z) = g(u, v, w), whereu=s-x, v=s-y, andw=s-z. Since of ox

og OU

og ov

og ow'

2-=--+-+-

2 of = og _ og

oy

ou

ov

+ og for u - x = y ow

- x,

we find

(x _ y)( of _ of) = (u _ v)(Og _ Og). ox oy ou ov Thus f is Schur-convex (Schur-concave) on ~ ~ iff g is Schur-convex (Schur-concave) . In the following we provide some examples for the purpose of illustration. Since the function g(u, v, w) = VIi + VV + YW is Schurconcave, we have, for LVs - x defined similarly, y'S -s LYs

-.x; ~ V3S

on ilaCs) or {T 1 , To}

y'S ~ '2:.Ys - x ~ (0 - 1 + 2Y0 - 1)y'S

on ilo(s) or {T 1 , T 3 }

Furthermore, the function g(u, v, w) = u- 1 + V-I Thus

1 s-x

(11.16) (11.17')

+ w-1 is Schur-convex.

9 on ilaCs) \ {T 1 } or {To}, s

' 2 : . - - ~ -

(11.17) (11.18)

Note that in many examples f(To) lies surprisingly close to f(T3 ) . However, we should be cautioned not to conclude that this is always the case, as shown in the prior example. We conclude this section by considering a classic problem of finding max f(T),

where f(T) = min{lx - yl, Iy - z], [z -xl}·

T E ~ ( S )

Although this is often encountered as an extreme value problem in real numbers or in the class of (R, r)-triangles, or R-triangles, it is most suitably formulated as a maximization problem in il(s). The function f vanishes on L 1 U L; U L 3 U L 4 , and it is neither Schur-convex nor

314

11. Convex Functions and Geometric Inequalities

Schur-concave on ~ ( s ) . However, on every level line L T , first it is increasing, then decreasing, always having its maximum value at y = 2s/3, attainable only when x - y = y - z. Indeed, on ~ ( s ) f can be simplified as f(T) = min{x - y, y - z} and, on each LT , x - Y is Schurconcave, while y - z and x - z are each Schur-convex. Thus Maxf = (x - z )/2 on each level line, and x - z is Schur-convex on the whole set ~ ( s ) . It then follows that Max minj]» - YI, Iy - z], [z - xl} = s/3,

(11.19)

attained uniquely at T= (s, 2s/3, s/3) (an old but beautiful result). Moreover, for T E ~ a ( s ) we have Maxf(T) = 5/6, attained uniquely at T = (5/6)(5, 4, 3), a nice right triangle. 11.2.

Concavity via Hyperbolic Forms

11.2. Definition. A real n-ary form is a homogeneous polynomial in n indeterminates with real coefficients. A real form P of degree m is said to be hyperbolic with respect to some a E IRn (or a-hyperbolic) if the equation in t E IR P(ta + x) = 0

(11.20)

has previsely m real roots for each given real vector x E IRn. Let us denote these roots by tl(a, x):s; ... :s; tm(a, x) (for fixed a they are all continuous in x). Let 'Jtp be the set of vectors a E IR n such that Pis a-hyperbolic. Note that for every a E 'Jtp the function Ha(x) = maXlsksm tk(a, x) and ha(x) = minlsksm tk(a, x) are continuous, positive, and homogeneous on IR n with H a ( -x) = -ha(x). Also note that the open cone CCP, a) = {b E IR n: Ha(b) < O} is the desired neighborhood of a in 'Jtp that is an open subset of IRn. Moreover, this cone is convex. (More results on hyperbolic form can be found in Garding, 1959, and Tanasescu, 1990). Throughout this section it will be assumed P is a hyperbolic form of degree m ~ 2 and LP = <jJ, where LP is the set of all common zeros of the partial derivatives of P of order m - 1. Furthermore, let us assume that the cone C = CCP, a) £; 'Jtp satisfies a E IR':-+ and pea) > 0 (hence P>O on C). Now let K be a convex cone included in the nonnegative orthant IR':-. For every s > 0 let K denote the set of all x' = { X : } 7 ~ 1such that x = { X ; } 7 ~ E1 K. Of course, under the prior assumptions we have K S £; S

11.2. Concavity via Hyperbolic Forms ~ ~ . Take

315

p = lIs and define on K the positively homogeneous function

1';, by (11.21)

Clearly, we have tacitly assumed that KlIp ~ C = C(P, a). In the following theorem P, and ~ j denote partial derivatives of the first and second order, respectively.

11.3. Theorem. Let K some real number p 2:: 1.

~ ~ :

be a convex cone such that Kl1p

~C

for

(a) If P; 2:: 0 on Kl/p n {Xi> O} for every i = 1, ... , n, then the function 1';, defined in (11.21) is strongly concave on K. (b) If ~ j 2:: 0 on KlIp n {Xi> 0, Xj > O} for 1:s; i <j:s; nand p < m, then the same conclusion as in (a) holds.

11.4. Theorem. If 1';, is strongly concave on some convex cone K c ~ : , then it is strongly concave on cl(K)+ (the closure of K) except for the trivial case 1';,(x) = 1';,(y) = 1';,(x + y) = 0 for x, y E cl(K). The following result is a consequence of Theorems 11.3 and 11.4.

11.5. Theorem. If T I = (Xl, xz, X3) and T z = (YI, Yz, Y3) are two triangles with areas Al and A z, respectively, then for each p 2:: 1 there exists another triangle with sides (xf + yf)lIP , i = 1, 2, 3, and with area A, such that: (a) If 1 :s;p :s; 4, then (11.22)

and equality holds iff (i) the triangles are similar or (ii) they are both degenerate, with the longest side on the same position (in (ii) it occurs only for p = 1); (b) if both triangles are nonobtuse, then the inequality in (11.22) holds for all p 2:: 1, and equality holds iffthe triangles are similar.

11.6. Remarks. (a) Theorem 11.5 was first stated as a conjecture by Oppenheim (1971), and later proved by him (Oppenheim, 1974) for p = 1, 2, 4 only (which is part (a». Carroll (1982) gave a complete proof for both parts of the theorem.

316

11. Convex Functions and Geometric Inequalities

(b) In fact, the inequality (11.22) is a consequence of Jensen's inequality for n = 2. Using the weighted Jensen's inequality, we can obtain known results for n triangles (see Mitrinovic and Pecaric, 1988a). (c) If a hyperbolic form P in Theorems 11.3 and 11.4 is also symmetric, then the function F;,(x) defined in (11.21) is also Schurconcave on K. This is a generalization of a related result in Mitrinovic and Pecaric (1988a). (Note that the results in their paper are given in Mitrinovic, Pecaric, and Volenec, 1988.) 0 Theorem 11.5 may be nicely generalized in an obvious manner by letting (for n 2= 2)

k= 1, ... , n,

P=Tt···Tn

and

Q = ToP.

Note that the forms P, Q are positive in IR':- whenever the Tk's are. Thus it seems natural to define C = {x E 1Il':-: P(x»O} = C(P, In)

n IR':- = C(Q, In) n IR':-.

where In = (1, ... , 1). This follows from the obvious fact that P, Q are In-hyperbolic and Tk(ln) > 0 for k = 0, 1, ... , n. Theorem 11.3(a) assures that F;" Gp are strongly concave on K p = CP for every p 2= 1, where

Consequently, the following theorem is valid.

11.7. Theorem.

(a) If 1:s p :s n, then

F;,(x + y) 2= Fp(x) + F;,(y)

for every x, y E cl(Kp),

(11.24)

and equality holds iffeither x, yare proportional, or p = 1, FI(x) = FI(y) = F1(x + y) = 0, and the largest component ofx, yare similarly placed. (b) If1:sp:sn+1, then Gp(x

+ y) 2= Gp(x) + Gp(y) for every x, y E cl(Kp),

(11.25)

and equality holds under the same conditions as in (a).

For n = 4, each point x = (Xl' Xz, X3' X4) can be considered as representing an inscribable quadrilateral of sides Xl,"" X4 with area Z A(XI' xz, X3, X4) = 2- (p (X)) lIZ, where P = T; T Z ~ 4 and x E C =

11.2. Concavity via Hyperbolic

FOnDS

317

C(P, 14 ) , Thus Theorem 11.6(a) yields a perfect analog of Theorem 11.5. This is precisely what the third theorem in Carroll (1982) says (also see Theorem 0 in Mitrinovic and Pecaric, 1988a). Theorems 11.3 and 11.4 give many other nongeometric applications. For example, Tanasescu (1989) used them to prove Minkowski's inequality, Bellman's inequality (the discrete case of Theorem 4.29), and a generalization of Theorem 10.48. A geometric implication of Theorem 10.48 is:

11.8. Theorem. If T 1 = (Xl, X2, X3) and T 2 = (Y1' Y2,Y3) are two arbitrary triangles, degenerate or not, with radii R 1, r1 and R 2, r2, respectively, then for every p ~ 1 there exists a triangle T with sides z, = (xf + yf)1/p for

i = 1,2,3,

and radii R, r such that

(11.26) Furthermore, equality holds iffthe triangles T 1 , T 2 (even if degenerate) are similar. .

11.9. Remark. Note that Remarks 11.6(b), (c) are also valid for Theorems 11.7 and 11.8. D

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Chapter l2

Convexity, Majorization, and SchurConvexity

In this chapter we describe some results on convexity, majorization, and Schur-convexity (Schur-concavity). The notion of majorization arose as a measure of the diversity of the components of an n-dimensional vector (an n-tuple) and is closely related to convexity. It is formally discussed in Hardy, Littlewood, and P6lya (1934, 1952) and is treated most comprehensively by Marshall and Olkin (1979). In this chapter we shall restrict our attention to results in majorization and Schur-convexity theory that directly involve convex functions. Most of the results are already given in Marshall and Olkin (1979); thus to avoid duplication of effort we refer to their book for most of the proofs.

12.1.

Majorization and Convex Functions

For fixed n

I2

let x=(x1,.

* *

>X,),

y = ( y , , ... ,y,)

(12.1)

. . . 2&,I, Y(1)5 Y ( 2 ) 5 . . S Y ( , )

(12.2)

denote two n-tuples. Let X [ l l ? X[2] 2 X(1) S X ( 2 ) 5

a

-

a

*

?X[n]

Y [ l ]2 Y [ 2 ] 2

9

SX(n),

*

(12.3)

be their ordered components.

121.. DeJinifion. y is said to majorize x (or x is said to be majorized by y ) , in symbols y > x, if m

m

i=l

i=l

2 x [ ; I ~ y[;l

holds for rn = 1, 2, . . . , n - 1

319

(12.4)

320

U. Convexity, Majorization, and Schur-Convexity

and n

n

2: Xi = i=l 2: Yi' i=l

(12.5)

Note that (12.4) is equivalent to n

2: i=n-m+l

n

XU)::5

2:

holds for

Yu)

m

= 1, 2, ... , n -

1.

(12.6)

i=n-m+l

This definition provides a partial ordering, namely, y > x implies that, for a fixed sum, the y;'s are more diverse than the x;'s. To illustrate this point, we see that (i) y > y always holds for y = (Y, ... ,y), where y = (lin) I:7=1 Yi' and (ii) ( I : 7 ~ Y 1 i, 0, ... , 0) > Y holds for all y such that Yi ~ 0 (i = 1, ... , n). This notion is closely related to convex functions as shown in the following theorem. 12.2. Theorem. Let I be an interval in such that Xi' YiE I (i = 1, ... ,n). Then n

~ ,

and let x, y be two n-tuples

n

2: cp(x;}::5 2: CP(Yi) i=l i ~ l

holds for every continuous convex function cp: I -

~

iff Y > x holds.

12.3. Remark. Theorem 12.2 is well-known as the majorization theorem, and a convenient reference for its proof is Marshall and Olkin (1979, p. 11). It is due to Hardy, Littlewood, and P6lya (1934, 1952, p. 75), and can also be found in Karamata (1932); for a discussion 0 concerning the matter of priority see Mitrinovic (1970, p. 169). The partial ordering of majorization defined in Definition 12.1 assumes that the sums of the components of x and yare the same. When this condition is replaced by a weaker one, we have the notions of weak submajorization and weak supermajorization (see, e.g., Marshall and Olkin, 1979, p. 10):

12.4. Definition. (a) y is said to weakly submajorize x, in symbols Y>wx, if (12.4) and n

n

;=1

;=1

2: Xi::5 2: Yi

(12.7)

12.1. Majorization and Convex Functions

321

hold. (b) Yis said to weakly supermajorize x, in symbols y>w x, if m

m

i=1

i=1

2: xU) ~ 2: Y(i)

for

m

= 1, 2, ... , n

- 1

(12.8)

and n

n

2: Xi ~ i=1 2: Yi i=1

(12.9)

hold. It is known that (see, e.g., Marshall and Olkin, 1979, p. 11, p. 22):

12.5. Fact. (a) Ify > x holds, then there exist a finite number of n-tuples Z1 , •.. , ZN such that y

= Z1 > Z2 > ... ZN-1 > ZN = x,

(12.10)

and that for all j, Zj and Zj+1 differ in two coordinates only.

(b) y > x holds iff there exists a doubly stochastic matrix Q = (qiJ (i.e., qij ~ 0 and Li qij = 1 for all j, l: j qij = 1 for all i) such that x = yQ.

12.6. Fact. (a) y >w x holds iff there exists an n-tuple Z such that y > Z and z e x (i.e., z, ~ x d o ir = 1, ... , n) hold. (b) y > W x holds iff there exists an n-tuple Z such that Z > x and Z ~ y. In view of Fact 12.5(a) we can prove Theorem 12.2 simply by proving it holds for n = 2. Similarly, by combining Theorem 12.2 and Facts 12.5(a) and 12.6 we can obtain (see e.g., Marshall and Olkin, 1979, p. 10):

12.7. Theorem. Let I be an interval in ~ , and let x, y be two n-tuples such that Xi, YiE I (i = 1, ... ,n). Then (a) n

n

2: cf>(Xi) :5 2: cf>(Yi) i=1

(12.11)

i ~ 1

holds for every continuous increasing convex function cf> iff Y> w x holds; and (b) the inequality in (12.11) holds for every continuous decreasing convex function cf> iff Y>W x holds.

12.8. Remark. Theorem 12.7 is due to Weyl (1949) and Tomic (1949). Weyl obtained the result by assuming Xi> 0 and Yi> 0 (i = 1, ... , n),

322

12. Convexity, Majorization, and Schnr-Convexity

whereas Tomic did not assume this condition. Earlier Polya (1947) proved Theorem 12.7 by using Theorem 12.2 for the special case I = IR, and a similar proof for an arbitrary interval is given in Mirsky (1959). In fact, Mirsky proved a more general result which includes Theorem 12.7 as a special case. 0 In the following we discuss related results for Wright-convex functions and for functions with increasing increments. In Theorems 12.9, 12.11, and 12.12 we assume that x, yare n-tuples such that Xi' Yi E I (i = 1, ... , n) for some interval I.c: IR.

12.9. Theorem. For k that Xk + Ck E I for all k.

= 2, ...

,n let Ck

= I:7,;:l

(Xi - Yi), and assume

(a) If Yk:s Xk+l for k

= 1, ...

, n - 1,

(12.12)

k

2: Xi;:::: 2: Yi i ~ 1

k

for

i=1

k = 1, ... , n - 1,

(12.13)

and n

n

2: Xi = i=1 2: Yi'

(12.14)

i ~ 1

then (12.11) holds for every Wright-convex function cf> :1- IR. Furthermore, (12.11) holds for every Wright-convex function cf> if the reverse of the inequalities in (12.12) and (12.13) hold.

(b) if (12.12) and (12.14) hold and the reverse of the inequality in (12.13) holds, then the reverse of the inequality in (12.11) holds for every Wright-convex function cf>: I -IR. Furthermore, the same is true if (12.13) and (12.14) hold and the reverse of the inequality in (12.12) holds.

Proof. The proof follows by changing X2k-l to x; and X2k to Yk (k = 1, ... , n) in Theorem 5.46 and the fact that z;:::: maxl05i05n {Xi' Yi}, where z is the value of X2m+l in (5.64). 0 12.10. Remark. The following result is a special case of Theorem 12.9: Assume that a l 2: a3;:::: ... ;:::: a 2n+l' a2k;:::: 0 (k = 1, ... , n), al, a2k+l E I,

12.1. Majorization and Convex Functions

ak + ak+l E I (k f/>(al

= 1, ... , 2n).

323

Then we have

+ a2) + ... + f/>(a2n-l + a2n) + f/>(a2n+l) ~ f/>(al) + f/>(a2 + a3) + ... + f/>(a2n + a2n+l)

for all Wright-convex functions f/>: I

0

~ ~ .

The following theorem is a weighted version of Theorem 12.2. It can be regarded as a generalization of the majorization theorem in Theorem 12.2 and is given in Fuchs (1947).

12.11. Theorem. Let x, y be two decreasing n-tuples, and let p = (PI' ... ,Pn) be a real n-tuple such that k

k

i=1

i=1

2- PiXi:=; 2- PiYi

for

k

n

n

i±1

i=1

= 1, ... , n -

1;

2- p.x, = 2- PiYi'

Then for every continuous convex function f/>: I n

n

i=1

i=1

(12.15)

(12.16) ~ ~ ,

we have

2- Pif/>(X;) :=; 2- Pif/>(Yi)'

(12.17)

Similarly, a weighted version of Theorem 12.9 is given in Bullen, Vasic, and Stankovic (1973):

12.12. Theorem. n-tuple. If k

Let x, y be two decreasing n-tuples, and p be a real k

2- p.x, :=; 2- PIYi i ~ 1

for

k

= 1, ... , n -

1, n

(12.18)

i=1

holds, then (12.17) holds for every continuous increasing convex function f/> :I ~ ~ . If x, yare increasing n-tuples and the reverse inequality in (12.18) holds, then (12.17) holds for every decreasing convex function f/> : I ~ ~ . Proof of Theorems 12.11 and 12.12. Without loss of generality, assume that Xi =1= Yi' and define d, = (f/>(Xi) - f/>(y;))/(x; - Y;) (i = 1, ... , n). Then

324

12. Convexity, Majorization, and Schur-Convexity

from (12.7) we have d,

n

n

~ di+1

for i:5 n - 1, and the proof follows from

n

2.: P;CP(Xi) - ;=1 2.: PiCP(Yi) = ;=1 2.: Pi(Xi ;=1

Yi)di n-I

= (Xn - Yn)dn +

2.:

(Xk

-

Yd(dk

-

dk+I),

k=1

12.13. Remarks. (a) In Mitrinovic (1970, p. 165), it is stated that (12.15) and (12.16) hold iff (12.17) holds. Imoru (1974) presented what he claimed was a proof for this statement, but Cheng (1977) and Pecaric (1980a) independently showed, by a counterexample, that (12.15) and (12.16) are sufficient, but not necessary, for (12.17) to hold. Furthermore, they showed that (12.15) and (12.16) become necessary (and (12.18) is necessary in Theorem 12.12) when the components of p are all nonnegative. (b) Theorem 12.11 can be used to prove a number of inequalities for convex functions, including Petrovic's inequality (see Remark 5.38(e» and the Jensen-Steffensen Inequality and its reverse inequality (see Peearic, 1981c). 0 The definition of majorization stated in Definition 12.1. involves the comparison of the diversity of the components of two n-tuples. In the following we state a similar definition for integrable functions. Let x(t), y(t) be real-valued functions defined on an interval [a, b] such that f ~ x ( t dt, ) f ~ Y ( td ) t both exist for all s E [a, b].

12.14. Definition. y(t) is said to majorize x(t), in symbols, y(t) > x(t), for t E [a, b], if they are decreasing in t E [a, b] and s

s

J x(t) dt:5 Jy(t) dt o

0

and equality in (12.19) holds for s

= b.

for s

E

[a, b],

(12.19)

12.1. Majorization and Convex Functions

325

An integral analog of Theorem 12.2 is the following: 12.15. Theorem. y(t) > x(t) for t [a,b] and b

f

[a, b] iff they are decreasing in

E

b


a

f

(12.20)


a

holds for every
An inequality that is related to the majorization theorem is given in Fan and Lorentz (1959) (see, e.g., Mitrinovic, 1970, p. 304, or Roberts and Varberg, 1973, pp. 157-58). In the following we state a more general form of that result (Pecaric, 1981a and 1981c). 12.16. Theorem. for (t; Ul, ... , un) k, His

Let H(t; Ul' ... , un) be a real-valued function defined E [a, b] X [al , btl X . . . X [an, b n] such that, for fixed

(i) a (1, I)-convex function of (Ui' uj) for 1:5 i, j:5 k and k + 1:5 i, j:5 n when the remaining variables are kept fixed; (ii) a (1, I)-concave function of (Ui' Uj) for 1:5 i s: k and k + 1:5 j :5 n; (iii) a (1, I)-concave function of (t, Ui) for 1:5 i z: k and a (1, I)-convex function of (t, uj) for k + 1 :5 j :5 n; (iv) a convex function of u, for 1 :5 i :5 k. Let Ii, gi : [a, b] ~[ai' bJ (i = 1, ... , n) be real-valued continuous functions which are decreasing for 1:5 i :5 k and increasing for k + 1 :5 i :5 n; and let G : [a, b] ~ ~ be a function of bounded variation.

(a) If x

f

x

Ii(t) dG(t):5

a

gi(t) dG(t)

for all x

E

[a, b]

1:5 i :5 k,

and

a

b

f

f

(12.21)

b

Ii(t) dG(t):5

x

f

gJt) dG(t)

for all x

E

[a, b]

and

k

+ 1:5 i :5 n,

x

f b

a

f

(12.22)

b

h(t) dG(t):5

a

gi(t) dG(t)

for all

1:5 i:5 n,

(12.23)

326

12. Convexity, Majorization, and Schur-Convexity

then b

b

JH(t;fI(t), ... ,fn(t» dG(t):::; JH(t;gl(t), . . . ,gn(t» dG(t). a

(12.24)

a

(b) If H satisfies (i)-(iv), is an increasing function of u, (1:::; i :::; n), and x

x

J/;(t) dG(t):::; J gi(t) dG(t) a

for all x

E

[a, b]

1:::;i :::; k,

and

a

b

(12.25)

b

J/;(t) dG(t):::; Jgi(t) dG(t)

for all x

E

[a, b]

and

+ 1:::; i :::; n,

k

x

x

(12.26) then (12.24) is also valid.

Proof. Since H is continuous, it can be uniformly approximated by Bernstein's polynomials. Furthermore, if H satisfies the conditions of Theorem 12.16, then the polynomials satisfy the conditions

a2 H

--:::;0 for

l:::;i:::;k

ataui

a2H aUi su, a2 H - - :::; 0

- - 2: 0 for for

aUi Bu,

and

a2 H

--2:0 for ataui

k+1:::;i:::;n,

1:::; i, j :::; k

and

k

+ 1 :::; i, j

1:::; i :::; k

and

k

+ 1 -s i :::; n;

:::; n,

which follows by repetitively applying the formula

-dxd 2: (n). . aix'(l- xt-'.= n 2:i ~ O (n -. 1) (ai+l n-l

n

i=O

l

.

..

a;)x'(l - xt-1 -

,.

l

Thus, without loss of generality, we can assume that the partial second derivatives of H exist and satisfy those conditions. Let us define ui(t, A)

= Agi(t) + (1 -

H(t;

U 1, . . . ,

A)/;{t) for' 1:::; i :::; n,

un) = H(t, u(t, A»,

b

F(A) =

J H(t, u(t, A» dG(t), a

A E [0,1]

12.1. Majorization and Convex Functions

Since u;(t, A) are decreasing in t for i = k + 1, ... , n, we have

E

[a, b] for i

aH(t'a:;t,

A» dG(t)

b

r(A)

J(g;(t) - /;(t»

= ~1

= 1, ...

, k and increasing

a k

b

= ~aH(b'a:;b,

A» J (g;(t) - h(t» dG(t) a

stu», neb, A» J b

k

= ~

au;

(g;(x) - h(x» dG(x)

a k

b

t

- ~J(J (g;(x) - h(X» dG(x») a a

2

H ~ ~; ~ ' A»

dt

a

b

+

; ~ ~ + 1a H(a'a:;a,

A» J(g;(x) -

h e x » d~ G(x)

a

b

+

b

i J(J (g;(x) -

;=k+l

a

t

b

+

2H(t, net, at au;

A» dt

b

; = ~ + 1 j ~ J(J (g;(x) - h(x» dG(x») a 2 H ~ ~ ~ A» ~ , dUj(t, A) a

:2: O.

h e x » d~ G(x») a

t

327

'\.- J

328

12. Convexity, Majorization, and Schur-Convexity

Therefore F(A) is increasing on [0,1], and from the inequality F(O)::::; F(l) we obtain (12.24). This completes the proof under the conditions in (a). The proof under conditions in (b) is similar.

0

The following result is a simple consequence of Theorem 12.14 (Pecaric, 1984b): 12.17. Theorem. Let x(t) = (XI(t), ... ,Xk(t» and y(t)=(YI(t), ... ,Yk(t» denote the mapping from the real interval [a, b] into an interval I in the k-dimensional Euclidean space IRk such that the components of x(t) and y(t) are continuous and increasing, and let G:[a, b ] ~ 1 be R a function of bounded variation. (a) If b

b

J Xi(t) dG(t)::::; J yJt) dG(t) x

for all x

E

[a, b],

(12.27)

x

and b

b

J Xi(t) dG(t) = J Yi(t) dG(t) a

(12.28)

a

hold for i = 1, ... , k, then for every continuous function f with increasing increments on I we have b

b

Jf(x(t» dG(t)::::; Jf(y(t» dG(t). a

(12.29)

a

(b) If (12.27) holds for i = 1, ... ,k, then (12.29) holds for every continuous increasing function f with increasing increments on I.

12.18. Remarks. (a) Assume that in Theorem 12.17 G is an increasing function. Then (12.29) holds for every continuous function f with increasing increments on I iff (12.28) holds. Similarly, (12.29) holds for every continuous increasing function f with increasing increments on I iff (12.27) holds for i = 1, ,k. If (12.29) holds for f(x) = Xi and f(x) = -Xi for each i = 1, ,k and x = (Xl, ... ,Xk), then (12.28) holds. Furthermore, let c" = max{O, c} and for u E [a, b] let f(x) = (Xi Yi(U»+ for a given i. Then f(x) is continuous with increasing increments

12.1. Majorization and Convex Functions

on I, and f(x)?: 0, f(x) ?: X; b

f

b

x;(t) dG(t) - y;(u)

u

f

-

y;(u). Since G is increasing, we have b

f =f

dG(t)::;

u

329

b

f(x(t)) dG(t)::;

a

f

f(y(t)) dG(t)

a

b

f b

y;(t) dG(t) - y;(u)

u

dG(t),

u

and thus we obtain (12.27). Similarly, we can prove a result when f is increasing. For k = 1, we then have the known result for convex functions. (b) In the proof of Theorem 12.15 we actually proved more general results such that the function F(A) is increasing. Therefore, for 0::; u ::; v::; 1 we obtain the following refinement of the result in (12.24): F(O)::; F(u)::; F(v)::; F(1).

(12.30)

In a similar fashion we can obtain a corresponding refinement of the result in (12.29). (c) As a special case of Remark (b) we obtain the following result: Let f, p, and x satisfy the conditions of Theorem 2.19; then

. hen p is a positive n-tuple, is decreasing on [0,1], where P; = ~ 7 = l P ; W this result is valid for convex functions f on a real linear vector space (see Theorem 3.22 and Remark 3.23(b)). For some related results, see Marshall and Olkin (1979, p. 134). (d) It is clear from the proofs of Theorems 12.16 and 12.17 that the statement in (c) also holds when f is a discrete convex function. (e) Fan and Lorentz (1959) proved the result in Theorem 12.16 for k = nand G(t) = t. It should be noted that, as shown in Theorem 11.3, 0 the results for k = n and for any k are equivalent.

The following result, due to Boland and Proschan (1986), depends on a result in Fan and Lorentz (1959) and has an application in reliability theory: Let F(t), G(t) be two continuous and increasing functions for t e: such that F(O) = G(O) = 0, and define

°

F(t)

= 1-

F(t),

G(t) = 1 - G(t)

for

t e: O.

(12.31)

330

U. Convexity, Majorization, and Schar-Convexity

By taking [a, b) e [0,00) if

= [0,00)

in Definition 12.14, we say that F(t) > G(t) for

t

s

s

JF(t) dt ~JG(t) dt o

for all s > 0

(12.32)

0

and =

=

J F(t) dt = J G(t) dt < 00. o

(12.33)

0

The following theorem is the integral inequality of Fan and Lorentz (1959) in a slightly different form:

12.19. Theorem. Let : [0, 1l" - [0, 00) be function, and assume that for i = 1, ... , n, F; for t E [0,00). If 4> is nonnegative decreasing (concave) in each variable separately, and property:

a continuous increasing and G; satisfy F;(t) > G;(t) (increasing), is convex satisfies the following

(u;

+ h,

Uj

+ k) - (u; + h,

Uj) -

(u;,

Uj

+ k) + (u;,

Uj)

((u;

+ h,

Uj

+ k) - (u; + h,

Uj) -

(u;,

Uj

+ k) + (u;,

Uj):5

for all i =1= j, 0:5 U; :5 U; + h :51, 0:5 Uj:5 integrals are finite,

0)

+ k:5 1, then providing the

=

co

J4>(t)(G

Uj

~0

1(t),

... , Gn(t)) dt s;

( ~ )J4>(t)(F 1(t), ... , F;.(t)) dt.

o

0

(12.34)

As a consequence of Theorem 12.19, Boland and Proschan (1986) obtained the following results:

12.20. Corollary. Assume that co

co

Jt dF(t) < 00,

J t dG(t) < 00.

o

o

Then F(t) > G(t) for t E [0,00) holds iff for all nonnegative increasing continuous convex (concave) functions 4> and nonnegative decreasing

12.1. Majorization and Convex Functions

331

(increasing) functions , 00

00

J (t)( G(t» dt s;

J (t)(F(t»

(2=)

o

dt,

(12.35)

0

provided the integrals are finite.

Proof. The "only if' part follows immediately from Theorem 12.19. To prove the "if" part, assume now that (12.35) holds. Letting (u) = u and x(t) = I[x, (i.e., the indicator function of the interval [x, (0», it follows that f; G(t) dt 2= f; F(t) dt for all x 2= 0 since x is nonnegative and increasing. Taking (t) == 1, it also follows that f ~t dF(t) = f ~ t dG(t), and hence F(t) > G(t) for t E [0, (0). D 00)

12.21. Corollary.

F(t) > G(t) for t E [0, (0) holds iff 00

00

J(t) dG(t) J(t) dF(t)

(12.36)

2=

o

0

holds for all convex functions , provided the integrals are finite.

Proof. The "if" part of the result is immediate. Now suppose F(t) > G(t) for t E [0, (0). It suffices to prove (12.36) for the case where has derivative 1jJ and (0) = O. Then, by Theorem 12.19, 00

00

J (t) dG(t) = J 1jJ(t)G(t) dt o

0 00

=

00

J[1jJ(t) - 1jJ(O)]G(t) dt + 1jJ(0) Jt dG(t) o

0

00

2=

00

J[1jJ(t) - 1jJ(0) ]F(t) dt + 1jJ(0) Jt dF(t) o

0

00

= J (t) dF(t).

D

o

12.22. Remark. Another approach to proving the necessity of (12.36) in Corollary 12.21 is as follows (see Bhattacharjee, 1981): Suppose

332

12. Convexity, Majorization, and Schur-Convexity

pet) > G(t) for t E [0, (0). Let ZG and ZF be the random variables with [5;;' t dG(t)]-l fO G(t) dt and respective distribution functions [f;;' t dF(t)t 1 S ~Pet) dt (these are the equilibrium distributions of G and F respectively). Then ZG >st ZF (ZG is stochastically larger than ZF) and hence E( 'lJ!(ZG» 2 E('lJ!(ZF» for all increasing functions 'lJ!. But 00

00

00

f 'lJ!(t)G(t) dt = [f t dG(t) ]E( 'lJ!(ZG» 2 [f t dF(t)] E('lJ!(ZF» o

0

f

0

00

=

'lJ!(t)P(t) dt;

o

hence the statement follows.

12.2.

D

Schur-Convex Functions

The following notion of Schur-convexity and Schur-concavity, due to Schur (1923), generalizes the definition of convex functions and concave functions via the notion of majorization. 12.23. Definition. A real-valued function f defined on a set A c said to be Schur-convex (Schur-concave) on A if y > x on A =? fey) 2 (::5)f(x).

~ n is

(12.37)

If equality in (12.37) holds only when x is a permutation of y, then said to be strictly Schur-convex (Schur-concave) on A.

f is

We note the trivial fact that f is a Schur-concave function on A iff -f is a Schur-convex function on A; thus results for Schur-convex functions immediately apply to Schur-concave functions, and vice versa, under obvious modifications. Furthermore, we note in the following theorem that all Schur-convex functions are permutation invariant. 12.24. Theorem. Assume that A c

~ n is

permutation invariant, that is, x E A implies that Px E A for every n X n permutation matrix P. If f is a Schur-convex function in A, then f(x) = f(Px) for all P and all x E A.

Proof.

For an arbitrary but fixed permutation matrix P, we have x > Px

and

Px > x

for all x E A;

U.2. Schur-Convex Functions

333

thus we have f(x) "2:.f(Px) and f(Px) "2:.f(x).

This implies that f(x) = f(Px)

for all x E A

and all P.

D

A useful condition for verifying the Schur-convexity property of f is known as the Schur-Ostrowski condition, and is stated in the following (see, e.g., Marshall and Olkin, 1979, p. 57): 12.25. Theorem. Let f : A ~Iffi be a real-valued function where A c Iffi n is permutation-invariant, and assume that the first partial derivatives of f exist in A. Then f is Schur-convex in A iff f)"2:. (x; - xJ( af - a 0 for all x E A ax; aXj holds for all 1 :5 i =1= j

:5

(12.38)

n.

The next theorem is a restatement of Theorem 12.2 and illustrates how one-dimensional convex functions and n-dimensional Schur-convex functions are related (see, e.g., Marshall and Olkin, 1979, p. 64). 12.26. Theorem. Let I be an interval in Iffi. Let x, y be two n-tuples such that x., Y; E I(i = 1, ... , n) and definef(x): I">« Iffi bea real-valuedfunction such that f(x) = ~ 7 = 1¢ (x;)for some continuous function ¢: I ~Iffi. Then f is a Schur-conuex function on I" iff ¢ is a convex function on I. A corresponding result can be given for functions of the form f(x) = n 7 ~ 1¢ (x;) when ¢ is a log-concave function (see Fact 13.24), and it has important statistical applications when the random variables are i.i.d. with a common density function ¢(x). To see how n-dimensional convex functions and n-dimensional Schurconvex functions are related, we observe the following result (Marshall and Olkin, 1979, pp. 68-69). 12.27. Theorem.

Let f :A

~Iffi

where A c Iffi n is permutation invariant.

(a) If f is permutation invariant and convex in A, then it is Schurconvex in A.

334

U. Convexity, Majorization, and Schur-Convexity

(b) More generally, if f is permutation invariant and convex in each pair of arguments (where the remaining arguments are held fixed) in A, then f is Schur-convex in A.

By combining Schur-convexity and monotonicity, useful results can be obtained via weak majorization. For example, a restatement of Theorem 12.7 is (Marshall and Olkin, 1979, p. 59): U.28. Theorem. Assume that f is defined as in Theorem 12.26. If 4J is an increasing (decreasing) convex function on I, then f is an increasing (decreasing) Schur-convex function on I", Similarly, we observe an analog of Theorem 12.27 for monotonic convex functions defined in IR n (marshall and Olkin, 1979, p. 68): U.29. Theorem. Let f be defined as in Theorem 12.27 where A c IR n is permutation invariant. If f is permutation invariant, increasing (decreasing), and convex in A, then f is increasing (decreasing) and Schur-convex in A. Marshall and Olkin (1979, p. 61) contains a detailed study of closure properties of compositions of Schur-convex and Schur-concave functions of the form

'ljJ(x) = h(4Jl(x), ... ,4Jk(X)),

X

E IR n •

For example, they illustrate that (i) If each of 4Ji (i = 1, ... ,k) is Schur-convex and if h : IR k ~IR is increasing (decreasing), then 'ljJ: IR n ~ IR is Schur-convex (Schur-concave); and (ii) if each of 4Ji (i = 1, ... , k) is Schur-concave and if h is increasing (decreasing), then 1/J is Schurconcave (Schur-convex). Combining those results with Theorem 12.27, we immediately obtain useful results when each of 4Ji (i = 1, ... , k) is a permutation-invariant convex function in IR n • If each of 4Ji is also monotonic, then similar results in their Table 1 apply. For details, see Marshall and Olkin (1979, p. 61). The notions of Schur-convexity and Schur-concavity are useful in obtaining inequalities via majorization and weak majorization, and there exist numerous results in this area. In the following we describe two convolution theorems (Theorems 12.30 and 12.32) due to Marshall and Olkin (1974) and Proschan and Sethuraman (1977a), respectively (see also, Marshall and Olkin, 1979, pp. 100-101).

U.2. Schur-Convex Functions

335

12.30. Theorem. If and f are two Schur-concave functions defined on IR n, then the function tjJ defined on IR n by tjJ(a)

=

J (a - x)f(x) dx ~ n

is Schur-concave (whenever the integral exists).

An application of Theorem 12.30 yields the following Schur-concavity property of distribution functions when the density functions are Schurconcave. This result is due to Marshall and Olkin (1974, 1979, p. 101). 12.31. Corollary. Let f(x): IR n ~[0, 00) be a probability density function of a random variable X = (Xl' ... , X n ) such that the probability measure is absolutely continuous with respect to Lebesgue measure. If f is a Schur-concave function of x for x E IR n, then the distribution function of X

F(a)

=

l

~ L Q{Xi ~a;}

a

E

IR

n

is a Schur-concave function of a.

Proof.

For i = 1, ... , n, let I if a, - Xi 2= 0, I(-oc.ail = { 0 otherwise

be the indicator function of {x, :Xi of the set

~ ai}'

Since the indicator function (x)

A = {x: x E IR n , Xi 2= 0 for i = 1, ... , n}

is a Schur-concave function of x E IR n and, by (a - x) F(a) can be expressed as

F(a)

=

= II7=1 I(-OC,ail'

J (a - x)f(x) dx, ~

n

then the conclusion follows from Theorem 12.30.

0

12.32. Theorem. Suppose that either (i) .r = IR, e c IR is an interval and f.l is Lebesgue measure or (ii) .r is the set of all integers, e is an interval or an interval of integers, and f.l is the counting measure. If g: e X . r ~[0,00)

336

12. Convexity, Majorization, and Schur-Convexity

is totally positive of order two (TP2) and satisfies the semigroup property g(OI

+ O2, x) =

Jg(OI' t)g(02' X- t) dv(t) .1'

for some measure von x, and 1J(x) is Schur-convex for x E IR n. Then the function 1jJ defined for 0 == (0 1 , • . . , On) E ex· .. X e by 1jJ(O) ==

J}] g(

0i' x i)1J(x) }] d/l(xi)

is a Schur-convex function of O. Theorems 12.30 and 12.32 can be applied to obtain useful integral inequalities and probability inequalities. Some of the recent results derived from those two theorems using Schur-concavity can be found in the review article by Tong (1988).

12.3.

Multivariate Majorization and Convex Functions

As indicated in Definition 12.1, the notion of majorization concerns a partial ordering of the diversity of the components of two n-tuples x and yin IR n • A natural problem of interest is the extension of this notion from n-tuples (vectors) to k X n matrices. For example, let

X == (XI, X2, ... ,Xk)

,

=

Y==(YI,Y2"",Yk)'==

Xl2 X22

X2n

Xkl Xk2

Xkn

C X21

Yl2 Y22

C Y21

Ykl Yk2

be two k X n real matrices, where responding row vectors. If Xi == Yi

for

Xl, . . .

X'") , (12.39)

Y'Y2n") Ykn

,Xk; y, ... , Yk are the cor-

i == 1, ... , k

(12.40)

12.3. Multivariate Majorization and Convex Functions

337

where (12.41) then intuitively speaking the components of x are less diverse than those of y. The question of interest is, of course, how to find a useful definition so that meaningful results can be obtained. This involves a multivariate extension of Definition 12.1. To state a definition of multivariate majorization given in Marshall and Olkin (1979, Chapter 15), we first observe the definition of a T-transform matrix (Marshall and Olkin, 1979, p. 21):

12.33. Definition. An n it is of the form

X

n matrix is said to be a T-transform matrix if

T = a1+ (1 - a)Irs

(12.42)

for some a E [0, 13, where I is the n X n identity matrix and I, is obtained by interchanging the rth and sth columns of I for some r Zs.

12.34. Definition. Let A, B be two k x n real matrices for k 2 2, n 2 2. (a) Y is said to chain majorize X (in symbols, Y >X ' ) if there exists a finite number of T-transform matrices T I , . . . ,TN such that X = YIIEV=,Ti. (b) Y is said to majorize X in a multivariate sense (Y > m X) if there exists an n x n doubly stochastic matrix Q such that x = YQ. (c) Y is said to row-wise majorize X (Y >'X) if yi > x i holds for i = l , 2 , . . . , k. The implications of partial orderings defined in Definition 12.34 are given in the following theorem: 12.35. Theorem. Let X, Y be two k x n real matrices. Then (ii)

Y>'X$

Y>"X+Y>'X.

(12.43)

Furthermore, the implications (i) and (ii) are strict. Proof. (i) follows from the fact that the multiplication of a finite number of T-transform matrices is doubly stochastic; (ii) follows from Fact 12.5(b). TOshow that (i) is strict, by Fact 12.5(b) it suffices to find a

338

12. Convexity, Majorization, and Schur-Convexity

matrix Q that is doubly stochastic but it not the multiplication of a finite number of T-transform matrices; such a 3 x 3 matrix can be found in Marshall and Olkin (1979, p. 431). The proof for the strict implication (ii) follows immediately from Fact 12.5(b) and Definition 12.34. 0 Rinott (1973) considered the notions of multivariate majorization given in Definition 12.34 and derived some useful results in probability and statistics. He also provided a characterization of chain majorization and row-wise majorization. His result is stated in the following theorem, and a convenient reference for the proof is Marshall and Olkin (1979, pp. 434-435). 12.36. Theorem. Let f(X) =f(XI' ... ,Xk): I R k x n IR~ be a differentiable function. Then (a) f(Y)

for all Y >c X

~f(X)

holds iff (i) f(X) = f(XP) holds for all n x n permutation matrices P and (ii) ~ 7 ~ 1(Xjr - xjs)[fUr)(X) - f u ~ ) ( X ) ] ~0 holds for all r, s = 1, ... , n, where fUr)(X) = 0/ oajr!(U)/u=x. (b) f(Y) ~f(X) for all Y »: X holds ifffor every fixed j = 1, ... , k, (Xjr - xjs)[fur/ X) - ks)(X)] ~0 holds when the values of the other columns of X are held fixed.

12.37. Remark. Note that the statement of Theorem 12.36(b) is just that xj , Yj satisfy Theorem 12.25 for each j

= 1, ...

, k.

0

Marshall and Olkin (1979, p. 435) proved a result for majorization in a multivariate sense. Their result is related to symmetric convex functions: 12.38. Theorem. Let X, Y be two k x n matrices as defined in (12.39). If y>mx, then f ( Y ) ~ f ( Xholds ) for all f(X)=f(XI"" , x d : l R k x n ~ 1 R which are symmetric and convex in the sense that (i) f(X) = f(XP) for all n x n permutation matrices P and (ii) f(aU for all a

E

+ (1- a) V)

~ af(U)

+ (1- a)f(V)

[0, 1] and k x n matrices U and V.

Chapter 13 Convexity and Log-Concavity Related Moment and Probability Inequalities

In this chapter we discuss some moment and probability inequalities that arise from the applications of convexity, Jensen-type inequalities, and log-concavity. We shall assume without explicit statement that expectations and other integrals mentioned in theorems, corollaries, etc., exist. If an expectation or integral does not exist, the corresponding statement is to be considered vacuous.' The convexity-related results are given in Sections 13.1-13.3, and results that arise from the log-concavity property of probability density functions are discussed in Sections 13.4 -13.7.

13.1.

Jensen's Inequality

Jensen's inequality for one-variable functions and for functions of several variables, and its reversals, refinements, and converses have been treated extensively earlier in this book. In this chapter we first present a stochastic version of Jensen's inequality. It is a probabilistic analog of the previous results and has important applications in probability and statistics. For n 2 1 let X = (XI,. . . , X, ) be an n-dimensional random variable. Let F ( x ) = P [ X s x ] be the distribution function of X, and let p = ( p l , . . . , p,) denote the mean vector of X.

13.1. Theorem. For n 2 1 let @ : A + R be a continuous convex function where A c R" is an open convex set such that P[X E A] = 1. Then p E A 339

340

13. Convexity and Moment and Probability Inequalities

and E1'(X)

=

J1'(x) dF(x)

2:

(13.1)

1'(r.a).

A

Proof. The fact that r.aEA is easy to verify. To show that (13.1) holds we use the proof in Marshall and Olkin (1979, p. 454). For Z E A and i = 1, , n, let 1't)(z) = lim.!. [1'(Zl'

,

dO E

+ E, Zi+l ,

Zi-l' Zi

... ,

zn) - 1'(Zl' ... , zn)],

and denote

V+ l' (x) = (1'(l)(x), ... , 1't.ix)),

x EA.

Then V+ l' (x) is Borel-measurable and, more importantly, 1'(x) - 1'(z) 2: [V+ l' (z)][x - z),

x,zEA

holds (Marshall and Olkin, 1979, p. 451). Choosing z = u, we have 1'(X) - 1'(r.a) 2: [V+1'(r.a)](x- u)

a.s.

(13.2)

The proof follows by taking expectations on both sides of (13.2).

0

Note that if l' is differentiable, then V+l' (x) is just the gradient vector of l' at x. Furthermore, note that if A is an interval in IR and the distribution of X is discrete, this result reduces to the result treated in Theorem 2.1. For n = 1, a more general result is known (Chow and Teicher, 1978, pp. 102-103): 13.2. Theorem. c E IR and r : IR

Let 1': IR ~IR be a continuous convex function. Let be Borel measurable. If r(x) and

~IR

6(x) == (x - EX) - (r(x) - Er(X))

are both increasing for x

E

IR, then

E1'«X - EX) + c)

2:

E1'« reX) - Er(X)) + c).

(13.3)

Proof. If P[ 6(X) = 0] = 1, then (13.3) is immediate. Otherwise, by the convexity of l' we have 1'«X - EX) + c) - 1'«r(X) - Er(X)) + c) 2:

6(X)V+1'«r(X) - Er(X)) + c)

a.s.,

(13.4)

13.1. Jensen's Inequality

341

where

and (by the convexity of 1J) 'y+ 1J(z) is increasing in z. By the mono tonicity of <5(x), there exists a real number Xo such that <5 (x){ :s;O ~ O

for all x < Xo for all x > Xo .

Furthermore, the monotonicity of r(x) and V+ 1J implies <5(x)V+1J«r(x) - Er(X))

for all x

E

+ c)

~ <5(x)V+ep«r(xo)

- Er(X))

+ c) (13.5)

IR. Thus (13.4) and (13.5) together imply

1J«X - EX)

+ c) -1J«

reX) - Er(X)) ~ <5 (X)V+ 1J«

+ c)

r(xo) - Er(X))

+ c) a.s. (13.6)

The conclusion follows from taking expectations on both sides of (13.6). 0 13.3. Remarks. (a) If 1J:A ~IR is a continuous convex function on an interval A such that P[X E A] = 1 and A is a proper subset of IR, then

Theorem 13.2 can be modified accordingly. (b) For n = 1 and A = IR, (13.1) follows from (13.3) as a special case by letting r(x) = 0 and c = EX. (c) If r } : I R ~ 1and R r 2 : 1 R ~ 1 are R such that r}(x) and

are both increasing for x

E

IR, then

holds, as noted in Liu (1988). The proof for this result is similar. (d) Note that the proof of Theorem 13.2 cannot be modified in a routine fashion for n > 1. 0 A different generalization, given below, depends on a partial ordering of the distribution functions. Its proof can be found in Karlin and Novikoff (1963) or Stoyan (1983).

342

13. Convexity and Moment and Probability Inequalities

13.4. Theorem. Let X, Y be two univariate random variables with a common mean EX and distribution functions F(x), G(x), respectively. If for all x < (2::)EX,

F(x)::5 (2::)G(x)

(13.8)

then Ecp(X) 2:: Ecp(Y) holds for all convex functions cp.

13.5. Remark. For n = 1, Theorem 13.1 follows from Theorem 13.4 by 0 letting pry = EX] = 1.

13.2.

Moment Inequalities for Univariate Random Variables

Let Z be a univariate random variable with distribution function F(z).

13.6. Definition.

The fith moment of Z for fi E [0, (0) is /-l(3

/-lo= 1,

= EZ(3=

f

z(3 dF(z)

for

fi > O.

(13.9)

13.7. Theorem. If a', fi are positive real numbers such that fila' is a positive even integer, then /-lf3 2:: (/-la )(3la. In particular, /-In 2:: (/-lnlzf and /-In 2:: (/-lIt hold for all positive even integers n.

Proof.

Immediate by choosing cp(z) = (zj(3la in Theorem 13.1.

0

The following results concern moments of nonnegative random variables.

13.8. Theorem. If Z 2:: 0 a.s., then log /-lf3 is a convex function of fi for fi E [0, (0). Consequently (13.10) /-l(3 2:: (/-la )f3la for all 0 < a' < fi, /-lf3

2:: /-la/-l(3-a

(13.11)

for all 0 < a' < fi,

and, more generally, /-la,/-l(3-at

2:: /-la 2J.t(3-a2

for all

0<

a'i

< a'z::5

en.

(13.12)

13.3. Dimension-Related Inequalities

Proof (see, e.g., Loeve, 1963, p. 156). inequality, Il~

343

By the Cauchy-Schwarz

= [E(Z
holds for all 0 < a < {3. Thus

holds for all 0 < {31 < {32' and this is equivalent to the log-convexity of 1l13' 0 Theorem 13.8 may also be proved by using total positivity arguments. A consequence of Theorem 13.8 is that 13.9. Corollary (see Loeve, 1963, p. 156). an increasing function of {3 for (3 E [0, (0).

13.3.

13.3.1.

If Z ~ 0 a.s., then (1l13)1113 is

Dimension-Related Inequalities for Exchangeable Random Variables

Exchangeable Random Variables and de Finetti's Theorem

The notion of exchangeable random variables involves changeability of an infinite sequence of random variables.

the ex-

13.10. Definition (Loeve, 1963, p. 364). An infinite sequence of random variables {Xn::l is said to be exchangeable if for every fixed n and every permutation (Jrl""" Jr n ) of (1, ... , n), the two ndimensional random variables ( X ~ ., .. ,X:) and ( X ~ I " .. , X ~ Jare identically distributed. For a finite integer n > 1, let X = (Xl' ... , X n ) be an n-dimensional random vector. 13.11. Definition. Xl"'" X n are said to be exchangeable if there exists an infinite sequence of exchangeable random variables { X n ~ ~ such that X and ( X ~, ... , X:) are identically distributed.

l

344

13. Convexity and Moment and Probability Inequalities

It is obvious that the joint distribution of exchangeable random

variables must be permutation invariant. But the converse is false. In fact, it is known that the common correlation coefficient of any pair of exchangeable random variables must be nonnegative, thus if the joint distribution of X is permutation invariant and if the common correlation coefficient is negative, then XI, ... , X; are not exchangeable. The following theorem, known as de Finetti's theorem (see e.g., Loeve, 1963, p. 365), states that exchangeable random variables must be a mixture of conditionally independent and identically-distributed (i.i.d.) random variables: 13.ll. Theorem. Let the distribution function F(x.) of (XI, ... Xn) be X; are exchangeable random permutation invariant. Then X I variables iff F(x) is of the form I

I

F(x.) =

J !I.

•••

I

n

(13.13)

Gv(x;) dH(v) ,

where H(v) is the distribution function of an r-dimensional random variable V taking values in A, and Gv(XI) is the conditional distribution of Xl given V = v.

An equivalent condition for exchangeability is (see, e.g., Shaked, 1977, or Tong, 1980, Section 5.3): 13.13. Theorem. Xl ,Xn are exchangeable random variables iff there exist i. i. d. random variables UI , • • . , Un' an r-dimensional random variable V (r ~ 1) independent of the U;'s, and a Borel-measurable function g: I W + I ~~ such that (XI X n) and (g(UI V), ... ,g(Un, V» are identically distributed. I

•••

I'

••

I

I

Note that Gv(XI) in (13.13) is in fact the conditional distribution function of g(UI V) defined in Theorem 13.13, given V = v. Because of this fact, exchangeable random variables have been called conditionally i.i.d. random varibles (Tong, 1977) or positively-independent-by-mixture (PDM) random variables (Shaked, 1977), and events defined by those random variables are said to be events which are almost independent (Dykstra, Hewett, and Thompson, 1973). This mixing process also I

13.3. Dimension-Related Inequalities

345

creates some positrve dependence properties of exchangeable random variables. For example, it is known that if V is a random variable such that {Gv(x): v E A} is a stochastically increasing family, then Xl, ... , X; are associated random variables in the sense of Esary, Proschan, and Walkup (1967).

13.14. Example. The random variables Xl, ... ,Xn given below are exchangeable random variables which are useful in statistics: (a) Multivariate normal. (Xl'.'" X n ) has a multivariate normal distribution with a common mean, a common variance, and a common correlation coefficient p E [0, 1]. t. (X 1, . . . , X n ) 4 (Y1 / S, ... , Yn / S), where (b) Multivariate (Yl , . . . , Yn ) has a multivariate normal distribution with a common mean 0, a common variance a 2 , and a common correlation coefficient p E [0, 1]; S is independent of (Y1 , ••• , Yn ) , and vS2 /aZ has a X2(v) distribution.

(c) Multivariate binomial. (Xl' ... ,Xn ) 4 (U 1 + V, ... , U; + V), where U1 , . . • , Un are i.i.d. binomial variables with parameters n1 and p, and V is independent of the U;'s and is a binomial variable with parameters no andp. (d) Multivariate Poisson. Similar to (b) except that U1 , • • • , Un are Poisson variables with parameter Al and V is a Poisson variable with parameter Ao. (e) Multivariate exponential. (Xl' ... , X n ) 4 (min(U1 , V), ... , mineU;; V)), where U1 , ••• , Un are i.i.d. exponential variables with common mean AI' and V is an exponential variable with mean Ao and is independent of the U;'s (Marshall and Olkin, 1967). More general forms of the multivariate exponential are also available. (f) Multivariate gamma (chi-squared) variables. Similar to (c) except that U1 , ••• , U; are i.i.d. gamma variables with parameters (Y1 and {3 (chi-squared variables with degrees of freedom VI) and V is a gamma variable with parameters (Yo and {3 (a chi-squared variable with degrees of freedom Yo). (g) Multivariate F. (Xl' ... ,Xn) 4 (UdV, ... , Un/V), where 2 m 1U1 , ... , m1 Un are i.i.d. x ( m 1) variables and mo V is a x2(m o) variable independent of the V;'s. 0

346

13. Convexity and Moment and Probability Inequalities

13.3.2. Inequalities In the following we describe some dimension-related inequalities for exchangeable random variables. The inequalities can be obtained by combining Theorem 13.12 (or 13.13) with the moment inequalities in Section 13.2. The first result (Tong, 1970) deals with exchangeable normal variables only and follows from an application of Theorems 13.7 and 13.8. 13.15. Theorem. Let (Xl"'" X n ) have a multivariate normal distribution with a common mean u, a common variance 0 2, and a common correlation coefficient p ? O. Let A( n) be defined as

Then A(n)? (A(k)t/k and A(n)? A(k)A(n - k) hold for all positive integers k < n. Proof. Consider A(n) = p[n7=1 {Xi 5:: a}], and let V, VI, ... ,Vn be i.i.d. K(O, 1) variables. Then (Xl' ... , X n ) and

are identically distributed. Thus, after first conditioning on V = v and then unconditioning,

A(n) =

E[J] P[o(yT=p U; + VP v) + fl5::a IV= v]]

= Ern(V) ?

[Erk(V)r/ k = [A(k)r/\

where r( v) = P[ 0(yT=p (U; + VP v) + fl 5:: a I V = v]. equality now follows from Theorem 13.8. Similarly,

The proof for the case A(n) =

p [ n 7 ~ {!X 1 i ! 5::

The

a}] is similar.

first

D

In-

13.4. Brunn-Minkowski Inequality

347

The result in Theorem 13.15 was generalized by Sidak (1973) to all exchangeable random variables and all Borel-measurable sets:

13.16. Theorem. Let B c

~

Let Xl' ... ,Xn be exchangeable random variables. be a Borel-measurable set. Define y(k) =

pLo

l

{X; E B}

k = 1, 2, ... , n.

(13.15)

Then y(n) ~ (y(kW/k and y(n) ~ y(k)y(n - k) hold for alII:::::: k < n.

Proof.

Let V, U1 ,

U; and g be defined as in Theorem 13.13, and

••• ,

let I for x E B, I B () X = { o otherwise. Then, by Theorem 13.8 and an argument similar to that given in the proof of Theorem 13.15, y(n) = E

D

IB(X;) == E[ ED IB(g(U;, v)) I V = v]

~ ( E [ E n I B ( g (v)) U ;I v=v])n/k , = (y(k)t/k • The second inequality follows similarly from (13.11).

D

13.17. Remark. A generalization of Theorem 13.16 from exchangeable random variables to a class of positively dependent random variables with a common marginal distribution will be given in Theorem 14.18. The proof of that theorem depends on an application of a moment inequality via Muirhead's theorem. D

13.4. Bmnn-Minkowski Inequality In the rest of this chapter we study inequalities via the log-concavity property of certain probability measures. In particular, we treat the Brunn-Minkowski Inquality and the theorems of Prekopa (1971), Borell (1975), and their generalizations (Rinott, 1976, and Das Gupta, 1980). We then describe some applications of these results in statistics.

348

13. Convexity and Moment and Probability Inequalities

Let 53 denote the class of all Borel-measureable sets in R", and let B be a measure defined on 53. For arbitrary but fued ( Y E [0, 11 and B , , B2 E 53, let us define &I+

(1 - a)B,

+ (1 - a ) x 2 for some x1 E B1 and x2 E B2}. (13.16) We assume that B , , B2 are in 53, are nonempty, and (aB, + (1 - a ) B Z )E = {x:x E R", x = ax,

53.t Let X = (X, , . . . , X n ) have a probability density function f (x) that is absolutely continuous w.r.t. Lebesgue measure. The inequalities we study here concern lower bounds on P(crB1+ (1- a)Bz) in terms of P ( B , ) and P(B2), where, if B is a probability measure, then P ( B ) stands for P(X E B). The classical Brunn-Minkowski Inequality states: 13.18. Theorem. If 9 si the Lebesgue measure, then

+

P((YB1 (1 - a)&)

holds for all B 1 , B2 E 53, all

2

[a(P(B1)),'"+ (1- (Y)(P(B2))""]"

(Y

E

[0, 11, and all n = 1, 2, . . . .

When B 1 , B2 are convex sets (which implies that aB1 + (1 - a ) B 2 is also convex), this inequality was first proved by Brunn (1887). Minkowski (1910) derived conditions for equality to hold when B , , B2 are convex. Later Lusternik (1935) generalized this result to any Borel-measurable set, and his conditions for equality to hold were corrected by Henstock and Macbeath (1953). A treatment of the historical developments related to this inequality can be found in Das Gupta (1980). Das Gupta (1980) also provided a generalization that includes a previous generalization of Henstock and Macbeath (1953) as a special case.

13.5. A Class of Log-Concave Probability Measures If 9 is a probability measure such that P(B)=

If

(x) dx

for all B E 53,

(13.17)

B

t Karlin (1983) provided a counterexample showing that B , , B , E 93 does not necessarily imply (CUB,+ (1 - (Y)B,) E B for all (YE [0, 11. Thus this assumption is needed.

13.5. A Class of Log-Concave Probability Measures

349

then (W, 00, gp) is a probability space. Prekopa (1971) considered probability measures with the following property: Ct' E [0, 1]. (13.18) If all the probabilities are positve, then (13.18) is equivalent to

log P(Ct'B I

+ (1- Ct')B 2 ) 2: Ct' log P(B I ) + (1- Ct')log P(B 2 ) ,

which is just the log-concavity of gp. He studied conditions on f(x) for (13.18) to hold, and proved that a sufficient condition is the log-concavity of f.

13.19. Definition. Let

f ( x ) : [ R n ~ [(0) o , be a probability density function such that gp is absolutely continuous w.r.t. Lebesgue measure. f is said to be log-concave if

(13.19) holds for all Ct' E [0, 1] and all Xl' X2 E [Rn. When f(x) > 0 for all XE [Rn, then (13.19) is equivalent to 10gf(Ct'xl + (1- Ct')X2) 2: Ct' log j'(x.) + (1- Ct')logf(x2)'

(13.20)

The main theorem of Prekopa (1971) states: 13.20. Theorem. Let gp be defined as in (13.17). Iffis log-concave, then (13.18) holds for all B I' B 2 C [Rn and all Ct' E [0, 1]. An immediate generalization of Theorem 13.20 is: 13.21. Coronary. Let X be an n-dimensional random vector with probability density function f(x), and let B I , . . . , B; (k 2: 2) be subsets of [Rn. Iff(x) is a log-concave function of x, then (13.21) holds for all Ct'; such that Ct'; >

0 and

I : 7 ~ 1Ct';

= 1.

350

13. Convexity and Moment and Probability Inequalities

Proof,

p[ X

E

~ 1a;B;]

=

p[X

E

~ {P[X E Bd}"" { P [ X E ~ {P[X E

X

p[x

~ 2( aJ(l-

a1 B1 + (1- (1) k

(1))B;]

~(aJ(l- ( 1))B;

]}1-"'1

Bd}"'l{P[X E B 2 ]}"'2 E

~(a;/(l-

~

...

~

TI {P[X E B

a1 - (

2

»BT-"'1-"'2

k

0

i ]} "" .

i=l

Prekopa's original result is given when B 1 , B 2 are convex sets only; his original proof depends on an application of the Brunn-Minkowski Inequality and is quite lengthy. His result was generalized to a larger class of probability measures for convex sets by Borell (1975). Borell considered measures with the property that

(13.22) n

holds for all convex sets B 1 , B 2 C IR , all a E [0, 1], and for some s E [-00, lin). When s = O,then by continuity (13.22) reduces to (13.18). When s = -00, then the right-hand side of (13.22) is just min{P(B 1 ) , P(B 2 )} . Borell (1975) first showed that 13.22. Theorem. If the probability measure !!J satisfies (13.22) for all B 1,B2clR n , all aE[O, 1], and for some SE[-OO, lin), then it is absolutely continuous w.r.t. Lebesgue measure. As a consequence of Theorem 13.22, the class of measures satisfying (13.22) for some s E [-00, lin) must be of the form (13.17) for some f. Borell's (1975) main theorem concerns a characterization of the class of probability measures satisfying (13.22). His original result concerns convex sets only. In the following theorem the convexity condition on B 1 , B 2 is removed: 13.23. Theorem. Let f: IR n ~[0, (0) be a probability density function, and let !!J be the probability measure defined in (13.17). Then the following

351

13.5. A Class of Log-Concave Probability Measures

statements are equivalent: (a) $9' satisfies (13.22) for all sets B , , B, E 93, all a E [0, 11 such that B E 3, and for some s E [-a, l / n ) . (b) There exists a Borel-measurable function g:Rn+R such that f(x) = g(x) almost everywhere and (i) i f s E [-cot 0), then (g(x))"'('-") si convex, (ii) i f s = 0, then logg(x) is concave, (iii) i f s E (0, U n ) , then (g(x))""l-"' 1s. concave.

Sketch of Proof. The original proof of Borell (1975) is lengthy and is for convex sets only. In the following we adopt the proof given by Rinott (1976). Rinott's proof involves replacing integrals on f over sets in R" by a certain measure Y of epigraphs in Fin+'. Thus his proof is valid only for s E [-m, l / ( n 1)). Let B* be any set in R"+'. Consider a measure Y given by

+

=(

dv(B*)

for s # 0, s E [-m, l / ( n

+ l)),

for s = 0.

(13.23) Then it can be shown that, for B:, B; c Rn+l,

+

V ( ~ B T ( 1 - (Y)B;)

2[

" ~ s f 0, s E [-m, l / ( n + l ) ) , [a(v(B:))s+ ( 1 - a ) ( ~ ( B 2 * ) ) " ]for (Y(BT))a(4B;))1-a for s = O . (13.24)

The inequality in (13.24) can be obtained by first showing it for rectangular sets, then extending it to any Bore1 measurable sets by using the argument in Borell (1975). (b) .$ (a): For arbitrary but fixed B c R" and g :R" + R, let us define the epigraph in Rn+l: A ( B , g ) = {(x, A) : x E B , a E R , g(x)

5

A).

352

13. Convexity and Moment and ProbabiJity Inequalities

Let us first consider the case s = 0 and assume that f satisfies (13.19). Denoting B* = A(B, -log!) c IR n +\ we can easily verify that for measure v defined in (13.23) we have PCB) = v(B*)

(13.25)

for all Be IR n and [aB 1 + (1- a)B 2)* ~ aBr

for all B 1 , B2 c;; IR

n

•

+ (1- a)Bi

(13.26)

Thus (13.24), (13.25), and (13.26) together imply

P[aB 1+ (1- a)B 2l = v[aB 1 + (1- a)B 2*l 2:

2:

v[aBr

+ (1 -

a)Bn

(v(Bt))"'(V(B;))l-'" = (P(B 1))"'(P(B 2))1-""

which completes the proof for s = O. For s < 0 we can repeat the above arguments with B* = A(B, rl(l-ns»), and for s E (0, 1/(n + 1)) we can define B* = H(B, r /(1-ns») where the hypograph H(B, g) is given by H(B, g)

= {(x, A):X E B, AE IR, and 0$ A$g(x)}.

(a):::;> (b): First consider the case s =0 and assume that (13.18) holds. Let Sk(X)= {y: Iy - x] $ IIk} and define !k(X) = [

f

fey) dy

Sk(X)

J/ f

dy.

Sk(X)

Then (13.18) implies !k( axl

+ (1 - a)x2) 2: (!k(x 1))"'(!k(x2))1-", for a E [0, 1].

Consequently the function g(x) = limk->oo inf!k(x) satisfies

g( ax1 + (1- a)x2) 2: (g(x 1))"'(g(X2))1-", for

a E [0, 1].

By differentiation of the integral argument, we have f = g almost everywhere. Thus the proof is complete for s = O. For s = -00, a similar argument yields g( axl + (1 - a)x 2) 2: min{g(x1)' g(X2)}' For s E ( -00, 0), we can consider the sets C 1, C 2 C IR n + 1 defined by C; = B; X (c., (0), where the B;'s are spheres in IR n and c, >0 (i = 1, 2). If B 11 B 2 are chosen to satisfy

(f fI dx,/ f fI dX;) = (Ct/C2r, 8,

1

8

2

1

then C 1 , C 2 satisfy (13.24) for some s E (-00,0). Suppose rl(l-ns) is not convex. We choose C 1, C 2 in this fashion to satisfy C; c A(B;, r/(l-ns))

13.6. Some Properties of Log-Concave Density Fnnctions

353

and

Thus we have

P[aB 1+ (1- a)Bzl < v(aC 1 + (1- a)C z)

= [a(v(C1))S + (1- a)(v(Cz)Yp/s< [a(P(B1))S + (1- a)(P(B 2)Yllls, which is a contradiction. For s E (0, 1/(n by choosing C; = B; x (0, c;) (i = 1, 2).

13.6.

+ 1)) the proof follows similarly 0

Some Properties of Log-Concave Density Functions

Log-concave density functions which satisfy (13.19) play an important role in statistics and probability. In the following we observe some known facts concerning this class of densities. 13.24. Fact. Let Xl' ... , X; be i.i. d. univariate random variables with a common density function hex). If hex) is a log-concave function ofx for x E [R, then the joint density function of (Xl' ... , X n ) is a log-concave function of x for x E [Rn. 13.25. Fact. If f(x) = g(T(x)) where g: [R ~[0,00) is decreasing and T(x) is a convex function of x for x E IRn, then f is a log-concave function ofxfor XE [Rn. The following theorem, due to Prekopa (1971) and Brascamp and Lieb (1975), shows that the integral of a log-concave function is log-concave: ~ be a log-concave function 13.26. Theorem. Let f(x, y): [ R n + m[0,00) of (x, y) for x E [Rn and y E [Rm. Then the function g: [Rn ~[0, 00) given by

g(x)

=

I

f(X, y) dy

(13.27)

[I;lm

is log-concave. Proof. We adopt the proof in Brascamp and Lieb (1975). First note that it suffices to prove the theorem for m = n = 1 because the general case

354

13. Convexity and Moment and Probability Inequalities

follows by Fubini's theorem and induction. Let Xl' X2 be two points in such that g(X1)g(X2) 0. For convenience we may assume that

~

'*

supf(x, y)

= supf(x', y);

y

y

because otherwise we can replace f(x, y) by ebXf(x, y) for suitably chosen b and the problem remains unchanged. For each fixed A> 0, denote C1(A) = {(x, y) :f(x, y) Cix, A) = {y :f(x, y)

2= A} C

2= A}

~ 2 ,

c ~ .

Then, by log-concavity of t. C1(A) is convex and C 2 (x , A) is an interval. (For the convexity of C 1(A), see Fact 13.28). Letting v(x, y) = f C2(X.).) dy be the Lebesgue measure of the set C 2 (x, A), we have, by Theorem 13.18, V(ax1 + (1- a)x2' A)

+ (1- a)v(x2' A) for all a E [0,1]. Since g(x) can be expressed as g(x) = fO' v(x, A) ds, we have g(ax1 + (1- a)x2) 2= ag(x1) + (1- a)g(x2) 2= (g(X1))"'(g(X2))1-", 2=

av(x1' A)

for all a E [0, 1], where the second inequality follows from the arithmetic mean-geometric mean inequality. 0 A simple application of Theorem 13.26 is (Brascamp and Lieb, 1975; see also Barlow and Proschan, 1981, p. 104): 13.27. Corollary. The convolution of two log-concave density functions 0 in ~ n is log-concave.

Proof. Let f1' f2 be log-concave density functions. Then h(x - y)fz(y) is jointly log-concave in (x, y) E ~ 2 n . Thus by Theorem 13.26 g(x)

=

f

ft(x - y)fiy) dy

JR"

is log-concave.

0

A density function f is said to be unimodal if the set

D).

= {x:x E

~ n , f ( x 2= ) A}

(13.28)

is a convex set in ~n for all A> 0. The following facts show how log-concavity and unimodality are related.

13.7. Some Statistical Applications

355

13.28. Fact. If f: ~ n- [0, (0) is a probability density function, then log-concavity offimplies unimodality off.

Proof. Let XI' Xz E (13.20),

~ n be

in D)... Then for every a

E

[0, 1], we have, by

log j'(ux, + (1- a)xz) 2: a 10gf(xI) + (1 - a)logf(xz) 2:

Thus axl

a log A + (1 - a)log A 2: log A.

+ (1 - a)xz is also in D)...

(13.29)

D

A function f is said to be Schur-concave if y > X implies f(x) 2: f(y) for all x, y E ~ n (see Definition 12.23). It is known that all Schur-concave functions are permutation invariant (see Theorem 12.24). Furthermore, it is known that

13.29. Fact. If f: ~ n [0, _ (0) is a permutation-invariant and logconcave function ofx E ~ n , then it is a Schur-concave function ofx E ~ n .

Proof. Assume y > x and, without loss of generality, it may be assumed that x, yare of the form

I and XI + Xz= YI + Yz· Let y* = (». YI, X3' ... , x n ) . where Yz<X2 ~ X
logf(x) = logf(ay + (1 - a)y*)

= logf(y).

2:

a logf(y) + (1 - a)logf(y*)

D

13.7. Some Statistical Applications In this section we describe some examples of applications of Prekopa's theorem (Theorem 13.20) and Borell's theorem (Theorem 13.23) in statistics and reliability theory. Application 13.30 is easy. Applications 13.31-13.33 were given in Rinott (1976), and Application 13.36 was given independently by Karlin and Rinott (1983) and Tong (1983, 1989).

356

13. Convexity and Moment and Probability Inequalities

13.30. Application. Let X = (XI , ... ,Xn ) have a probability density function [(x) and distribution function F(x). If [(x) satisfies the condition in Theorem 13.23(b) for some S E [-00, lin), then F( axl + (1

Proof.

-

) )

a X2

~

{[a(F(xI))S + (1- a)(F(x2)Y]lIs for S';tO I (F(XI))£>:(F(X2)) -£>: for s = o.

Choosing B, = {x:x E aB I

~ n , x:5

+ (1- a)B 2 = {x:x E

Xi} for i

= 1, 2, we have

~ n , x es aXI

+ (1- a)x 2}.

Thus the inequality follows from Theorem 13.23.

0

For s = 0 this result states that the distribution function of an n-dimensional random vector is log-concave if its density function is log-concave.

13.31. Application. Let (XI, ... , X n) have density function [(x) such satisfies the condition in Theorem 13.23. Let that t T/J(tl, ... , tn-I), 112(t1, ... , tn-I) be the implicit functions given by the equations tn-I, X n :5 T/J(tl,

, tn-I)]

= aI,

P[X I ~ t 1 , ... , X n_1 ~ tn-I, X; ~ T12(tI,

, tn-I)]

= a2'

P[X I :5 t l

for

arbitrary

1}j(t1' ... , tn-I)

defined in

, .•• ,

X n_l

:5

but fixed aj E (0, 1) (j = 1, 2). Then for j = 1, 2, is a concave function on the convex region where it is

~ n - I .

When n = 2 and [ is the bivariate normal density function, this result was given earlier by Tihanski (1972).

13.32. Application. Consider the problem of hypothesis testing concerning a location parameter 0 E ~ n of a density function [(x - 0). If (i) the acceptance region is a convex set in ~ n , (ii) the test is ao-similar on the boundary of H o (under the null hypothesis), and (iii) [(x) satisfies the conditions in Theorem 13.23 for some S E [-00, lin), then the test is unbiased with level ao' 13.33. Application. Let

[(x),

(XI' ... ,Xn), satisfy (13.19).

the density Let eJ>(x): ~ n-

function of X = be a log-concave

~

13.7. Some Statistical Applications

357

function of x and define G(t) = P[ ep(X) 2: t]. Then Theorem 13.20 implies G(at l

+ (1- a)t z) 2: (G(tl»"'(G(tzW-'"

for all t l , t z 2: 0 and all a E [0,1]. concave function, this result implies the life lengths of n independent log-concave, then the distribution of increasing failure rate property.

Since ep(x) = min{xl' ... ,xn } is a that if the joint density function of components in a series system is the life length of the system has the

The final application (Application 13.36) given later is for the special case when f(x) is both log-concave and permutation invariant. We first observe

13.34. Fact. Let the probability density function of X = (Xl, ... , X n ) be permutation invariant and log-concave. Then

P[X E (aB + (1- a)3t(B))] 2: P[X E B], where 3t(B)

= {y: y = 3t(x), X E B}

and 3t(x) is any given permutation of x. By applying this result to n-dimensional rectangles, we obtain the following result: Let B be an n-dimensional rectangle given by B

= B( c., cz) = {x: x E

~ n , Cli :5 Xi :5 CZi

for i

= 1, ...

,

n},

where Cz

= (CZI ,

•.. ,

czn ) ,

and Cli < CZi (i = 1, ... ,n). Let 3t = (Jrl' ... , Jr n ) be a permutation of (1, ... , n) and let b be the inverse permutation of 3t. Then it is easy to see that 3t(B)

== {x: x E

~",

= {x: x E

~ n , CI6 t :5 Xi :5 CZ6 i

CIi :5 X n,

:5 CZi

for i = 1, for i

= 1,

, n} , n}.

Thus aB

+ (1 -

a)3t(B)

= {x: x E ~ n , aCti

+ (1 -

a)C 16i :5 Xi :5 aCZi

+ (1 -

a)c Z6 i

for i

=

1, ... ,

n},

358

13. Convexity and Moment and Probability Inequalities

which is also an n-dimensional rectangle. Now let {:It!, ... , :lt n !} be the group of n! permutations, and for a'i ~ 0, ~ 7 ~ 1a'i = 1 consider the rectangle given by n!

S

=L

(13.30)

a'iJri(B).

i=l

Then we have (Karlin and Rinott, 1983, and Tong, 1983, 1989):

13.35. Fact. S is of the form (13.30) iff there exists an n stochastic matrix Q such that d,

= (d ll ,

..• ,

dIn)

= ClQ,

X

n doubly

d z = (d zl , ... , d zn) = czQ

and

S = {x: x E IR n, d li

::; Xi::;

dzJor i = 1, ... , n}.

For notational convenience we may write S = Q(B). Note that when < Cli < CZi < 00 (i = 1, ... , n), the perimeter of S is equal to the perimeter of B, but S is closer to being an n-dimensional cube. Using Theorem 13.20 and Fact 13.34, Karlin and Rinott (1983) and Tong (1983, 1989) independently obtained: -00

13.36. Application. Let f(x), the density function of X= (Xl' ... ,Xn ) , be permutation invariant and log-concave. Then for every given n-dimensional rectangle B, we have P[X E B]::; P[X E Q(B)]

for each n x n doubly stochastic matrix Q. In particular,

(13.31) holds, where Cj

= (lin)

n

~ cji

is the arithmetic mean (j = 1, 2).

i=l

13.37. Remarks. (a) With the definition of multivariate majorization stated in Definition 12.34, an equivalent condition on A, B in Application 13.36 is that the 2 x n matrix (Cji) majorizes (d ji ) in the multivariate sense. (b) When -00 < Cli < CZi < 00 for each i = 1, ... ,n, the region on the right-hand side of (13.31) is a cube; and the inequality holds when Q is chosen to be the matrix with all elements being lin.

13.7. Some Statistical Applications

359

(c) When Cli = -00 (i = 1, ... , n), Application 13.36 states that the distribution function of an n-dimensional random vector with a permutation invariant log-concave density is Schur-concave. In view of Fact 13.29, this also follows from a result in Marshall and Olkin (1974) (see Corollary 12.31). (d) When Cli = -CZi < 0, the statement in Application 13.36 yields a result in Tong (1982). D

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Chapter 14 Muirhead’s Theorem and Related Inequalities

Muirhead’s theorem is one of the most well-known results involving convexity and majorization and is described in detail in Hardy, Littlewood, and P6lya (1934, 1952, pp. 44-49). In this chapter we first state Muirhead’s Theorem and some analogous and more general results, then describe some useful moment and probability inequalities that follow from their applications. The moment and probability inequalities involve a majorization ordering of the dimension vectors, and have a significant implication in the study of positive dependence of random variables.

14.1 Muihead’s Theorem and Generalizations Muirhead’s theorem, in it original form, deals with an algebraic inequality for positive real numbers:

14.1. Theorem. Let y i > O for i = 1,. . . , n ; a = ( a l , . . . , a,), b = ( b l , . . . , b,) be two n-tuples of real numbers; and let C! denote the summation over the n ! permutations of (yl , . . . , y,). If a > b, then (14.1) i=l

t=l

Marshall and Proschan (1965) proved an inequality for the expectation of permutation-invariant and convex functions of permutation-invariant random variables. Their result, which follows, implies Theorem 14.1 as a special case. 361

362

14. Muirhead's Theorem and Related Inequalities

14.2. Theorem. Let X = (Xl' ... , Xn ) be an n-dimensional random vector with a joint distribution that is permutation invariant. Let a, b be two n-dimensional n-tuples, and 1>: ~ n - - - + ~ be a continuous convex function that is permutation invariant in its n arguments. If a > b, then (14.2)

If 1> is strictly convex, then equality occurs only when a = b, possibly after reordering components, or when the X;'s are all zero with probability one. A convenient reference for the proof of Theorem 14.2 is Marshall and Olkin (1979, p. 287).

14.3. Remark. A special consequence of Theorem 14.2 is that: If a, b, and 1> satisfy the conditions in Theorem 14.2, then (14.3) where I:! is defined as in Theorem 14.1. By choosing i = 1, ... , n,

for each permutation (Jrl, ... ,

Jr n )

Xi

= tny,

for

of (1, ... ,n), and

(14.3) implies (14.1) as a special case.

0

As applications of Theorem 14.2, Marshall and Proschan (1965) gave the following results: 14.4. Theorem. If the joint distribution of X = (Xl' ... ,Xn ) is permutation invariant, then 1

- E max{O, Xl' ... , Xn } is decreasing in n for n = 1,2, . . .. n

(14.4)

Proof. Immediate by letting 1>(Xl,'" ,xn)=max{O,xl,··· ,xn }, a= «n -1)-1, ... , (n _1)-1, 0) and b = (n-1, ... ,n- l) in Theorem 14.2. 0

14.1 Muirhead's Theorem and Generalizations

363

14.5. Theorem. Let Xl' ... , X; be independent and identically distributed random variables with a common distribution function F(x), and let 1J : ~ ~ ~ be continuous and convex. Then (14.5)

is decreasing in n

= 1, 2, ... ,

where F(n) is the nth convolution of F.

Proof. This follows from Theorem 14.2 by letting a = «n1)-1, ... , (n -1)-1, 0), b = (n- 1 , ••• ,n- 1 ) , and Xl, ... ,Xn be i.i.d. D random variables with distribution function F. A different generalization of Theoerm 14.1, given in Proschan and Sethuraman (1977b), concerns the multiplication of log-convex functions. Their result states: 14.6. Theorem. Let x = (Jt1 , ••• , Jr n) be a permutation of (1, 2, ... , n) and let ~ ! denote the summation over all such n! permutations. Let a, b be two n-tuples, 'ljJ(i, z) be a log-convex function in z for each i E {I, 2, ... ,n}, and define (14.6)

Then g(a) 2: g(b) holds for all such 'lJ1 iff a > b. For a continuous analog of Muirhead's theorem, Ryff (1967) gave the following result: 14.7. Theorem. Let a(s), b(s) be two bounded measurable functions defined on [0, 1], and let y(t) be a positive function defined on [0, 1] such that (y(t)Y' ELl for all p E (-00, 00). Let 1

1

g(u, a) = J log{J [y(t)]a(s) dt} ds. o

(14.7)

0

Then g(y,a)2:g(y,b)¢::>a(s»b(s)

for

sE[O,l].

(14.8)

364

14. Muirhead's Theorem and Related lnequalities

A generalization of Theorem 14.7 along the direction of Theorem 14.6 can be found in Proschan and Sethuraman (1977b):

14.8. Theorem. Let 'ljJ(t, z) defined on [0, 1] x (-00,00) be a log-convex function of z for each fixed t. Also let SUPlzlsk 'ljJ(t, z) belong to L 1 for each k < 00. For any bounded measurable function a(s) on [0, 1], define I

g",(a)

I

= f 10g{f 'ljJ(t, o

a(s)) dt} ds.

(14.9)

0

Then g",(a) "2=g",(b)

(14.10)

holds for all such 'ljJ iff a(s) > b(s).

14.9. Remark. Theorem 14.1 follows from Theorem 14.6 by choosing 'ljJ(i, z) = y ~ for i = 1, ... ,n. Similarly, Theorem 14.7 follows from Theorem 14.8 by choosing 'ljJ(t, z) = [y(t)y. 0 14.10. Remark. Examples of log-convex functions 'ljJ(t, x) that arise in probability, statistics, and analysis are as follows: (i)

Laplace transforms. Let f E L1[0, 00) and f(z) "2=0 for all z. Then the Laplace transform f*(x) = f ~fez )e- X Z dz is log-convex where-

ver finite. (ii) Examples of log-convex functions arising in analysis are presented in Mitrinovic (1970, pp. 18-20). (iii) Moments of distributions. Let f be a probability density on [0,00). Then the moment Il<x = f ~ z<Xf(z)dz is log-convex wherever finite (see e.g., Theorem 13.8). (iv) Additional examples of log-convex functions arising in stochastic 0 processes are given in Keilson (1971).

14.2.

Moment Inequalities

Applying Theorem 14.1 to permutation-invariant nonnegative random variables, we obtain the following moment inequality (Proschan and Sethuraman, 1977a; Tong, 1977):

14.2 Moment Inequalities

365

14.11. Theorem. Let (Zl' ... , ZN) have a permutation invariant density function such that p [ n ~ {Z, l 2: O}] = 1, and let a = (ai, ... , aN) and b = (b 1 , ••. , b N) be two N-tuples. If a > b, then E

Proof.

N

N

j=l

j=l

IT Zji 2: E IT ZJi.

(14.11)

By Theorem 14.1, for every w in the sample space the inequality N

N

IT

~ ! ~ Z';;lw) 2 : ~ ! j=l

(14.12)

Z~lw)

j ~ l

holds, where I:! denotes the summation taken over all permutations of (1, ... , N). The conclusion follows by taking expectations on both sides of (14.12) and the permutation invariance property of the distribution function. 0 A special case of Theorem 14.11 is 14.12. Corollary. For {3 E [0, 00) let !l(3 denote the 13th moment of the random variable Z(!lo == 1). If Z 2: 0 a.s. and a> b, then N

N

IT !lo} 2: IT !lbi'

j=l

First Proof. Immediate from Theorem Z, Z, , Z2' ... , ZN be i.i.d. random variables For the special case in which aj is possible:

(14.13)

j ~ l

2:

14.11

by

letting

o

0 and b, 2: 0 for all j, a different proof

Second Proof. Without loss of generality, we may assume that a 1 < b, :s: b 2 < a2, a l + a2 = b l + b 2 and a, = b; (i = 3, ... , n). The conclusion then follows immediately from Theorem 13.8. 0 14.13. Remark. Corollary 14.12 asserts that the function tjJ(a) = IIf=l !loi is a Schur-convex function of a. This result, in fact, is equivalent to Theorem 13.8 and is closely related to a known classic result in Hardy, Littlewood, and P6lya (1934, 1952, p. 72). It will be used in the rest of the chapter to derive probability inequalities for a class of positively dependent random variables. 0

366

14. Muirhead's Theorem and Related Inequalities

14.3. Additional Inequalities for Exchangeable Random Variables Applying Corollary 14.12 to exchangeable random variables (defined in Definition 13.11), we have the following result (Tong, 1977): 14.14. Theorem. Let X I , . . . ,Xn be exchangeable random variables and let a = (ai, ... , aN) and b = (b l , . • • , b N) be vectors of nonnegative integers such that I:;':I a; = I:;':I b, = n. Let B c ~ be an arbitrary but fixed Borel-measurable set, and denote

y(k) =

p[O {X;

E

J.

k= 1, ... , n,

B}

as defined in (13.15), where y(O) == 1. If a > b, then N

N

Il y(a;) ~Il y(b;). ;=1

(14.14)

; ~ I

Proof. Following the line of argument given in the proof of Theorem 13.16, from Corollary 14.12 we have

N

= Il ErQj(V) j ~ 1

Jl N

~

Jl N

Erbj(V)

=

rt b

E[ E

IB(g(V;, v)) I V

=

v]

N

=

n y(bJ

j=1

where r( v) = E {IB(g( VI, v)) I V

= v} is the

conditional expectation.

o

Note that Theorem 14.14 implies Theorem 13.16, but the converse is false. To see this, consider the inequaltiy y(n - 1)y(1)

~ y(n

- 2)y(2),

n

~ 3,

which follows from (n - 1, 1) > (n - 2,2) and Theorem 14.14. But Theorem 13.16 fails to apply.

14.3. Additional Inequalities

367

A special result from Theorem 14.14 and (n, 0, ... ,0) > (1, 1, ... , 1) is that

p[O {Xi

E

B}]

2:

D

P[Xi E B].

(14.15)

Since the right-hand side of (14.15) corresponds to the case of independent Xl' ... , X n , this result can be restated as:

14.15. Fact. Let Xl, ... , X; be exchangeable random variables and let ¥t, ... , Yn be i.i.d. random variables such that Xi' 1'; have a common marginal distribution. Then

p[O {X;

E

B}] 2:

p[O {1';

E

B}

J

(14.16)

holds for all Borel-measurable sets Be IR. A question of interest is whether the inequality in (14.16) also holds when the 1';'s are less positively dependent than the X;'s in a certain fashion. This leads to the problem of a partial ordering of positive dependence of exchangeable random variables, a natural extension from the comparison between positively dependent random variables and i.i.d. random variables to the comparison between two sets of exchangeable random variables. Rinott and Pollak (1980) and Shaked and Tong (1985), among others, studied this problem recently and obtained results under reasonable assumptions. A special result they obtained is for exchangeable normal variables (Rinott and Pollak, 1980, for n = 2, Shaked and Tong, 1985, for general n):

14.16. Fact. Let X I , . . . , X; be exchangeable normal variables with a common mean fl, a common variance o", and a common correlation coefficient pz. Let YI , . . . , Yn be exchangeable normal variables with a common mean fl, a common variance o", and a common correlation coefficient Pl' If 0 ~ PI < Pz, then E r r 7 ~ 14>(X;) 2: r r 7 ~ 41 >(1';) holds for all Borel-measurable functions 4>: IR - [0,00) such that the expectations exist. Consequently (14.16) holds for all Borel-measurable sets B. 14.17. Remark.

For n = 2, E

m=1 4>(X;) 2: E

r r ~ ~ 14>(1';) holds iff

Corr(4>(X I ) , 4>(Xz)) 2: Corr(4>(¥t), 4>(Yz))

(14.17)

368

14. Muirhead's Theorem and Related Inequalities

holds, which is the motivation given by Rinott and Pollak (1980) for studying this problem. Furthermore, if
14.4. Inequalities for a Class of Positively Dependent Random Variables We now show how to use the Muirhead-related moment inequality given in Theorem 14.11 to obtain more general results. We consider ndimensional random vectors X == (Xl' ... ,Xn ) and Y == (YI , . . . , Yn ) such that the X/s are not necessarily exchangeable and the Y;'s are not necessarily exchangeable, but they all have a common marginal distributon. We show that if the X;'s are more positively dependent than the Y;'s in a certain fashion, then the inequality in (14.16) holds. To obtain sufficient conditions for such a partial ordering, we first consider a sequence of i.i.d. random variables {V;}7=1> another inde' an independent pendent sequence of i.i.d. random variables { V ; } 7 ~ 1and random variable W as "building blocks." Then for a given Borel measurable function g: [ffi3 ~[ffi and a fixed n-tuple of nonnegative integers k == (k l

, ••• ,

k., 0, ... , 0),

1:s r z: n, r

k, ~ 1 for

j:S rand

2: k, == n,

(14.18)

j=l

we define an n-dimensional random vector

~ ==

(;], ... , ;n) given by

and ;k t+---+k,_1+ 1

== g(Vk 1+ ---+ k , _ I+ I , V" W), ... ,

;n == g(Vn, V"

W).

(14.19)

That is, each of the ;;'s depends on the common variable Wand on a different variable Vi' Furthermore, the first k 1 of them depend on the common variable VI' the next k z of them depend on the common variable Vz , and so on. The vector (';1, ... , ';n) will be denoted by l;(k).

14.4. Inequalities via Positive Dependence

369

It is obvious that ;1' ... , ;n have a common marginal distribution. Furthermore, the vector k plays an important role in the positive dependence of ;(k). Consider the following two special cases. Case 1: (i) W is a singular random variable and (ii) k = (1, 1, ... ,1). Clearly ;1, , ;n are i.i.d. random variables. Case 2: (i) P[ U; = u] = 1 (i = 1, , n) and (ii) k = (n, 0, ... ,0). Then P[;! = ... = ;n] = 1, so that Corr(;;, ;;') = 1 for i =1= i', This illustrates the fact that for given random variables {U;}, {V;}, and W, the "strength" of the positive dependence of the components of !;(k) can be partially determined via the diversity of the components of k. In the following theorem (Tong, 1989) we make this idea more precise by using Corollary 14.12.

14.18. Theorem. For fixed n 2:2 assume that (i) {U;}7, {V;}7, and W are stochastically independent, U1 , ••• , U; are i. i. d., V1 , ••• , Vn are i.i.d., (ii) g: ~ 3 _~ is any Borel-measurable function, and (iii) k and k* are two vectors of the form given in (14.18). Let X = !;(k) and Y = !;(k*) be two random vectors as defined in (14.19). If k > k*, then E

n

n

;=1

;=1

n eJ>(X;) 2: E n eJ>(Y;)

(14.20)

holds for all Borel-measurable functions eJ>: ~ - [0, 00) such that the expectations exist. Consequently (14.16) holds.

Proof. write

For every given eJ> 2: 0 such that the expectations exist, we can

r

= En E[ rki(l-) , W) I W],

(14.21)

j=l

where (14.22) denotes the conditional expectation. Now for every given W = w the random variables r(V1 , w), .... , r(Vn , w) are i.i.d. and are 2:0 a.s. By defining Ilkj to be the krth moment of r(l-), w) and applying Corollary l w) 2: TIj:1 Erkt(l--j, w) holds for every 14.12, we see that T I j ~ Erkj(l--j,

370

14. Muirhead's Theorem and Related Inequalities

fixed w. Thus E

n

r

i=1

j=1

f1 (Xi ) = E f1 E[ rkj(l-j, 2:: E

I

W) W]

r"

n

j ~ 1

i ~ 1

II E[ rkt(l-j, W) I W] = E Il (1';).

D

In the following corollary we show that if the elements in k and k* are even integers (including 0), then the condition that eJ> 2:: 0 can be dropped. 14.19. Corollary. Let {Uin {V;E, W, and g satisfy the conditions stated in Theorem 14.18. Let k, k" be two n-tuples such that their components are nonnegative even integers. If k > k", then (14.20) holds for all Borel-measurable functions : IR ~IR such that the expectations exist. Proof,

The proof is as for Theorem 14.18 since 1k> 1k*.

D

In certain applications the special case k" = (1, ... ,1) is of great interest. In the following corollary we show that if the vector k contains only even integers, then again the condition that 2:: 0 can be removed. 14.20. Corollary. Let {Ui H, {V;H, W, and g satisfy the conditions in Theorem 14.18. Let k, k" be two n-tuples such that k" = (1, ... ,1) and the components of k are nonnegative even integers such that ~ 7 = k1 , = n. Then (14.20) holds for all Borel-measurable functions : IR ~IR. Proof. For every fixed W = w the function r defined in (14.22) satisfies (again from Corollary 14.12) r

Il Erkj(l-j, w) 2:: [Er

2

(V1 '

wW/2 2:: [Er(V1 , wW·

j=1

The conclusion then follows after unconditioning.

D

As a special consequence, we observe 14.21. Corollary. Let {UiH, {V;}'i, W, and g satisfy the conditions in Theorem 14.18, and let s(k) be the random vector defined in (14.19). Let

14.5. Applications to Special Families

371

k = (n, 0, ... , 0), k* = (1, ... ,1), and let n be an even positive integer. Then (14.16) holds for all Borel-measurable functions cf>: ~ ~ ~ .

In certain applications to be discussed in Section 14.5 we restrict our attention to a family of random variables such that ;(k) is obtained by choosing k = (s, 1, ... , 1,0, ... ,0) in (14.18). For notational convenience we shall denote this random vector by ;(s). The following corollary shows how the components of ;(s) depend on s. Its proof follows immediately from Theorem 14.18 and is omitted. 14.22. Coronary. Let {U;}7, { ~ H ,W, and g satisfy the conditions in Theorem 14.18. For given s 2: 1, let ;(s) = (;1' ... , ;n) be the random vector obtained according to (14.18) by choosing

k2

= ... = k n - s + l = 1,

kn -

s +2

= ... = k; = O.

Let X = ;(s + 1) and Y = ;(s). Then (14.20) holds for all such cf> all n, and all s < n.

14.5.

(14.23) 2:

0, for

Applications to Special Families of Random Variables and Distributions

In this section we state some applications of the main results in Section 14.4 for obtaining inequalities via this partial ordering of positive dependence for several families of random variables and distributions (see Tong, 1989).

14.5.1.

Exchangeable Random Variables

Consider the random variables defined for i X;=g(lf;, VI' W),

= 1, ... , n by

Y;=g(U;,

~ , W).

Then Xl' ... ,Xn are exchangeable and Y\, ... , Yn are exchangeable. But for k = (n, 0, ... ,0) and k" = (1, ... ,1), we have X 4 ;(k) and y 4 ;(k*). Thus a partial ordering of positive dependence can be

372

14. Muirhead's Theorem and Related Inequalities

obtained by applying Theorem 14.18 and the related results given in Section 14.4 By applying this result to the exchangeable normal, t, chi-square, gamma, F, and exponential variables, we obtain many useful inequalities as special cases. The multivariate normal variables will be treated separately in this section. Exchangeable exponential variables can be obtained by taking feu, v, w) = min(u, v, w) as considered previously by Marshall and Olkin (1967), and have an important application in reliability theory.

14.5.2.

Distributions with the Semigroup Property

Let {fe(x): () E Q} denote a family of density functions, and assume that Q is an interval of real numbers or an interval of integers. It is said to possess the semigroup property (see, e.g., Proschan and Sethuraman, 1977a) if ()', ()" E Q implies ()' + ()" E Q and the convolution fe'(x) * fe"(x) = feo+e'.(x),

14.23. Application. Let X e,I' ... , Xe,n denote i.i.d. random variables with density fe(x), and for fixed ()o and ()o - () E Q let X eo-e denote another independent random variable with density feo-eCx). Next define an n-dimensional random vector X( () = (XI' ... , X n) such that Xi = Xe,i + X eo-e for i = 1, ... ,n. If {fe(x): () E Q} possesses the semigroup property and if ()1, ()z E Q, ()1 =1= ()z implies 10 1 - Ozl E Q, then (a) E e II7=1 ¢(Xi ) is a decreasing function of () for () < ()o for all Borelmeasurable functions ¢ ~ 0 (provided that the expectation exists); (b) E e II7=1 ¢(X;) is a decreasing function of () for () < ()o for all even positive integers n and all Borel-measurable functions ¢; (c) Pe[XI E B, ... , X; E B] is a decreasing function of () for () < ()o for all Borelmeasurable sets B c IR . Proof. For every fixed ()1' ()z E Q such that ()1 < ()z < ()o, define U; = X elo i' ~ = X e 2 - e l o i ( i =... 1 , ,n) and W=XeO-Xe2. The conclusions follow from Theorem 14.18. D Note that Application 14.23 applies to the binomial, gamma, Poisson distributions, the Poisson process, and several other distributions.

14.5. Applications to Special Families

14.5.3.

373

The Multivariate Normal Distribution

Applying Theorem 14.18 we now show how the positive dependence of a multivariate normal variable with a common marginal distribution can be partially ordered via the correlation coefficients.

14.24. Application. Let 0::; PI < pz::; 1, k and k* be two n-tuples of nonnegative integers given as in (14.18), and R =R(k) = (Pij) be a correlation matrix such that for i =1= j, Pij

= ,-I

pz if 1::;i,j::;k l,k l+1::;i,j::;kl+k z , ... , :!=/m+ 1::;i,j::;n; {

PI

otherwise.

[That is, the random variables X I, . . . , X; are partitioned into r groups of sizes k«. ... , k, respectively; the correlation coefficients of the variables within the same group are pz, and the correlation coefficients between groups are Pl'] Let X - . N " n ( crR(k)) ~ , and Y ~ K n ( ~aZR(k*)), , where ~ = ( t J "... , tJ,). (a) If k>k*, then (14.20) holds for all such