Advances on Fractional Inequalities

SpringerBriefs in Mathematics

For further volumes: http://www.springer.com/series/10030

George A. Anastassiou

Advances on Fractional Inequalities

123

George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152 USA [email protected]

ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4614-0702-7 e-ISBN 978-1-4614-0703-4 DOI 10.1007/978-1-4614-0703-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011932688 Mathematics Subject Classification (2010): 26A33, 26D10, 26D15 © George A. Anastassiou 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my wife Koula and my daughters Angela and Peggy The measure of success for a person is the magnitude of his/her ability to convert negative conditions to positive ones and achieve goals —The author

Preface

This short monograph is a spin-off of the author’s “Fractional Differentiation Inequalities,” a research monograph published by Springer, New York, 2009. It continues and complements the earlier book to various interesting and important directions. In this short monograph we use primarily the Caputo fractional derivative, as the most important in applications, and we present first fractional differentiation inequalities of Opial type where we involve the so-called balanced fractional derivatives. We continue with right and mixed fractional differentiation Ostrowski inequalities in the univariate and multivariate cases. Then we present right and left, as well as mixed, Landau fractional differentiation inequalities in the univariate and multivariate cases. The inequalities are given for various norms. Fractional differentiation inequalities are by themselves an important and great mathematical topic for research. Furthermore they have many applications, the most important ones are in establishing uniqueness of solution in fractional differential equations and systems and in fractional partial differential equations. Also they provide upper bounds to the solutions of the above equations. In this brief monograph we give several applications. Each chapter is self-contained and can be read independently of the others and several graduate courses and seminars can be taught out of this monograph. The final preparation of this book took place in Memphis, USA, during 2010– 2011. Fractional calculus has become very useful over the last 40 years due to its many applications in almost all applied sciences. We see now applications in acoustic wave propagation in inhomogeneous porous material, diffusive transport, fluid flow, dynamical processes in self-similar structures, dynamics of earthquakes, optics, geology, viscoelastic materials, biosciences, bioengineering, medicine, economics, probability and statistics, astrophysics, chemical engineering, physics, splines, tomography, fluid mechanics, electromagnetic waves, nonlinear control, signal processing, control of power electronics, converters, chaotic dynamics, polymer

vii

viii

Preface

science, proteins, polymer physics, electrochemistry, statistical physics, rheology, thermodynamics, neural networks, etc. By now almost all fields of research in science and engineering use fractional calculus as better describing them. So as expected this book being a part of fractional calculus is useful for researchers and graduate students for research, seminars and advanced graduate courses, in pure and applied mathematics, engineering, and all other applied sciences. I would like to thank my family members for their support and for their tolerance to accept my continuous mathematics habit. Also I am greatly indebted and thankful to my Ph.D. student Razvan Mezei for the heroic and fantastic technical preparation of the manuscript in a very short time. Memphis, TN, USA

George A. Anastassiou

Contents

1

Opial-Type Inequalities for Balanced Fractional Derivatives . . . . . . . . . 1.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 2 3 18

2

Univariate Right Caputo Fractional Ostrowski Inequalities . . . . . . . . . . 2.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

21 21 22 27

3

Multivariate Right Caputo Fractional Ostrowski Inequalities . . . . . . . . 3.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

29 29 30 39

4

Univariate Mixed Fractional Ostrowski Inequalities . . . . . . . . . . . . . . . . . . . 4.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

41 41 42 48

5

Multivariate Radial Mixed Fractional Ostrowski Inequalities . . . . . . . . 5.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51 51 52 60

6

Shell Mixed Caputo Fractional Ostrowski Inequalities . . . . . . . . . . . . . . . . 6.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63 63 64 75

7

Left Caputo Fractional Uniform Landau Inequalities . . . . . . . . . . . . . . . . . 7.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

77 77 78 ix

x

Contents

7.3 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

83 84

8

Left Caputo Fractional Lp -Landau-Type Inequalities .. . . . . . . . . . . . . . . . 8.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

85 85 86 91

9

Right Caputo Fractional Lp -Landau-Type Inequalities . . . . . . . . . . . . . . . 9.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93 93 94 99

10 Mixed Caputo Fractional Lp -Landau-Type Inequalities . . . . . . . . . . . . . . 10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

101 101 102 110

11 Multivariate Caputo Fractional Landau Inequalities . . . . . . . . . . . . . . . . . . 11.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

113 113 114 121

Chapter 1

Opial-Type Inequalities for Balanced Fractional Derivatives

Here we study Lp ; p > 0; fractional Opial-type inequalities subject to high-order boundary conditions. They involve the right and left Caputo, Riemann–Liouville fractional derivatives. These derivatives are blended together into the balanced Caputo, Riemann–Liouville, respectively, fractional derivative. We give an application to a balanced fractional boundary value problem by proving uniqueness of the solution. This chapter relies on [7].

1.1 Introduction The presented material here is related to [5]. This chapter is motivated by the famous theorem of Z. Opial [14], 1960, which follows. Theorem 1.1. Let x.t/ 2 C 1 .Œ0; h/ be such that x.0/ D x.h/ D 0, and x.t/ > 0 in .0; h/: Then Z h Z h ˇ ˇ ˇx.t/x 0 .t/ˇ dt  h .x 0 .t//2 dt: () 4 0 0 In ./; the constant optimal function

h 4

is the best possible. Inequality ./ holds as equality for the ( x.t/ D

ct; 0  t  h=2 c.h  t/;

h 2

 t  h;

where c > 0 is an arbitrary constant. To prove easily Theorem 1.1, Beesack [8] gave the following well-known Opialtype inequality which is used very commonly. This is another inspiration to this chapter.

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 1, © George A. Anastassiou 2011

1

2

1 Opial-Type Inequalities for Balanced Fractional Derivatives

Theorem 1.2. Let x.t/ be absolutely continuous in Œ0; a, and x.0/ D 0: Then Z a Z a ˇ ˇ ˇx.t/x 0 .t/ˇ dt  a .x 0 .t//2 dt: 2 0 0

()

Inequality ./ is sharp, it is attained by x.t/ D ct; c > 0 is an arbitrary constant. See also [13]. Opial-type inequalities are used a lot in proving uniqueness of solutions to differential equations and also to give upper bounds to their solutions. They have become a great subject of intensive research and there exists a great literature about them. Typical and important sources on them are the monographs [1, 3].

1.2 Background We need Definition 1.3 ([4, 10–12, 15]). Let f 2 AC m .Œa; b/ (space of functions from Œa; b into R with m  1 derivative absolutely continuous function on Œa; b; m 2 N; where m D d˛e .de is the ceilling of the number), ˛ > 0: We define the right Caputo fractional derivatives of order ˛ > 0 by ˛ f .x/ D Db

.1/m  .m  ˛/

Z

b x

.  x/m˛1 f .m/ ./d:

(1.1)

0 We set Db f .x/ D f .x/; 8x 2 Œa; b:

We mention the right Caputo fractional Taylor formula with integral remainder. Theorem 1.4 ([4]). Let f 2 AC m .Œa; b/; x 2 Œa; b; ˛ > 0; m D d˛e. Then f .x/ D

m1 X kD0

1 f .k/ .b/ .x  b/k C kŠ  .˛/

Z

b x

˛ .  x/˛1 Db f ./d:

(1.2)

We make Definition 1.5 ([9, p. 38]). Let f 2 AC m .Œa; b/; m 2 N;where m D d˛e, ˛ > 0: We define the left Caputo fractional derivative of order ˛ > 0 by ˛ Da f .x/ D

1  .m  ˛/

Z a

x

.x  t/m˛1 f .m/ .t/dt:

(1.3)

0 We set Da f .x/ D f .x/; 8x 2 Œa; b: We mention the left Caputo fractional Taylor formula with integral remainder.

1.3 Main Results

3

Theorem 1.6 ([9, p. 40]). Let f 2 AC m .Œa; b/; m 2 N;where m D d˛e; ˛ > 0; x 2 Œa; b: Then f .x/ D

m1 X kD0

f .k/ .a/ 1 .x  a/k C kŠ  .˛/

Z

x a

˛ .x   /˛1 Da f . /d:

(1.4)

Definition 1.7 ([5]). Let f 2 AC m .Œa; b/; m 2 N;where m D d˛e; ˛ > 0; x 2 Œa; b: We define the balanced Caputo fractional derivative, ( ˛

D f .x/ WD

˛ Db f .x/; for

aCb 2

 x  b;

˛ f .x/; for ˛  x < Da

aCb : 2

(1.5)

1.3 Main Results We give Theorem 1.8. Let ˛ > 0; m D d˛e, x 2 Œa; b; f 2 AC m .Œa; b/; f .k/ .a/ D f .k/ .b/ D 0; k D 0; 1; :::; m  1I 0 < p < 1 and q < 0 W p1 C q1 D 1:     ˛ ˛ and of fixed sign, a.e.; Db f ¤ 0 on a; aCb f ¤ 0 on aCb ;b Assume Da 2 2 and of fixed sign, a.e. Then Z

b a

jf .w/j jD ˛ f .w/j dw 0

1

..˛1/C p2 /

.b  a/ C B  @ 1=p A 1 2.˛C p /  .˛/..p.˛  1/ C 1/.p.˛  1/ C 2/ 2 !2=q !2=q 3 Z aCb Z b 2 ˛ ˛ 5: 4 f .t/jq dt C f .t/jq dt jDa jDb aCb 2

a

(1.6)

Proof. Here ˛ > 0; N 3 m D d˛e; x 2 Œa; b; f 2 AC m .Œa; b/; f .k/ .a/ D f .k/ .b/ D 0; for k D 0; 1; :::; m  1I 0 < p < 1 and q W p1 C q1 D 1: Then by (1.4), 1 f .x/ D  .˛/

Z

x a

˛ .x  t/˛1 Da f .t/dt;

8x 2 Œa; b:

4

1 Opial-Type Inequalities for Balanced Fractional Derivatives

Also by (1.2), 1 f .x/ D  .˛/

Z

b x

˛ .t  x/˛1 Db f .t/dt;

8x 2 Œa; b:

  ˛ ˛ f ¤ 0 over ˛; aCb f ¤0 Assume Da and of fixed sign, a.e., and assume Db 2  aCb  over 2 ; b and of fixed sign, a.e. Then we have   Z w 1 aCb ˛1 ˛ .w  t/ ; jf .w/j D jDa f .t/j dt; 8 w 2 a;  .˛/ a 2 and 1 jf .w/j D  .˛/

Z

b

˛1

aCb ; b : 8w 2 2 

˛ f .t/j dt; jDb

.t  w/ w

Therefore by H¨older’s inequality .q D p=.p  1/ < 0/ we obtain 1 jf .w/j   .˛/

Z a

w

p.˛1/

.w  t/

1=p

Z

w

dt a

˛ f .t/jq jDa

1=q :

dt

Hence 1

.w  a/.˛1/C p 1 jf .w/j   .˛/ .p.˛  1/ C 1/1=p

Z

w a

˛ f .t/jq jDa

1=q dt

;

  aCb 8w 2 a; : 2

;

aCb ;b : 8w 2 2

Similarly we derive 1

.b  w/.˛1/C p 1 jf .w/j   .˛/ .p.˛  1/ C 1/1=p Set

Z z1 .w/ WD

w

a

Z

b

w

!1=q ˛ f .t/jq jDb

dt



˛ f .t/jq dt; z1 .a/ D 0; jDa

˛ ˛ f .w/jq and jDa f .w/j D .z01 .w//1=q ; all a  w < so that z01 .w/ D jDa

Z z2 .w/ WD

b w

˛ f .t/jq dt; jDb

aCb : 2

Set

z2 .b/ D 0;

ˇ ˛ ˇq ˇ ˛ ˇ so that z02 .w/ D  ˇDb f .w/ˇ and ˇDb f .w/ˇ D .z02 .w//1=q ; aCb  w  b: 2 Therefore 1

˛ f .w/j  jf .w/j jDa

1=q 1 .w  a/˛1C p  ; z1 .w/z01 .w/ 1=p  .˛/ .p.˛  1/ C 1/

1.3 Main Results

all a  w < Z

aCb 2

a

Z

aCb 2

a

5

aCb : 2

Hence 1  .˛/.p.˛1/ C 1/1=p

a f .w/j dw  jf .w/j jDa 1

.w  a/˛1C p .z1 .w/z01 .w/1=q dw

1   .˛/.p.˛1/C1/1=p

Z

aCb 2

a

!1=p Z

.w  a/p.˛1/C1

aCb 2

a

!1=q z1 .w/z01 .w/dw

0 1 ˇ aCb 1=q 

1 b  a .˛1/C.2=p/ @z21 .w/ ˇˇ 2 A D  .˛/.p.˛1/C1/1=p .p.˛1/C2/1=p 2 2 ˇa 1

D





 .˛/.p.˛1/C1/1=p .p.˛1/

C

ba 2

2/1=p



˛1C p2



0 @

z21



aCb/ 2

 11=q A

2

:

Therefore Z a

aCb 2

˛ f .w/j dw jf .w/j jDa

c



Z

aCb 2

a

!2=q ˛ f .t/jq jDa

;

dt

(1.7)

where c  WD

1 21=q  .˛/ ..p.˛  1/ C 1/ .p.˛  1/ C 2//1=p





ba 2



˛1C p2



:

(1.8)

Similarly 1

˛ f .w/j jf .w/j jDb

and Z b aCb 2

1 .b  w/.˛1/C p  .z2 .w/.z02 .w///1=q  .˛/ .p.˛  1/ C 1/1=p

˛ f .w/dwj  jf .w/j jDb

1  .˛/.p.˛  1/ C 1/1=p Z b  1=q   .b  w/.p.˛1/C1/=p z2 .w/ z02 .w/ dw aCb 2



21=q  .˛/.p.˛



ba 2

1  1/ C 1/1=p .p.˛  1/ C 2/1=p !2=q 2 Z

.˛1/C p

b

aCb 2

˛ f .t/jq dt jDb

;

6

1 Opial-Type Inequalities for Balanced Fractional Derivatives

i.e., Z

b aCb 2

˛ f .w/j dw jf .w/j jDb

c

Z



!2=q

b aCb 2

˛ f .t/jq jDb

dt

:

(1.9) t u

adding (1.7) and (1.9) we get (1.6). We present and need Theorem 1.9. Let x  s  0 and f 2 L1 .Œx; 0/; r > 0: Define Z G.s/ D Then there exists G 0 .s/ D r

0

s

Z s

.t  s/r f .t/dt: 0

.t  s/r1 f .t/dt:

(1.10)

(1.11)

Proof. Fix s0 2 Œx; 0 and observe that Z G.s0 / D

0 s0

We call

r

.t  s0 / f .t/dt D

Z

0 x

Œs0 ;0 .t/.t  s0 /r f .t/dt:

g.s; t/ WD Œs;0 .t/.t  s/r f .t/;

which is a Lebesgue integrable function for every s 2 Œx; 0: That is, g.s0 ; t/ D R0 Œs0 ;0 .t/.t  s0 /r f .t/; all t 2 Œx; 0 and G.s0 / D x g.s0 ; t/dt: We would like to study if there exists 

.t/Œs0 Ch;o .t  s0  h/r  .t/Œs0 ;0 .t  s0 /r @g.s0 ; t/ : D f .t/ lim h!0 @s h

(1.12)

We distinguish the following cases: (1) Let x  t < s0 , then there exists small enough h > 0 such that t < s0 ˙ h: That is, Œs0 ˙h;0 .t/ D Œs0 ;0 .t/ D 0: Hence, there exists @g.s0 ; t/ D 0; all t W x  t < s0 : @s (2) Let s0 < t  0; then there exists small enough h > 0 such that t > s0 ˙ h: That is, Œs0 ˙h;0 .t/ D Œs0 ;0 .t/ D 1:

1.3 Main Results

7

In that case 

.t  .s0 C h//r  .t  s0 /r @g.s0 ; t/ D f .t/ lim h!0 @s h D r.t  s0 /r1 f .t/ exists for almost all t W s0 < t  0: (3) Let t D s0 : Then we observe that @gC .s0 ; s0 / D f .s0 / @s



0  .h/r  1  0r h!0C h lim

 D 0:

Also we get  1  .h/r  1  0r lim h!0 h



 r .h/ .h/r D f .s0 / lim D f .s0 / lim h!0 h!0 .h/ h



  D f .s0 / lim .h/r1 D f .s0 / lim hr1 :

@g .s0 ; s0 / D f .s0 / @s



h!0

h!0C

The last limit does not exist if 0 < r < 1; equals f .s0 / if r D 1 and may not exist, and equals 0 if r > 1: 0 ;t / In general as a conclusion we get that #g.s exists for almost all t 2 Œx; 0: #s Next we define the difference quotient at s0 :

Ds0 .h; t/ WD f .t/

.t/Œs0 Ch;0 .t  s0  h/r  Œs0 ;0 .t/.t  s0 /r h

 ;

for h ¤ 0; and Ds0 .0; t/ WD 0: Again we distinguish the following cases: (1) Let x  t < s0 ; then there exists small enough h > 0 such that t < s0 ˙ h: Clearly then Ds0 .h; t/ D 0: (2) Let s0 < t  0; then there exists small enough h > 0 such that t > s0 ˙ h: In that case 

.t  .s0 ˙ h/r  .t  s0 /r : Ds0 .˙h; t/ D f .t/ ˙h Call  WD t  s0 > 0I clearly 0 <   jxj : Define '.h/ WD

.t  s0  h/r  .t  s0 /r .h/r r D ; for h close to zero; r > 0: h h

8

1 Opial-Type Inequalities for Balanced Fractional Derivatives

That is, Ds0 .h; t/ D f .t/'.h/: If r D 1; then '.h/ D 1 and Ds0 .h; t/ D f .t/: We now treat the following subcases: (2.i /) If r > 1 and jhj small, then by mean value theorem we get jDs0 .h; t/j D jf .t/j j'.h/j  jf .t/j r2r1 jxjr1 : That is, for r  1 and small jhj we derive jDs0 .h; t/j  r2r1 jxjr1 jf .t/j : (2.i i // If 0 < r < 1 and jhj small we get the following: The function  ./ D r ; 0    jxj is concave and increasing. Let h > 0; then j'.h/j D

r  .  h/r < r1 D .r  s0 /r1 ;   .  h/

and for h < 0; again '.h/ D

.  h/r  r < r1 D .t  s0 /r1 : .  h/  

Therefore we obtain jDs0 .h; t/j  jf .t/j .t  s0 /r1 ; for 0 < r < 1 and jhj small. (3) Case of t D s0 ; then Ds0 .h; s0 / D f .s0 / jhjr1 ; for h < 0; and Ds0 .h; s0 / D 0; for h > 0: So, if r  1 we obtain jDs0 .h; s0 /j  jf .s0 /j jxjr1 ; for small jhj. If 0 < r < 1; then for small jhj with h < 0; the function Ds0 .h; s0 / may be unbounded.

1.3 Main Results

9

In conclusion we get: (I) For r  1; that jDs0 .h; t/j  r2r1 jxjr1 kf k1 < C1; for almost all t 2 Œx; 0: (II) For 0 < r < 1; that jDs0 .h; t/j  .t/; for almost all t 2 Œx; 0; where 0; x  t  s0 ; .t/ WD jf .t/j .t  s0 /r1 ; s0 < t  0: Clearly is integrable on Œx; 0: Then by Theorem 24.5, pp. 193–194 of [2], we get that integrable function and there exists Z

defines an

0

@g.s0 ; t/ dt @s x Z 0 Z r1 D r .t  s0 / f .t/dt C

0

@g.s0 ;/ @s

G .s0 / D

s0

Z D r

s0

0dt

x

0

.t  s0 /r1 f .t/dt:

s0

t u

That proves the claim. We use

Theorem 1.10 ([3] Theorem 7.7, p. 117). Let 0  s  x and f 2 L1 .Œ0; x/; r > 0: Define Z s

F .s/ WD Then there exists F 0 .s/ D r

0

Z 0

s

.s  t/r f .t/dt:

(1.13)

.s  t/r1 f .t/dt;

(1.14)

8s 2 Œ0; x: We make Remark 1.11. Let f 2 AC m .Œa; b/; m 2; N; m D d˛e ; ˛ > 1: Suppose f .k/ .a/ D f .k/ .b/ D 0; k D 0; 1; :::; m  1: Then we get by (1.4) Z x 1 ˛ .x  t/˛1 Da f .t/dt; 8x 2 Œa; b; (1.15) f .x/ D  .˛/ a also (by (1.2)) 1 f .x/ D  .˛/

Z

b x

˛ .t  x/˛1 Db f .t/dt;

8x 2 Œa; b:

(1.16)

10

1 Opial-Type Inequalities for Balanced Fractional Derivatives

Suppose that ˛ ˛ f; Db f 2 L1 .Œa; b/: Da

(1.17)

Here ˛ … N and m  1 < ˛ < m; m 2 N: So from ˛ > m  1 we get .˛  1/  .m  2/ > 0: Thus for ` D1; :::; m  1; we obtain 0  `  1  m  2; and .˛1/  .`  1/  .˛1/  .m  2/ > 0: Hence .˛  1/  .`1/ > 0 and ˛  ` > 0; ` D 1; :::; m  1: Therefore by Theorem 1.8 and (1.15) we get f 0 .x/ D

.˛  1/  .˛/

Z

1 D  .˛  1/

x

a

˛ .x  t/˛2 Da f .t/dt

Z

x

a

˛ .x  t/˛2 Da f .t/dt:

Similarly, we obtain f

.`/

1 .x/ D  .˛  `/

Z

x a

˛ .x  t/˛`1 Da f .t/dt;

for ` D 1; :::; m  1; 8x 2 Œa; b:

(1.18)

Using Theorem 1.9 and (1.16), we derive f

.`/

.1/` .x/ D  .˛  `/

Z

b x

˛ .t  x/˛`1 Db f .t/dt;

` D1; :::; m  1; x 2 Œa; b: (1.19)

We prove (p D 1; q D 1 case). Theorem 1.12. Let ˛ > 0; m D d˛e; x 2 Œa; b; f 2 AC m .Œa; b/; f .k/ .a/ D f .k/ .b/ D 0; for k D 0; 1; :::; m  1: Assume ˛ f Da





  aCb aCb ˛ 2 L1 a; ; Db f 2 L1 ;b : 2 2

Then Z a

b



 1 b  a ˛C1  jf .x/j jD f .x/j dx  2  .˛ C 2/

˛ ˛  kDa f k2 h aCb  C kDb f k2 ˛

1; a;

2

i h 1; aCb 2 ;b

 (1.20)

1.3 Main Results

11

Proof. Let ˛ > 0; m D d˛e; x 2 Œa; b; f 2 AC m .Œa; b/; f .k/ .a/ D f .k/ .b/ D 0; for k D 0; 1; :::; m  1: Suppose 

 aCb ; 2 L1 a; 2

  aCb ˛ Db f 2 L1 ;b : 2 ˛ Da f

Then jf .x/j 

aCb .x  a/˛ ˛ f k1;a; aCb  ; a  x < ; kDa  .˛ C 1/ 2 2

jf .x/j 

.b  x/˛ aCb ˛  x  b: f k1; aCb ;b ; kDb  .˛ C 1/ 2 2

and

Thus ˛ f .x/j  jf .x/j jDa

  .x  a/˛ aCb ˛ f k2 h aCb  ; 8x 2 a; : kDa 1; a; 2  .˛ C 1/ 2

Hence Z

aCb 2

a

˛ f .x/j dx jf .x/j jDa

˛ f k2 kDa

1;



h  a; aCb 2



 .˛ C 2/

.b  a/˛C1 : 2˛C1

Similarly ˛ f k1;h aCb ;b i jf .x/j  kDb 2

.b  x/˛ ;  .˛ C 1/

 8x 2

aCb ;b 2



and ˛ f .x/j  jf .x/j jDb

˛ 2

D f h aCb i b 1; ;b 2

 .˛ C 1/

Hence Z

b aCb 2

˛ f .x/j dx  jf .x/j jDb

.b  x/˛ ; 8x 2

˛ 2 h

D aCb i b 1; ;b 2

 .˛ C 2/



aCb ;b : 2

.b  a/˛C1 : 2˛C1

12

1 Opial-Type Inequalities for Balanced Fractional Derivatives

Consequently it holds Z

 .b  a/˛C1 2 h ˛ ˛ h i  CkDb f k1; aCb ;b ; kDa f k jf .x/j jD f .x/j dx  ˛C1 1; a; aCb 2  .˛ C 2/ 2 2

b a

˛

t u

establishing the claim. We continue with

Theorem 1.13. Let f 2 AC m .Œa; b/; N 3 m D d˛e; ˛  1: Assume f .k/ .a/ D ˛ ˛ f .k/ .b/ D 0; k D 0; 1; :::; m  1: Assume Da f; Db f 2 L1 .Œa; b/; p; q > 1 W 1 1 1 C D 1; ˛ > m  : p q p Then (i) Case of 1 < q  2: Then Z b ˇ .`/ ˇ ˛ ˇf .w/ˇ jD f .w/j dw a



2

   ˛`C p1



.b  a/

˛`1C p2

 1

 .˛  `/Œ.p.˛  `  1/ C .p.˛  `  1/ C 2/ p !2=q Z b



a

jD ˛ f .w/jq dw

;

` D 1; :::; m  1:

(1.21)

(ii) Case of q > 2: Then Z b ˇ .`/ ˇ ˛ ˇf .w/ˇ jD f .w/j dw a



2

   ˛`C q1

  ˛`1C p2

.b  a/

1

 .˛  `/Œ.p.˛  `  1/ C 1/.p.˛  `  1/ C 2/ p !2=q Z b



a

jD ˛ f .w/jq dw

;

` D 1; :::; m  1:

(1.22)

(iii) When p D q D 2; ˛ > m  12 ; .` D 1; :::; m  1/: Then Z

b a

ˇ .`/ ˇ ˛ ˇf .w/ˇ jD f .w/j dw 

2.˛`C 2 / .b  a/˛` p  .˛  `/Œ 2.˛  `/.2˛  2`  1/ ! Z 1

b



a

.D ˛ f .w//2 dw :

(1.23)

1.3 Main Results

13

Proof. By the assumption of the theorem we have Z x 1 ˛ .x  t/˛`1 Da f .t/dt; f .`/ .x/ D  .˛  `/ a for ` D 1; :::; m  1; 8x 2 Œa; b: Similarly it holds .`/

f

.1/` .x/ D  .˛  `/

Z

b x

˛ .t  x/˛`1 Db f .t/dt;

` D l; :::; m  1; 8x 2 Œa; b: Hence ˇ .`/ ˇ ˇf .x/ˇ 

1  .˛  `/

Z

x a

˛ .x  t/˛`1 jDa f .t/j dt;

` D 1; :::; m  1; 8x 2 Œa; b and ˇ .`/ ˇ ˇf .x/ˇ 

1  .˛  `/

Z

b x

˛ .t  x/˛`1 jDb f .t/j dt;

` D 1; :::; m  1; 8x 2 Œa; b: Let p; q > 1 W p1 C q1 D 1: Acting now as in the proof of Theorem 9 of [5], and t u replacing f by f .`/ and ˛ by ˛  ` we prove our theorem. We further give Theorem 1.14. Let ˛  1; ` D 1; :::; m1I f 2 AC m .Œa; b/ ; m D d˛e; f .k/ .a/ D f .k/ .b/ D 0; k D 0; 1; :::; m  1:   ˛ ˛ ˛ ; and of Assume Da f; Db f 2 L1 .Œa; b/: Assume Da f ¤ 0 over a; aCb 2  aCb  ˛ fixed sign, a.e.; Db f ¤ 0 over 2 ; b and of fixed sign, a.e., 0 < p < 1; q < 0I 1 1 p C q D 1: Then Z

b a

ˇ .`/ ˇ ˛ ˇf .w/ˇ jD f .w/j dw 0

B @



.b  a/

  ˛`C p1

2 2

4

Z a

aCb 2

.˛`1/C p2

 .˛  `/

1



C .p.˛`1/C1/1=p A .p.˛`1/C2/

! q2 ˛ f .t/jq dt jDa

Z C

b aCb 2

˛ f .t/jq dt jDb

! q2 3

5:

(1.24)

14

1 Opial-Type Inequalities for Balanced Fractional Derivatives

Proof. As in the proof of Theorem 1.6, by replacing f by f .`/ and ˛ by ˛  `; for ` D 1; :::; m  1I ˛  1: t u Theorem 1.15. The p D 1; q D 1 case again. Let ˛  1; ` D 1; :::; m  1I f 2 AC m .Œa; b/; m D d˛e, f .k/ .a/ D f .k/ .b/ D 0I k D 0; 1; :::; m  1: Assume ˛ ˛ Da f; Db f 2 L1 .Œa; b/: Then Z a

b

˛`C1 ˇ .`/ ˇ ˛ 1 ˇf .x/ˇ jD f .x/j dx  b  a  2  .˛  ` C 2/

˛ ˛  kDa f k2 h aCb  C kDb f k2 1; a;

i h 1; aCb 2 ;b

2

 : (1.25)

Proof. As in the proof of Theorem 1.12, by replacing f by f .`/ and ˛ by ˛  `, for ` D 1; :::; m  1: t u We need ([4, 9–11], p. 22) Definition 1.16. Let ˛ > 0, m D d˛e, f 2 AC m .Œa; b/: We define the right Riemann–Liouville fractional derivative by ˛ f .x/ Db

.1/m WD  .m  ˛/



d dx

m Z

b

x

.t  x/m˛1 f .t/dt;

(1.26)

0 Db f .x/ WD I .x/ (the identity operator). We also define the left Riemann–Liouville fractional derivative by

˛ f .x/ WD DaC

1  .m  ˛/



d dx

m Z a

x

.x  t/ma1 f .t/dt;

(1.27)

0 DaC f .x/ WD I.x/:

We further define the balanced Riemann–Liouville fractional derivative ([5]) ( ˛

D f .x/ WD

˛ f .x/; for Db

aCb 2

 x  b;

˛ f .x/; for ˛  x < DaC

aCb : 2

(1.28)

Remark 1.17. Let f 2 C m .Œa; b/; m D d˛e ; ˛ > 0: In [6] we have proved that ˛ ˛ Db f .x/; Da f .x/ are continuous functions in x 2 Œa; b: Of course C m .Œa; b/  m AC .Œa; b/; so that f 2 AC m .Œa; b/:

1.3 Main Results

15

˛ Thus by Theorem 9 of [4], we get that also Db f .x/ exists and continuous for .k/ every x 2 Œa; b: Furthermore if f .b/ D 0; k D 0; 1; :::; m  1 we obtain ˛ ˛ f .x/ D Db f .x/; Db

(1.29)

8x 2 Œa; b: ˛ Similarly, by [9], p. 39, we get that DaC f .x/ exists and continuous in x 2 Œa; b: .k/ Furthermore if f .˛/ D 0; k D 0; 1; :::; m  1 we get ˛ ˛ f .x/ D Da f .x/; DaC

(1.30)

8x 2 Œa; b: So if f .k/ .˛/ D f .k/ .b/ D 0; k D 0; 1; :::; m  1 we find D˛ f .x/ D D ˛ f .x/;

(1.31)

8x 2 Œa; b: So by Theorem 1.13 we obtain the corresponding results for the balanced Riemann–Liouville fractional derivative. Theorem 1.18. Let f 2 C m .Œa; b/; N 3 m D d˛e ; ˛  1: Assume f .k/ .a/ D f .k/ .b/ D 0; k D 0; 1; :::; m  1: Suppose p; q > 1 W p1 C q1 D 1; ˛ > m  p1 : (i) Case of 1 < q  2: Then Z

b a

ˇ .`/ ˇ ˛ ˇf .w/ˇ jD f .w/j dw



2

   ˛`C p1



.b  a/

˛`1C p2

 1

 .˛  `/Œ.p.˛  `  1/ C 1/.p.˛  `  1/ C 2/ p ! q2 Z b



a

jD˛ f .w/jq dw

;

` D 1; :::; m  1:

(1.32)

(ii) Case of q > 2: Then Z a

b

ˇ .`/ ˇ ˛ ˇf .w/ˇ jD f .w/j dw   ˛`C 1



˛`1C 2



q p .b  a/ 2   .˛  `/Œ.p.˛  `  1/ C 1/.p.˛  `  1/ C 2/1=p !2=q Z

b



a

jD˛ f .w/jq dw

;

` D 1; :::; m  1:

(1.33)

16

1 Opial-Type Inequalities for Balanced Fractional Derivatives

(iii) When p D q D 2; ˛ > m  12 ; .` D 1; :::; m  1/: Then Z

b a

2.˛`C 2 / .b  a/˛` p .w/ jD f .w/j dw   .˛  `/Œ 2.˛  `/.2˛  2`  1/ ! Z b  .D˛ f .w//2 dw : (1.34) 1

.`/

jf j

˛

a

Remark 1.19. For f 2 C m .Œa; b/ m D d˛e, ˛ > 0, one can rewrite also Theorems 1.8, 1.12, 1.14, and 1.15 in the language Riemann–Liouville fractional derivatives, as we did in Theorem 1.18. Application 1.20. Uniqueness of solution to fractional boundary value problem   .D f /.t/ D F t; f; f 0 ; :::; f .m1/ W 1 f .i / .a/ D ai ; f .i / .b/ D bi ; 0  i  m  1; m WD d e ;  1; > m  : 2 (1.35)



f; Db f 2 L1 .Œa; b/I F is a continuous function on Here f 2 AC m .Œa; b/; Da Œa; b  Rm ; t 2 Œa; b: Also F fulfills the Lipschitz condition

X ˇ ˇ m1 ˇ ˇ qi .t/ ˇzi  z0i ˇ ; m1 

ˇ ˇF .t; z0 ; z1 ; :::; zm1 /  F .t; z0 ; z0 ; :::; z0 0

1

i D0

where qi .t/  0; 0  i  m  1 are continuous functions on Œa; b: Call 1 m1 X 2. i C 2 / .b  a/ i .a; b/ WD p kqi k1  .  i / 2.  i /.2  2i  1/: i D0 Suppose that .a; b/ < 1:

(1.36)

Next we prove uniqueness to the solution of (1.35). Let f1 ; f2 as above fulfilling (1.35), that is,   .m1/ .i / .i / W fj .a/ D ai ; fj .b/ D bi ; 0  i  m1; .D v fj /.t/ D F t; fj ; fj0 ; :::; fj j D 1; 2: Call g WD f1  f2 : Then     .m1/ .m1/  F t; f2 ; f20 ; :::; f2 ; .D g/.t/ D F t; f1 ; f10 ; :::; f1 such that g .i / .a/ D 0; g.i / .b/ D 0; 0  i  m  1:

(1.37)

1.3 Main Results

17

Here ˇ    ˇ ˇ .m1/ .m1/ ˇ  F t; f2 ; f20 ; :::; f2 ˇ ˇF t; f1 ; f10 ; :::; f1 

m1 X

ˇ ˇ m1 ˇ ˇ ˇ .i / ˇ X .i / qi .t/ ˇf1 .t/  f2 .t/ˇ D qi .t/ ˇg.i / .t/ˇ :

i D0

i D0

From n    o .m1/ .m1/ .D g.t//2 D .D g.t// F t; f1 ; f10 ; :::; f1  F t; f2 ; f20 ; :::; f2 ; we obtain

2



.D g.t//  jD g.t/j

m1 X

! m1 X ˇ .i / ˇ ˇ ˇ qi .t/ ˇg .t/ˇ D qi .t/.jD g.t/j ˇg.i / .t/ˇ/:

i D0

i D0

Integrating the last inequality we get Z a

b

.D g.t//2 dt 

m1 XZ b i D0



m1 X

a

ˇ ˇ qi .t/ ˇg .i / .t/ˇ jD g.t/j dt Z

kqi k1

i D0

b

a

ˇ .i / ˇ

ˇg .t/ˇ jD g.t/j dt:

(1.38)

From inequalities (23) and (13) of [5] we find that Z a

b

ˇ .i / ˇ

ˇg .t/ˇ jD g.t/j dt

2. i C 2 / .b  a/ i  p  .  i / 2.2  2i  1/.  i / 1

Z

b a

..D g/.t//2 dt; for i D 0; 1; :::; m  1: (1.39)

Combining (1.38) and (1.39) we derive Z

b a



2

..D g/.t// dt  .a; b/

Z

b

..D g.t//2 dt:

(1.40)

a

Rb If a .D g.t//2 dt ¤ 0; then from (1.40) we get .a; b/  1; a contradiction by (1.36).

18

1 Opial-Type Inequalities for Balanced Fractional Derivatives

Thus

Z a

b

.D g.t//2 dt D 0;

which implies .D g.t/2 D 0; a:e: in t 2 Œa; b; that is, D g.t/ D 0; a:e; in t 2 Œa; b: Therefore

g.t/ D 0; for Db

aCb  t  b; a:e: 2

and

Da g.t/ D 0; for a  t <

aCb ; a:e: 2

But g .i / .a/ D g .i / .b/ D 0; for all 0  i  m  1:     So from (1.2) and (1.4) we get g.x/ D 0 on aCb , ; b , and g.x/ D 0 on a; aCb 2 2 respectively. Thus g.x/ D 0; 8x 2 Œa; b; giving us f1 .x/ D f2 .x/; 8x 2 Œa; b; that is, proving the uniqueness of solution to fractional boundary value problem (1.35).

References 1. R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Klower, Dordrecht, London, 1995. 2. C.D. Aliprantis and O. Burkinshaw “Principles of Real Analysis”, 3rd Edition , Academic Press, Boston, New York, 1998. 3. G. A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009. 4. G. A. Anastassiou, On Right Fractional Calculus, “Chaos, Solitons and Fractals”, 42(2009), 365-376. 5. G. A. Anastassiou, “Balanced fractional Opial inequalities”, Chaos Solitons Fractals, 42(2009), no.3, 1523-1528. 6. G. A. Anastassiou, Fractional Korovkin Theory, 42(2009), 2080-2094. 7. G. A. Anastassiou, Opial-type inequalities for functions and their ordinary and balanced fractional derivatives, submitted, 2011. 8. P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475. 9. Kai Diethelm, Fractional Differential Equations, online: http://www.tubs.de/˜ diethelm/lehre/ f-dg102/fde-skript.ps.gz. 10. A. M. A. El-Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12(2006), 81-95.

References

19

11. G. S. Frederico and D. F. M. Torres, Fractional Optimal Control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol.3, No. 10 (2008), 479-493. 12. R. Gorenflo and F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http://www.maphysto.dk/oldpages/events/LevyCAC2000/Mainardi Notes/fm2k0a.ps 13. C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math. 8 (1960), 61-63. 14. Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), 29-32. 15. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].

Chapter 2

Univariate Right Caputo Fractional Ostrowski Inequalities

Here we present general univariate right Caputo fractional Ostrowski inequalities. One of them is proved sharp and attained. Estimates are with respect to kkp , 1  p  1: This chapter is based on [4].

2.1 Introduction In 1938, A. Ostrowski [8] proved the following important inequality: Theorem 2.1. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/, i.e., kf 0 k1 WD sup jf 0 .t/j < C1: Then t 2.a;b/

ˇ ˇ "  2 # ˇ 1 Z b ˇ   x  aCb 1 ˇ ˇ 2  .b  a/ f 0 1 ; C f .t/ dt  f .x/ˇ  ˇ 2 ˇb  a a ˇ 4 .b  a/ for any x 2 Œa; b : The constant

1 4

(2.1)

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications to numerical analysis and probability. This chapter is greatly motivated and inspired also by the following result. Theorem 2.2 (see [1]). Let f 2 C nC1 .Œa; b/, n 2 N and x 2 Œa; b be fixed, such that f .k/ .x/ D 0, k D 1; :::; n: Then it holds ˇ ˇ  ˇ 1 Z b ˇ f .nC1/   ˇ ˇ 1  f .y/ dy  f .x/ˇ  ˇ ˇb  a a ˇ .n C 2/Š

.x  a/nC2 C .b  x/nC2 ba

! : (2.2)

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 2, © George A. Anastassiou 2011

21

22

2 Univariate Right Caputo Fractional Ostrowski Inequalities

Inequality (2.2) is sharp. In particular, when n is odd is attained by f  .y/ WD .y  x/nC1  .b  a/, while when n is even the optimal function is f .y/ WD jy  xjnC˛  .b  a/ ,

˛ > 1.

Clearly inequality (2.2) generalizes inequality (2.1) for higher order derivatives of f . Also in [2], see Chaps. 24–26, we gave a complete theory of left fractional Ostrowski inequalities.

2.2 Main Results We need Definition 2.3 ([3, 5–7, 9]). Let f 2 L1 .Œa; b/, ˛ > 0. The right Riemann– Liouville fractional operator of order ˛ by ˛ Ib f

1 .x/ D  .˛/

Z

b x

.J  x/˛1 f .J / dJ ,

(2.3)

0 WD I (the identity 8 x 2 Œa; b, where  is the gamma function. We set Ib operator).

Definition 2.4 ([3, 5–7, 9]). Let f 2 AC m .Œa; b/ (f .m1/ is in AC .Œa; b/), m 2 N, m D d˛e, ˛ > 0 (de the ceiling of the number). We define the right Caputo fractional derivative of order ˛ > 0 by ˛ Db f .x/ D

.1/m  .m  ˛/

Z x

b

.J  x/m˛1 f .m/ .J / dJ; 8 x  b:

(2.4)

If ˛ D m 2 N, then m f .x/ D .1/m f .m/ .x/; 8 x 2 Œa; b: Db ˛ f .x/ D 0. If x > b we define Db

We also need Theorem 2.5 ([3]). Let f 2 AC m .Œa; b/, x 2 Œa; b, ˛ > 0, m D d˛e. Then f .x/ D

m1 X kD0

1 f .k/ .b/ .x  b/k C kŠ  .˛/

Z

b x

˛ .J  x/˛1 Db f .J / dJ;

the right Caputo fractional Taylor formula with integral remainder.

(2.5)

2.2 Main Results

23

We give Theorem 2.6. Let ˛ > 0, m D d˛e, f 2 AC m .Œa; b/. Assume f .k/ .b/ D 0, ˛ k D 1; :::; m  1; and Db f 2 L1 .Œa; b/. Then ˇ  ˛  ˇ ˇ D f  ˇ 1 Z b b ˇ ˇ 1;Œa;b f .x/ dx  f .b/ˇ  .b  a/˛ : ˇ ˇ ˇb  a a  .˛ C 2/

(2.6)

Proof. Let x 2 Œa; b. We have Z

1 f .x/  f .b/ D  .˛/

x

b

˛ .J  x/˛1 Db f .J / dJ:

Then jf .x/  f .b/j 

1  .˛/

1   .˛/ 1 D  .˛/ D

Z

b x

˛ .J  x/˛1 jDb f .J /j dJ

Z

b x

! .J  x/

˛1

dJ

˛ f k1;Œa;b kDb

ˇb ! .J  x/˛ ˇˇ ˛ ˇ kDb f k1;Œa;b ˛ x

1 ˛ f k1;Œa;b : .b  x/˛ kDb  .˛ C 1/

Therefore jf .x/  f .b/j 

.b  x/˛ ˛ f k1;Œa;b ; 8 x 2 Œa; b: kDb  .˛ C 1/

Hence it holds ˇ ˇ ˇ ˇ ˇ ˇ 1 Z b ˇ ˇ 1 Z b ˇ ˇ ˇ ˇ f .x/ dx  f .b/ˇ D ˇ .f .x/  f .b// dx ˇ ˇ ˇ ˇb  a a ˇ ˇb  a a Z b Z b .b  x/˛ 1 1 ˛  f k1;Œa;b dx kDb jf .x/  f .b/j dx  ba a b  a a  .˛ C 1/  ˛  Z b D f  b 1;Œa;b .b  x/˛ dx D .b  a/  .˛ C 1/ a 0 0  ˛  ˇ 11 D f  ˛C1 ˇb b ˇ AA 1;Œa;b @ @ .b  x/ D  ˇ .b  a/  .˛ C 1/ ˛C1 ˇ a

24

2 Univariate Right Caputo Fractional Ostrowski Inequalities

 ˛  D f  b 1;Œa;b

! .b  a/˛C1 D .1/ 0  .b  a/  .˛ C 1/ ˛C1  ˛   ˛  D f  D f   .b  a/˛ b b 1;Œa;b 1;Œa;b ˛C1  .b  a/ ; D D .b  a/  .˛ C 2/  .˛ C 2/ t u

proving the claim. We present

Theorem 2.7. Let ˛  1, m D d˛e, f 2 AC m .Œa; b/. Assume that f .k/ .b/ D 0, ˛ k D 1; :::; m  1, and Db f 2 L1 .Œa; b/. Then ˇ  ˛  ˇ ˇ D f  ˇ 1 Z b b ˇ ˇ L1 .Œa;b/ f .x/ dx  f .b/ˇ  (2.7) .b  a/˛1 : ˇ ˇ ˇb  a a  .˛ C 1/ Proof. We have again jf .x/  f .b/j  

1  .˛/

Z x

b

˛ .J  x/˛1 jDb f .J /j dJ

1 .b  x/˛1  .˛/

Z x

b

˛ f .J /j dJ jDb

1 ˛ .b  x/˛1 kDb  f kL1 .Œa;b/ :  .˛/ Hence jf .x/  f .b/j 

 ˛  D f  b L1 .Œa;b/  .˛/

.b  x/˛1 ; 8 x 2 Œa; b:

Therefore ˇ ˇ Z b ˇ ˇ 1 Z b 1 ˇ ˇ f .x/ dx  f .b/  jf .x/  f .b/j dx ˇ ˇ ˇ ba a ˇb  a a  Z b D ˛ f  1 b L1 .Œa;b/ .b  x/˛1 dx  ba a  .˛/  ˛  Z b D f  b L1 .Œa;b/ .b  x/˛1 dx D .b  a/  .˛/ a  ˛   ˛  D f  D f  ˛ b b L1 .Œa;b/ .b  x/ L1 .Œa;b/ D D .b  x/˛1 ; .b  a/  .˛/ ˛  .˛ C 1/ proving the claim.

t u

2.2 Main Results

25

We continue with Theorem 2.8. Let p; q > 1 W p1 C q1 D 1, ˛ > 1  p1 , m D d˛e, f 2 AC m .Œa; b/. ˛ Assume that f .k/ .b/ D 0, k D 1; :::; m  1, and Db f 2 Lq .Œa; b/. Then  ˛  ˇ ˇ D f  ˇ ˇ 1 Z b 1 b Lq .Œa;b/ ˇ ˇ   .b  a/˛1C p : f .x/ dx  f .b/ˇ  ˇ ˇ  .˛/ .p .˛  1/ C 1/ p1 ˛ C 1 ˇb  a a p

(2.8) Proof. We have again jf .x/  f .b/j 

1  .˛/

1   .˛/

Z

b x

˛ .J  x/˛1 jDb f .J /j dJ

Z

b x

.J  x/

p.˛1/

! p1

Z

dJ x

Z

.˛1/C 1

p 1 .b  x/   .˛/ .p .˛  1/ C 1/ p1

b

x

b

q

˛ f .J /j dJ jDb

˛ f .J /jq dJ jDb

! q1

! q1

.˛1/C 1



p 1 .b  x/ ˛ f kLq .Œa;b/ : kDb  .˛/ .p .˛  1/ C 1/ p1

Therefore jf .x/  f .b/j  Hence

 ˛  D f  b Lq .Œa;b/  .˛/ .p .˛  1/ C 1/

1

1 p

.b  x/˛1C p ; 8 x 2 Œa; b:

ˇ ˇ Z b ˇ ˇ 1 Z b 1 ˇ ˇ f .x/ dx  f .b/ˇ  jf .x/  f .b/j dx ˇ ˇ ba a ˇb  a a  ˛  Z b D f  1 b Lq .Œa;b/ .b  x/˛1C p dx  1 .b  a/  .˛/ .p .˛  1/ C 1/ p a  ˛  1 D f  .b  a/˛1C p b Lq .Œa;b/   : D 1  .˛/ .p .˛  1/ C 1/ p ˛ C p1 t u 1 , 2

m

Corollary 2.9. Let ˛ > m D d˛e, f 2 AC .Œa; b/. Suppose f ˛ k D 1; :::; m  1; Db f 2 L2 .Œa; b/. Then

.k/

.b/ D 0,

 ˛  ˇ ˇ D f  ˇ ˇ 1 Z b 1 b ˇ ˇ L2 .Œa;b/   f .x/ dx  f .b/ˇ  .b  a/˛ 2 : ˇ ˇ  .˛/ p2˛  1 ˛ C 1  ˇb  a a 2

(2.9)

26

2 Univariate Right Caputo Fractional Ostrowski Inequalities

We finish the chapter with Proposition 2.10. Inequality (2.6) is sharp, namely it is attained by f .x/ D .b  x/˛ , ˛ > 0, ˛ … N, x 2 Œa; b: Proof. Notice that .b  x/˛ 2 AC m .Œa; b/. We observe that f 0 .x/ D ˛ .b  x/˛1 ; f 00 .x/ D .1/2 ˛ .˛  1/ .b  x/˛2 ; :: : f .m1/ .x/ D .1/m1 ˛ .˛  1/ .˛  2/    .˛  m C 2/ .b  x/˛mC1 ; and f .m/ .x/ D .1/m ˛ .˛  1/ .˛  2/    .˛  m C 2/ .˛  m C 1/ .b  x/˛m : Thus ˛ Db f

Z b .1/2m .x/ D ˛ .˛1/    .˛m C 1/ .J  x/m˛1 .b  J /˛m dJ  .m  ˛/ x Z b ˛ .˛  1/    .˛  m C 1/ .b  J /.˛mC1/1 .J  x/.m˛/1 dJ D  .m  ˛/ x D

˛ .˛  1/    .˛  m C 1/  .˛  m C 1/  .m  ˛/  .m  ˛/  .1/

D ˛ .˛  1/    .˛  m C 1/  .˛  m C 1/ D  .˛ C 1/: That is,

˛ f .x/ D  .˛ C 1/ , 8 x 2 Œa; b: Db

˛ f 2 L1 .Œa; b/. So f Also we see that f .k/ .b/ D 0, k D 0; 1; :::; m  1; and Db fulfills all assumptions. Next we see

 .˛ C 1/ .b  a/˛ .b  a/˛ D :  .˛ C 2/ .˛ C 1/ Z b 1 L:H:S: .2.6/ D .b  x/˛ dx ba a

R:H:S: .2.6/ D

D

1 .b  a/˛C1 .b  a/˛ D ; b  a .˛ C 1/ ˛C1

proving attainability and sharpness of (2.6).

t u

References

27

References 1. G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781. 2. G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009. 3. G.A. Anastassiou, On Right Fractional Calculus, Chaos, Solitons and Fractals, 42 (2009), 365-376. 4. G.A. Anastassiou, Univariate right fractional Ostrowski inequalities, CUBO, accepted, 2011. 5. A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95. 6. G.S. Frederico, D.F.M. Torres, Fractional Optimal Control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol. 3, No. 10 (2008), 479-493. 7. R. Gorenflo, F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http://www.maphysto.dk/oldpages/events/LevyCAC2000/ MainardiNotes/fm2k0a.ps. ¨ 8. A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funcktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227. 9. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].

Chapter 3

Multivariate Right Caputo Fractional Ostrowski Inequalities

Here we present general multivariate right Caputo fractional Ostrowski inequalities. Some of them are proved to be sharp and attained. Estimates are with respect to kk1 : This chapter relies on [5].

3.1 Introduction In 1938, A. Ostrowski [8] proved the following important inequality: Theorem 3.1. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/, i.e., kf 0 k1 WD sup jf 0 .t/j < C1: Then t 2.a;b/

ˇ ˇ "  2 # ˇ 1 Z b ˇ   x  aCb 1 ˇ ˇ 2  .b  a/ f 0 1 ; C f .t/ dt  f .x/ˇ  ˇ 2 ˇb  a a ˇ 4 .b  a/ for any x 2 Œa; b. The constant

1 4

(3.1)

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications to numerical analysis and probability. This chapter is greatly motivated and inspired also by the following result. Theorem 3.2 (see [1]). Let f 2 C nC1 .Œa; b/, n 2 N and x 2 Œa; b be fixed, such that f .k/ .x/ D 0, k D 1; :::; n: Then it holds ˇ ˇ  ˇ 1 Z b ˇ f .nC1/   ˇ ˇ 1  f .y/ dy  f .x/ˇ  ˇ ˇb  a a ˇ .n C 2/Š

.x  a/nC2 C .b  x/nC2 ba

! : (3.2)

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 3, © George A. Anastassiou 2011

29

30

3 Multivariate Right Caputo Fractional Ostrowski Inequalities

Inequality (3.2) is sharp. In particular, when n is odd is attained by f  .y/ WD .y  x/nC1  .b  a/, while when n is even the optimal function is f .y/ WD jy  xjnC˛  .b  a/ ,

˛ > 1.

Clearly inequality (3.2) generalizes inequality (3.1) for higher order derivatives of f . Also in [2], see Chaps. 24–26, we gave a complete theory of left fractional Ostrowski inequalities.

3.2 Main Results We need Remark 3.3. We define the ball B .0; R/ D fx 2 RN W jxj < Rg  RN , N  2, R > 0, and the sphere S N 1 WD fx 2 RN W jxj D 1g; where jj is the Euclidean norm. Let d! be the element of surface measure on S N 1 and let Z N 2 2 !N D d! D  N  :  2 S N 1 For x 2 RN  f0g we can write uniquely x D r!, where r D jxj > 0 and R N ! D xr 2 S N 1 ; j!j D 1. Note that B.0;R/ dy D ! NNR is the Lebesgue measure of the ball. Following [6, pp. 149–150, exercise 6] and [7, pp. 87–88, Theorem 5.2.2], we can write F W B .0; R/ ! R a Lebesgue integrable function that

B.0;R/

Z

Z

Z F .x/ dx D

S N 1

R

 F .r!/ r N 1 dr d!I

(3.3)

0

we use this formula a lot. Initially the function f W B .0; R/ ! R is radial, that is, there exists a function g such that f .x/ D g .r/, where r D jxj, r 2 Œ0; R, 8 x 2 B .0; R/. Here we assume that g 2 AC m .Œ0; R/ (means g.m1/ is in AC .Œ0; R/), m D d˛e (de ceiling of the number), ˛ > 0, and g .k/ .R/ D 0, k D 1; :::; m  1: By [3] we obtain g .s/  g .R/ D

1  .˛/

Z s

R

˛ .J  s/˛1 DR g .J / dJ;

(3.4)

3.2 Main Results

31

˛ 8 s 2 Œ0; R, where DR g is the right Caputo derivative. Further suppose that ˛ DR g 2 L1 .Œ0; R/ : We get

1 jg .s/  g .R/j   .˛/ 

1  .˛/

Z

R

s

Z

˛ .J  s/˛1 jDR g .J /j dJ R

s

 ˛ .J  s/˛1 dJ kDR gk1;Œ0;R

.R  s/˛ ˛ D gk1;Œ0;R : kDR  .˛ C 1/ That is, jg .s/  g .R/j 

 ˛  D g  R 1;Œ0;R  .˛ C 1/

.R  s/˛ ;

(3.5)

(3.6)

8 s 2 Œ0; R. Next observe that  R ˇ R ˇ ˇˇ ˇ R R ˇ ˇ ˇ g .s/ s N 1 ds d! ˇˇ N 1 f .y/ dy 0 S ˇ ˇ ˇ B.0;R/ ˇ  R ˇ D g .R/  R ˇf .R!/  ˇ R N 1 ˇ Vol .B .0; R// ˇ ˇˇ s ds d! ˇ N 1 S

0

ˇZ ˇ ˇ ˇ Z R ˇ ˇ ˇ N N ˇˇ R N 1 N 1 ˇ ˇ D ˇg .R/  N g .s/ s ds ˇ D N ˇ s .g .R/  g .s// ds ˇˇ (3.7) R R 0 0  ˛  Z Z R N DR g 1;Œ0;R R N 1 N N 1 .R/  g .s/j ds   N s s .R  s/˛ ds jg N R R  .˛ C 1/ 0 0  ˛  Z   R N DR g 1;Œ0;R .R  s/.˛C1/1 s N 1 ds D N R  .˛ C 1/ 0  ˛  N DR g 1;Œ0;R  .˛ C 1/ .N  1/Š ˛CN R D N R  .˛ C 1/  .˛ C N C 1/ ˛ D kDR gk1;Œ0;R

N ŠR˛ :  .˛ C N C 1/

(3.8)

So we have proved that ˇ ˇ ˇ R ˇ Z R ˇ ˇ ˇ ˇ N ˇ B.0;R/ f .y/ dy ˇ g .s/ s N 1 ds ˇˇ ˇf .R!/  ˇ D ˇˇg .R/  N ˇ Vol .B .0; R// ˇ R 0 ˛ gk1;Œ0;R  kDR

N ŠR˛ :  .˛ C N C 1/

(3.9)

(3.10)

32

3 Multivariate Right Caputo Fractional Ostrowski Inequalities

The last inequality (3.10) is sharp, it is attained by g .r/ D .R  r/˛ , ˛ > 0, r 2 Œ0; R. As in [4] we get ˛ DR g .r/ D  .˛ C 1/ , 8 r 2 Œ0; R :

 ˛  g 1;Œ0;R D  .˛ C 1/. And g .R/ D 0. Therefore Hence DR L:H:S: .3.10/ D

N RN

N D N R D

Z

R

0

Z

R

0

.R  s/˛ s N 1 ds .R  s/.˛C1/1 .s  0/N 1 ds

N  .˛ C 1/  .N / ˛CN  .˛ C 1/ N Š ˛ D R R : RN  .˛ C N C 1/  .˛ C N C 1/

And R:H:S: .3.10/ D

 .˛ C 1/ N ŠR˛ ;  .˛ C N C 1/

proving attainability of (3.10). We have established the following multivariate Ostrowski inequality. Theorem 3.4. Let f W B .0; R/ ! R which is radial, that is, there exists g such that f .x/ D g .r/, r D jxj, 8 x 2 B .0; R/. Suppose that g 2 AC m .Œ0; R/, ˛ m D d˛e, ˛ > 0, and g .k/ .R/ D 0, k D 1; :::; m  1; and DR g 2 L1 .Œ0; R/. Then ˇ ˇ ˇ R ˇ Z R ˇ ˇ ˇ ˇ N ˇ B.0;R/ f .y/ dy ˇ N 1 ˇ ˇ g .s/ s ds ˇ ˇ D ˇg .R/  N ˇf .R!/  ˇ Vol .B .0; R// ˇ R 0 ˛  kDR gk1;Œ0;R

N ŠR˛ :  .˛ C N C 1/

(3.11)

The last inequality is sharp, that is attained by g .r/ D .R  r/˛ , ˛ > 0, 8 r 2 Œ0; R : We also make Remark 3.5. Let the spherical shell A WD B .0; R2 /  B .0; R1 /, 0 < R1 < R2 , A  RN , N  2, x 2 A: Consider again that f W A ! R is radial, that is, there exists g such that f .x/ D g .r/, r D jxj, r 2 ŒR1 ; R2 , 8 x 2 A. Here again x can be written uniquely as x D r!, where r D jxj > 0, and ! D xr 2 S N 1 , j!j D 1. We can write for F W A ! R a Lebesgue integrable function that Z

Z A

F .x/ dx D

S N 1

Z

R2 R1

F .r!/ r

N 1

 dr d!:

(3.12)

3.2 Main Results

33

! .RN RN / Here Vol .A/ D N 2N 1 ; and we assume that g 2 AC m .ŒR1 ; R2 /, m D d˛e, ˛ > 0, and g.k/ .R2 / D 0, k D 1; :::; m  1. We get (see [3]) Z R2 1 g .s/  g .R2 / D .J  s/˛1 DR˛ 2  g .J / dJ; (3.13)  .˛/ s

8 s 2 ŒR1 ; R2 , where DR˛ 2  g is the right Caputo fractional derivative. Further suppose that DR˛ 2  g 2 L1 .ŒR1 ; R2 /. Hence jg .s/  g .R2 /j 

1  .˛/

Z

R2 s

Z

ˇ ˇ .J  s/˛1 ˇDR˛ 2  g .J /ˇ dJ

R2

   .J  s/˛1 dJ DR˛ 2  g 1;ŒR



1  .˛/

D

 1 .R2  s/˛  D ˛ g  : R2  1;ŒR1 ;R2   .˛/ ˛

Therefore jg .s/  g .R2 /j 

s

1 ;R2 

 ˛ D

 

R2  g 1;ŒR1 ;R2 

 .˛ C 1/

.R2  s/˛ ;

(3.14)

(3.15)

8 s 2 ŒR1 ; R2  : Next we observe that R ˇ ˇ ˇ ˇ   Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .R2 !/  A f .y/ dy ˇ D ˇg .R2 /  g .s/ s ds ˇ ˇ N N Vol .A/ ˇ ˇ R2  R1 R1 ˇ  ˇZ R2  ˇ ˇ N N 1 ˇ ˇ D .g .R2 /  g .s// s ds ˇ ˇ N N R2  R1 R1  Z R2  N  jg .R2 /  g .s/j s N 1 ds N R2  R1N R1    D ˛ g  N R2  1;ŒR1 ;R2    .˛ C 1/ R2N  R1N Z R2  .R2  s/˛ s N 1 ds DW ./ : (3.16) R1

We evaluate Z R2 .R2  s/˛ s N 1 ds R1

D

Z

R2 R1

.R2  s/˛ ..s  R1 / C R1 /N 1 ds

34

3 Multivariate Right Caputo Fractional Ostrowski Inequalities

Z D

D

R2 R1

.R2  s/˛

kD0

N 1  X kD0

D

N 1  X kD0

D

N 1 X kD0

N 1  X



N 1 R1N 1k k

!  N 1 .s  R1 /k R1N 1k ds k Z

R2

.R2  s/.˛C1/1 .s  R1 /.kC1/1 ds

R1

(3.17)

  .˛ C 1/  .k C 1/ N 1 R1N 1k .R2  R1 /˛CkC1 k  .˛ C k C 2/

.N  1/Š  .˛ C 1/ kŠ RN 1k .R2  R1 /˛CkC1 : kŠ .N  k  1/Š 1  .˛ C k C 2/

Therefore we get Z

R2 R1

.R2  s/˛ s N 1 ds D .N  1/Š .˛ C 1/

N 1 X kD0

R1N 1k .R2  R1 /˛C1Ck : .N  1  k/Š .˛ C 2 C k/ (3.18)

Consequently we find  ./ D

NŠ N R2  R1N



N 1 X

 ˛  D g  R2  1;ŒR

1 ;R2 

kD0

! R1N 1k .R2  R1 /˛C1Ck : .N  1  k/Š .˛ C 2 C k/ (3.19)

So we have established R ˇ ˇ ˇ ˇ   Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .R2 !/  A f .y/ dy ˇ D ˇg .R2 /  g .s/ s ds ˇ ˇ ˇ ˇ N N Vol .A/ R2  R1 R1 !   NX 1   ˛ NŠ R1N 1k .R2  R1 /˛C1Ck D g   : R2  N N 1;ŒR1 ;R2  .N  1  k/Š .˛ C 2 C k/ R2  R1 kD0

(3.20) The last inequality (3.20) is sharp, that is attained by AC m .ŒR1 ; R2 / 3 g .r/ D .R2  r/˛ , ˛ > 0, m D d˛e, r 2 ŒR1 ; R2  : Indeed DR˛ 2  g .r/ D  .˛ C 1/ ; 8 r 2 ŒR1 ; R2  ; and

 ˛  D g  R2  1;ŒR

1 ;R2 

D  .˛ C 1/ :

(3.21)

Also we have g .k/ .R2 / D 0, k D 0; 1; :::; m  1; and DR˛ 2  g 2 L1 .ŒR1 ; R2 /. So g fulfills all the assumptions here.

3.2 Main Results

35

We see that N N R2  R1N

L:H:S: .3.20/ D



N 1 X kD0

Z

R2 R1

.R2  s/˛ s N 1 ds

N Š .˛ C 1/ R2N  R1N

R1N 1k .R2  R1 /˛C1Ck .N  1  k/Š .˛ C 2 C k/

!  NX 1 R1N 1k .R2  R1 /˛C1Ck NŠ D .N  1  k/Š .˛ C 2 C k/ R2N  R1N kD0    DR˛ 2  g 1;ŒR ;R  

1

2

D R:H:S: .3.20/;

(3.22)

proving the optimality of (3.20). We have established the Ostrowski inequality. Theorem 3.6. Let f W A ! R be radial, that is, there exists g such that f .x/ D g .r/, r D jxj, 8 x 2 A; ! 2 S N 1 . Assume g 2 AC m .ŒR1 ; R2 / ; m D d˛e, ˛ > 0, and g .k/ .R2 / D 0, k D 1; :::; m  1, and DR˛ 2  g 2 L1 .ŒR1 ; R2 /. Then R ˇ ˇ ˇ ˇ   Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .R2 !/  A f .y/ dy ˇ D ˇg .R2 /  g .s/ s ds ˇ ˇ ˇ ˇ N N Vol .A/ R2  R 1 R1  

NŠ R2N  R1N

!  NX 1   ˛ R1N 1k .R2  R1 /˛C1Ck D g  : (3.23) R2  1;ŒR1 ;R2  .N  1  k/Š .˛ C 2 C k/ kD0

The last inequality (3.6) is sharp, that is attained by g .s/ D .R2  s/˛ ; ˛ > 0, s 2 ŒR1 ; R2  :

(3.24)

We need Definition 3.7. Let F W A ! R, ˛ > 0, m D d˛e such that F .!/ 2 AC m .ŒR1 ; R2 /, for all ! 2 S N 1 . We call the Caputo right radial fractional derivative the following function @˛R2  F .x/ @r ˛

D

.1/m  .m  ˛/

Z

R2 r

.t  r/m˛1

@m F .t!/ dt; @r m

(3.25)

where x 2 A, that is, x D r!, r 2 ŒR1 ; R2 , ! 2 S N 1 . Clearly @0R2  F .x/ @r 0

D F .x/ ;

(3.26)

36

3 Multivariate Right Caputo Fractional Ostrowski Inequalities

@˛R2  F .x/ @r ˛

D

@˛ F .x/ ; if ˛ 2 N: @r ˛

(3.27)

The above defined function exists almost everywhere for x 2 A. We justify this next. Remark 3.8. Call 1 WD r 2 ŒR1 ; R2  W

@˛R2  F .x/ @r ˛

does not exist :

We have that Lebesgue measure R .1 / D 0. Call N WD 1  S N 1 . So there    exists a Borel   set 1 ŒR1 ; R2 , such that 1 1 , R 1 D R .1 / D 0; thus, RN 1 D 0, see [2], pp. 419–422. N Consider now N WD 1  S N 1 A, which    is a Borel set of R f0g. Clearly then by Theorem 16.59, p. 420, [2], RN N D 0, but N N ; implying RN .N / D 0. Consequently the above radial derivative exists a.e. in x w.r.t. RN on A: We make Remark 3.9. We treat here the general, not necessarily radial, case of f: We apply last Theorem 3.6 to f .r!/, ! is fixed, r 2 ŒR1 ; R2 , under the following assumptions: f .!/ 2 AC m .ŒR1 ; R2 / ; for all ! 2 S N 1 , ˛ > 0, m D d˛e, where k f W A ! R is Lebesgue integrable; @@rfk , k D 1; :::; m  1 vanish on @B .0; R2 /, and ˛   @R  f 2 2 B A , along with DR˛ 2  f .!/ 2 L1 .ŒR1 ; R2 /, 8 ! 2 S N 1 : @r ˛ So we have ˇ ˇ   Z R2 ˇ ˇ N N 1 ˇ ˇf .R2 !/  f .s!/ s ds ˇ ˇ N N R2  R 1 R1  

NŠ N R2  R1N

!  NX 1  @˛R2  f R1N 1k .R2  R1 /˛C1Ck  .N  1  k/Š .˛ C 2 C k/  @r ˛ kD0

   

1;A

DW 1 : (3.28)

Consequently it holds ˇR Z R2  ˇˇ Z ˇ f .R !/ d! N N 1 2 ˇ ˇ S   N f .s!/ s N 1 ds d! ˇ  1 : ˇ N ˇ ˇ !N R2  R1 ! N S N 1 R1 (3.29) That is,

ˇ ˇ  Z R ˇ ˇ N f .x/ dx ˇ ˇ 2 A f .R !/ d!  ˇ  1 : ˇ 2 ˇ 2 N2 S N 1 Vol .A/ ˇ

(3.30)

3.2 Main Results

37

Therefore, it holds for x 2 A, that    Z R ˇ ˇ ˇˇ ˇ ˇ ˇ  N2  N2 f .x/ dx A ˇf .x/  ˇ D ˇf .x/  f .R2 !/ d! C N N ˇ Vol .A/ ˇ ˇ 2 2 S N 1 2 2 R ˇ Z ˇ A f .x/ dx ˇ  f .R2 !/ d!  Vol .A/ ˇ S N 1 ˇ ˇ  Z ˇ ˇ  N2 ˇ ˇ f .R2 !/ d! ˇ C 1 : (3.31)  ˇf .x/  N ˇ ˇ N 1 2 S 2 We have proved Theorem 3.10. Let f W A ! R be Lebesgue integrable with f .!/ 2 k AC m .ŒR1 ; R2 /, ˛ > 0, m D d˛e, 8 ! 2 S N 1 I @@rfk , k D 1; :::; m  1 vanish   @˛ f on @B .0; R2 / I @˛R2  f .!/ 2 L1 .ŒR1 ; R2 /, 8 ! 2 S N 1 I and R@r2 ˛ 2 B A (bounded functions on A). Then, for x 2 A, we have R ˇ ˇ ˇ ˇ ˇf .x/  A f .x/ dx ˇ ˇ Vol .A/ ˇ ˇ ˇ  Z ˇ ˇ  N2 ˇ ˇ  ˇf .x/  f .R2 !/ d! ˇ N ˇ ˇ N 1 2 S 2  C

NŠ N R2  R1N

!  NX 1  @˛R2  f R1N 1k .R2  R1 /˛C1Ck  .N  1  k/Š .˛ C 2 C k/  @r ˛ kD0

   

:

1;A

(3.32) We also make Remark 3.11. Let f W B .0; R/ ! R be a Lebesgue integrable function, that is, not necessarily a radial function. Assume f .!/ 2 AC 1 .Œ0; R/, 8 ! 2 S N 1 I ˛ 0 < ˛ < 1; and DR f .!/ 2 L1 .Œ0; R/, 8 ! 2 S N 1 : Clearly here we obtain 1 f .s!/  f .R!/ D  .˛/

Z

R s

˛ .J  s/˛1 DR f .J !/ dJ;

8 ! 2 S N 1 , 8 s 2 Œ0; R : We further suppose that ˛ f .J !/k1;.J 2Œ0;R/  K; 8 ! 2 S N 1 ; kDR

where K > 0:

(3.33)

38

3 Multivariate Right Caputo Fractional Ostrowski Inequalities

Applying the earlier Theorem 3.4 we get ˇ ˇ Z R ˇ ˇ N 1 ˇ ˇf .R!/  N f .s!/ s ds ˇ ˇ N R 0   KN ŠR˛ N ŠR˛ ˛ f .t!/k1;.t 2Œ0;R/  :  kDR  .˛ C N C 1/  .˛ C N C 1/

(3.34)

Consequently we obtain ˇR  ˇ Z R Z ˛ ˇ S N 1 f .R!/ d! ˇ N N 1 ˇ ˇ  KN ŠR  f .s!/ s ds d! ˇ ˇ  .˛CN C1/ : !N RN ! N S N 1 0 (3.35) Hence ˇ ˇ  Z R ˇ ˇ N KN ŠR˛ ˇ B.0;R/ f .x/ dx ˇ 2 : f .R!/ d!  ˇ ˇ N ˇ 2 2 S N 1 Vol .B .0; R// ˇ  .˛ C N C 1/ Consequently it holds ˇ ˇ R ˇ ˇ ˇ B.0;R/ f .x/ dx ˇ ˇf .R!/  ˇ ˇ Vol .B .0; R// ˇ ˇ  Z ˇ  N2 ˇ D ˇf .R!/  f .R!/ d! N ˇ 2 2 S N 1 ˇ R  Z ˇ  N2 B.0;R/ f .x/ dx ˇ C f .R!/ d!  ˇ N Vol .B .0; R// ˇ 2 2 S N 1 ˇ ˇ  Z ˇ ˇ  N2 KN ŠR˛ ˇ ˇ  ˇf .R!/  f .R!/ d! ˇ C : N ˇ ˇ  .˛ C N C 1/ 2 2 S N 1

(3.36)

(3.37)

So we have proved the Ostrowski inequality. Theorem 3.12. Let f W B .0; R/ ! R be a Lebesgue integrable function, not necessarily radial. Assume f .!/ 2 AC 1 .Œ0; R/, R > 0; 8 ! 2 S N 1 I ˛ 0 < ˛ < 1; and DR f .!/ 2 L1 .Œ0; R/, 8 ! 2 S N 1 : Suppose also that  ˛ D f .t!/  K; 8 ! 2 S N 1 ; where K > 0. Then R 1;.t 2Œ0;R/ ˇ ˇ R ˇ ˇ ˇ B.0;R/ f .x/ dx ˇ ˇ ˇf .R!/  ˇ Vol .B .0; R// ˇ ˇ ˇ  Z ˇ ˇ  N2 KN ŠR˛ ˇ ˇ : f .R!/ d! ˇ C  ˇf .R!/  N ˇ  .˛ C N C 1/ ˇ 2 2 S N 1

(3.38)

References

39

References 1. G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781. 2. G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009. 3. G.A. Anastassiou, On Right Fractional Calculus, Chaos, Solitons and Fractals, 42 (2009), 365-376. 4. G.A. Anastassiou, Univariate right fractional Ostrowski inequalities, CUBO, accepted, 2011. 5. G.A. Anastassiou, Multivariate right fractional Ostrowski inequalities, submitted, 2011. 6. W. Rudin, Real and Complex Analysis, International Student edition, Mc Graw Hill, London, New York, 1970. 7. D. Stroock, A Concise Introduction to the Theory of Integration, Third Edition, Birkha¨user, Boston, Basel, Berlin, 1999. ¨ 8. A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funcktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227.

Chapter 4

Univariate Mixed Fractional Ostrowski Inequalities

Here we give general univariate mixed Caputo fractional Ostrowski inequalities, one is proved sharp and attained. Estimates are with respect to kkp ; 1  p  1: This chapter is based on [4].

4.1 Introduction In 1938, A. Ostrowski [9] proved the following important inequality. Theorem 4.1. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/; i.e., kf 0 k1 WD supt 2.a;b/ jf 0 .t/j < C1: Then ˇ " ˇ  2 # ˇ ˇ 1 Z b   x  aCb 1 ˇ ˇ 2 f .t/dt  f .x/ˇ  C  .b  a/ f 0 1 ; ˇ 2 ˇ ˇa  b a 4 .b  a/ for any x 2 Œa; b:The constant

1 4

(4.1)

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications to numerical analysis and probability. This chapter is greatly motivated and inspired also by the following result. Theorem 4.2 (see [1]). Let f 2 C nC1.Œa; b/; n 2 N and x 2 Œa; b be fixed, such that f .k/ .x/ D 0; k D 1; :::; n: Then it holds ˇ  ˇ ˇ f .nC1/ ˇ 1 Z b   .x  a/nC2 C .b  x/nC2  ˇ ˇ 1 f .y/dy  f .x/ˇ  : (4.2)  ˇ ˇ ˇb  a a .n C 2/Š ba

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 4, © George A. Anastassiou 2011

41

42

4 Univariate Mixed Fractional Ostrowski Inequalities

Inequality (4.2) is sharp. In particular, when n is odd is attained by f  .y/ WD .y  x/nC1  .b  a/; while when n is even the optimal function is f .y/ WD jy  xjnC˛ .b  a/; ˛ > 1: Clearly inequality (4.2) generalizes inequality (4.1) for higher order derivatives of f: Also in [2], see Chaps. 24–26, we gave a complete theory of left fractional Ostrowski inequalities. Here we combine both right and left Caputo fractional derivatives and produce Ostrowski inequalities. For the concepts of right and left Caputo fractional Calculus we use here, we refer to [3–8, 10].

4.2 Main Results We make Remark 4.3. Let Œa; b  R; ˛ > 0; m D d˛e : Let f 2 AC m .Œa; b/; x0 2 Œa; b (i.e., f .m1/ 2 AC.Œa; b/). Thus f 2 AC m .Œa; x0 / and f 2 AC m .Œx0 ; b/: Consequently, by [5, p. 40], f .x/ D

m1 X kD0

f .k/ .x0 / 1 .x x0 /k C kŠ  .˛/

Z

x x0

˛ .x J /˛1 Dx f .J /dJ; 8x 2 Œx0 ; b: 0

And by [3]; f .x/ D

m1 X kD0

f .k/ .x0 / 1 .xx0 /k C kŠ  .˛/

Z x

x0

.J x/˛1 Dx˛0  f .J /dJ; 8x 2 Œa; x0 :

˛ f; Dx˛0  f are the left and right Caputo fractional derivatives of order ˛: Here Dx 0

Suppose f .k/ .x0 / D 0; k D 1; :::; m  1: Then f .x/  f .x0 / D

1  .˛/

Z

x x0

˛ .x  J /˛1 Dx f .J /dJ; 8x 2 Œx0 ; b 0

(4.3)

.J  x/˛1 Dx˛0  f .J /dJ; 8x 2 Œa; x0 :

(4.4)

and f .x/  f .x0 / D

1  .˛/

Z x

x0

4.2 Main Results

43

Hence 1 jf .x/  f .x0 /j   .˛/

Z

x x0

Z

ˇ ˇ ˛ ˇ dJ .x  J /˛1 ˇDx f .J / 0 x

   ˛ .x  J /˛1 dJ Dx f 1;Œx 0



1  .˛/

D

 1 .x  x0 /˛  D ˛ f  : x0 1;Œx0 ;b  .˛/ ˛

Hence jf .x/  f .x0 /j 

x0

 ˛  D f  x0 1;Œx

0 ;b

 .˛ C 1/

0 ;b

(4.5)

.x  x0 /˛ ; 8x 2 Œx0 ; b:

(4.6)

Similarly it holds jf .x/  f .x0 /j 

1  .˛/

Z x

x0

   .J  x/˛1 dJ Dx˛0  f 1;Œa;x

0

 1 .x0  x/˛  D ˛ f  x0  1;Œa;x0   .˛/ ˛   ˛ D f  x0  1;Œ˛;x0  D .x0  x/˛ ;  .˛ C 1/ D

that is,

jf .x/  f .x0 /j 

 ˛  D f  x0  1;Œa;x

0

 .˛ C 1/

.x0  x/˛ ; 8x 2 Œa; x0 :

(4.7)

Next we observe that ˇ ˇ ˇ ˇZ ˇ 1 Z b ˇ ˇ 1 ˇˇ b ˇ ˇ ˇ f .x/dx f .x0 /ˇ D ˇ ˇ .f .x/  f .x0 //dx ˇ ˇb  a a ˇ b  aˇ a ˇ Z b 1  jf .x/  f .x0 /j dx ba a (Z x0 1 D jf .x/  f .x0 /j dx ba a Z C

b x0

jf .x/  f .x0 /j dx

(4.8)

)

44

4 Univariate Mixed Fractional Ostrowski Inequalities

(   ˛ 1 D f   x0  1;Œa;x0  .b  a/ .˛ C 1/ Z

x0 a

x0 1;Œx0 ;b

1 D .b  a/ .˛ C 1/  ˛  C Dx f 1;Œx 0 D

Z

  .x0 x/˛ dx C D ˛  (

b x0

) .xx0 /˛ dx

  ˛ D f  x0  1;Œa;x

0

 0 ;b

.b  x0 /˛C1 ˛C1

)

˛C1

.x0  a/ ˛C1

!

n  1 D ˛ f  .x0  a/˛C1 x0  1;Œa;x0  .b  a/ .˛ C 2/ o  ˛  C Dx f 1;Œx ;b .b  x0 /˛C1 : (4.9) 0 0

We have established that Theorem AC m .Œa; b/; and  4.4. LetŒa;˛b  R; ˛ > 0; m D d˛e ; f 2 .k/  ˛ D f  ; Dx0 1;Œx ;b < 1; x0 2 Œa; b: Assume f .x0 / D 0; k D x0  1;Œa;x0  0 1; :::; m  1: Then ˇ ˇ ˇ ˇ 1 Z b n  1 ˇ ˇ D ˛ f  f .x/dx  f .x0 /ˇ  .x0  a/˛C1 ˇ x0  1;Œa;x0  ˇ .b  a/ .˛ C 2/ ˇb  a a o  ˛  C Dx f 1;Œx b .b  x0 /˛C1 0 0;

n  1  max Dx˛0  f 1;Œa;x  ; 0  .˛ C 2/ o  ˛    Dx f .b  a/˛ : 0 1;Œx ;b 0

(4.10) We also make Remark 4.5. As before we have .˛  1/ 1 jf .x/  f .x0 /j   .˛/

Z

x x0

ˇ ˇ ˛ ˇ dJ .x  J /˛1 ˇDx f .J / 0

.x  x0 /˛1   .˛/ ˛1



.x  x0 /  .˛/

Z

x

x0

ˇ ˛ ˇ ˇD f .J /ˇ dJ x0

 ˛  D  x0 L

1.Œx0 ;b/

:

4.2 Main Results

45

That is, we get jf .x/  f .x0 /j 

 .x  x0 /˛1  D ˛ f  ; 8x 2 Œx0 ; b: x0 L1.Œx0 ;b/  .˛/

(4.11)

Similarly we derive .˛  1/ jf .x/  f .x0 /j 

1  .˛/

Z

x0 x

ˇ ˇ .J  x/˛1 ˇDx˛0  f .J /ˇ dJ

1 .x0  x/˛1   .˛/

Z

x0 x

ˇ ˇ ˛ ˇD f .J /ˇ dJ x0 

  1  : .x0  x/˛1 Dx˛0  f L 1.Œa;x0 /  .˛/ That is, jf .x/  f .x0 /j 

 .x0  x/˛1  D ˛ f  ; 8x 2 Œa; x0 : x0  L1 .Œa;x0 /  .˛/

As before we have ˇ ˇ Z b ˇ ˇ 1 Z b 1 ˇ ˇ f .x/dx  f .x0 /ˇ  jf .x/  f .x0 /j dx ˇ ˇ ba a ˇb  a a (Z ) Z b x0 1 D jf .x/  f .x0 /j dx C jf .x/  f .x0 /j dx ba x0 a ( Z  x0   1 .x0  x/˛1 dx Dx˛0  f L .Œa;x /  1 0 .b  a/ .˛/ a ! ) Z b   ˛ ˛1 C .x  x0 / dx D f  x0

x0

D

n   1 .x0  a/˛ Dx˛0  f L .Œa;x / 1 0 .b  a/ .˛ C 1/   C .b  x0 /˛ D ˛ f  x0

(4.12)

(4.13)

L1 .Œx0 ;b/

L1 .Œx0 ;b/

o

:

(4.14)

We have established Theorem 4.6. Let ˛  1; m D d˛e ; and f 2 AC m .Œa; b/: Suppose that ˛ f .k/ .x0 / D 0; k D 1; :::; m  1; x0 2 Œa; b and Dx˛0  f 2 L1 .Œa; x0 /; Dx f 2 0 L1 .Œx0 ; b/: Then

46

4 Univariate Mixed Fractional Ostrowski Inequalities

ˇ Z ˇ 1 ˇ ˇb  a

b

a

ˇ ˇ ˇ f .x/dx  f .x0 /ˇ ˇ

n   1 .x0  a/˛ Dx˛0  f L .Œa;x / 1 0 .b  a/ .˛ C 1/ o   ˛  C.b  x0 /˛ Dx f 0 L .Œx ;b/



1  .˛ C 1/



n  max Dx˛0  f L

1 .Œa;x0

1

0

 ˛  ; Dx f L 0 /

 .b  a/˛1 :

o

1 .Œx0 ;b/

(4.15)

We further make Remark 4.7. Let p; q > 1 W p1 C q1 D 1 and ˛ > 1  p1 : Working as before on (4.3) and (4.4) and using H¨older’s inequality we obtain 1

jf .x/  f .x0 /j 

 1 .x  x0 /˛1C p  D ˛ f  ; 8x 2 Œx0 ; b: x 0 Lq .Œx0 ;b/  .˛/ .p.˛  1/ C 1/1=p (4.16)

And also it holds 1

 1 .x0  x/˛1C p  D ˛ f  ; 8x 2 Œa; x0 : jf .x/  f .x0 /j  x  0 Lq .Œa;x0 /  .˛/ .p.˛  1/ C 1/1=p (4.17) We observe that ˇ ˇ Z b ˇ ˇ 1 ˇ ˇ f .x/dx  f .x0 /ˇ ˇb  a ˇ a ) (Z Z b x0 1  jf .x/  f .x0 /j dx C jf .x/  f .x0 /j dx ba x0 a ( Z  x0 1 ˛1C p1 .x0  x/ dx  .b  a/ .˛/.p.˛  1/ C 1/1=p a ) ! Z b  ˛  ˛1C p1    D f C .x  x0 / dx kDx f k x0

Lq .Œa;x0 /

0

x0

Lq .Œx0 ;b/

(4.18) D

1

  .b  a/ .˛/.p.˛  1/ C 1/1=p ˛ C p1 n o  1  1  .x0  a/˛C p Dx˛0  f L .Œa;x / C .b  x0 /˛C p kDx0 f kLq .Œx0 ;b/ : q

0

(4.19)

4.2 Main Results

47

We have proved Theorem 4.8. Let p; q > 1 W p1 C q1 D 1; ˛ > 1  p1 ; m D d˛e ; ˛ > 0; and f 2 AC m .Œa; b/: Suppose that f .k/ .x0 / D 0; k D 1; :::; m  1; x0 2 Œa; b: ˛ Assume Dx˛0  f 2 Lq .Œa; x0 /; and Dx f 2 Lq .Œx0 ; b/: Then 0 ˇ ˇ ˇ 1 Z b ˇ 1 ˇ ˇ   f .x/dx  f .x0 /ˇ  ˇ ˇb  a a ˇ .b  a/ .˛/.p.˛  1/ C 1/1=p ˛ C 1 p n  1   .x0  a/˛C p Dx˛0  f L

q .Œa;x0 /

o

 1  ˛ C .b  x0 /˛C p Dx f L 0

(4.20)

q .Œx0 ;b/

1

   .˛/.p.˛  1/ C 1/1=p ˛ C p1



n  max Dx˛0  f L

;

q .Œa;x0 /

 ˛  D f  x0 L

o q .Œx0 ;b/

 .b  a/˛1=q : (4.21)

Corollary 4.9. Let p D q D 2; ˛ > 12 ; m D d˛e ; ˛ > 0; and f 2 AC m .Œa; b/: Assume f .k/ .x0 / D 0; k D 1; :::; m  1; x0 2 Œa; b: Assume Dx˛0  f 2 ˛ L2 .Œa; x0 /; and Dx f 2 L2 .Œx0 ; b/: Then 0 ˇ ˇ ˇ 1 Z b ˇ 1 ˇ ˇ   f .x/dx  f .x0 /ˇ  ˇ ˇb  a a ˇ .b  a/ .˛/.2˛  1/1=2 ˛ C 12  1   .x0  a/˛C 2 Dx˛0  f L

2.Œa;x0 /

  1   C .b  x0 /˛C 2 D˛x f  0



L2 .Œx0 ;b/



1

   .˛/ 2˛  1 ˛ C 12 n   ˛   max Dx˛0  f L .Œa;x / ; Dx f L 0 p

2

˛ 12

 .b  a/

:

0

o

2 .Œx0 ;b/

(4.22)

48

4 Univariate Mixed Fractional Ostrowski Inequalities

We further make Remark 4.10. Here ˛ > 0; a  x0  b: Let ( .x  x0 /˛ ; x 2 Œx0 ; b; f .x/ WD .x0  x/˛ ; x 2 Œa; x0 :

(4.23)

See f is in AC m .Œx0 ; b/; and in AC m .Œa; x0 /: .k/ .k/ See that f  .x0 / D f C .x0 / D 0; k D 0; 1; :::; m  1: .m1/

.m1/

Hence there exists f at x0 ; also f 2 AC Œa; b: That is, f 2 AC m Œa; b: We find that    ˛  D  .˛ C 1/ Dx0  f 

(4.24)

1;Œa;x0 

and

    ˛ Dx0 f 

1;Œx0 ;b

D  .˛ C 1/:

(4.25)

Consequently ˚   .˛ C 1/.˛ C 1/ .x0  a/˛C1 C .b  x0 /˛C1 .b  a/ .˛ C 2/.˛ C 1/ ˚  1 .x0  ˛/˛C1 C .b  x0 /˛C1 : D .b  a/.˛ C 1/

R:H:S: .4.10/ D

Also we observe that 1 L:H:S .4.10/ D ba

(Z

1 D b  a/ D



a

x0

˛

.x0  x/ dx C

Z

b x0

) ˛

.x  x0 / dx

.x0  a/˛C1 .b  x0 /˛C1 C ˛C1 ˛C1



˚  1 .x0  a/˛C1 C .b  x0 /˛C1 : .b  a/.˛ C 1/

(4.26)

Therefore inequality (4.10) is sharp and attained. We have proved Proposition 4.11. Inequality (4.10) is sharp, in particular it is attained.

References 1. G. A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781. 2. G. A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009.

References

49

3. G. A. Anastassiou, On Right Fractional Calculus, “Chaos, Solitons and Fractals”, 42 (2009), 365-376. 4. G. A. Anastassiou, Univariate mixed Caputo fractional Ostrowski inequalities, J. Comp. Anal. and Appl., accepted, 2011. 5. K. Diethelm, Fractional Differential Equations, online: http://www.tubs.de/˜ diethelm/lehre/fdg102/fde-skript.ps.gz. 6. A. M. A. El-Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12 (1006), 81-95. 7. G. S. Frederico and D. F. M. Torres, Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol. 3, No. 10 (2008), 479-493. 8. R. Gorenflo and F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http://www.maphysto.dk/oldpages/events/LevyCAC2000/Mainardi Notes/fm2k0a.ps ¨ 9. A. Ostrowski, (1938) Uber die Absolutabweichung einer differtentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10, 226-227. 10. S. G. Samko, A. A. Kilbas and O. I.Marichev, Fractional Integrals and Derivatives Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].

Chapter 5

Multivariate Radial Mixed Fractional Ostrowski Inequalities

Here we give general multivariate radial mixed Caputo fractional Ostrowski inequalities. One of them is proved sharp and attained. Estimates are with respect to kkp ; 1  p  1: This chapter relies on [4].

5.1 Introduction In 1938, A. Ostrowski [11] proved the following important inequality. Theorem 5.1. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/;i.e., kf 0 k1 WD supt 2.a;b/ jf 0 .t/j < C1: Then ˇ ˇ "  2 # ˇ 1 Z b ˇ   x  aCb 1 ˇ ˇ 2  .b  a/ f 0 1 ; C f .t/dt  f .x/ˇ  ˇ ˇb  a a ˇ 4 .b  a/2 for any x 2 Œa; b: The constant

1 4

(5.1)

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications to numerical analysis and probability. This chapter is greatly motivated and inspired also by the following result. Theorem 5.2 (see [1]). Let f 2 C nC1 .Œa; b/; n 2 N and x 2 Œa; b fixed, such that f .k/ .x/ D 0; k D 1; :::; n: Then it holds ˇ  ˇ ˇ f .nC1/ ˇ 1 Z b   .x  a/nC2 C .b  x/nC2  ˇ ˇ 1 f .y/dy  f .x/ˇ  : (5.2)  ˇ ˇ ˇb  a a .n C 2/Š ba

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 5, © George A. Anastassiou 2011

51

52

5 Multivariate Radial Mixed Fractional Ostrowski Inequalities

Inequality (5.2) is sharp. In particular, when n is odd is attained by f  .y/ WD .y  x/nC1  .b  a/; while when n is even the optimal function is f .y/ WD jy  xjnC˛  .b  a/; ˛ > 1: Clearly inequality (5.2) generalizes inequality (5.1) for higher order derivatives of f . Also in [2] see Chaps. 24–26, we gave a complete theory of left fractional Ostrowski inequalities. Here we combine both right and left Caputo fractional derivatives and produce Ostrowski inequalities in the multivariate setting for radial functions. For the concepts of right and left Caputo fractional calculus we use here, we refer to [3–8, 12].

5.2 Main Results We make Remark 5.3. We define the ball B.0; R/ WD fx 2 RN W jxj < Rg  RN ; N  2; R > 0 and the sphere S N 1 WD fx 2 RN W jxj D 1g; where jj is the Euclidean norm. Let d! be the element of surface measure on S N 1 and let Z 2 N=2 d! D : !N D  .N=2/ S N 1 For x2 RN  f0g we can write uniquely x D r!; where r D jxj > 0 and ! Dx=r2S N 1 ; j!j D 1: Note that Z B.0;R/

dy D

! N RN N

is the Lebesgue measure of the ball. Following [9, pp. 149–150, Exercise 6] and [10, pp. 87–88, Theorem 5.2.2] we can write for F W B.0; R/ ! R a Lebesgue integrable function that  Z R Z Z N 1 F .x/dx D F .r!/r dr d!I (5.3) B.0;R/

we use this formula often.

S N 1

0

5.2 Main Results

53

Here the function f W B.0; R/ ! R is radial, that is, there exists a function g such that f .x/ D g.r/; where rD jxj ; r 2 Œ0; R; 8x 2 B.0; R/: We further assume that g2AC m .Œ0; R/ (i.e., g.m1/ 2 AC.Œ0; R/), m D d˛e .de the ceiling of the number); ˛ > 0 and g.k/ .r0 / D 0; k D 1; :::; m  1; r0 2 Œ0; R fixed. We find by left Caputo Taylor’s formula, [5], p. 40, that jg.r/  g.r0 /j 

 ˛  D g  r0 1;Œr

0 ;R

 .˛ C 1/

.r  r0 /˛ ;

(5.4)

˛ g is the left Caputo fractional derivative, [5] p. 38. 8r 2 Œr0 ; R; where Dr 0 Similarly, by right Caputo Taylor formula, [3], we find that

jg.r/  g.r0 /j 

 ˛  D g  r0  1;Œ0;r  .˛ C 1/

0

.r0  r/˛ ;

(5.5)

8r 2 Œ0; r0 ; where Dr˛0  g is the right Caputo fractional derivative see [6–8]. Next we observe ˇ ˇ R ˇ ˇ B.0;R/ f .y/dy ˇ ˇ ˇf .r0 !/  ˇ ˇ Vol.B.0; R// ˇ  R ˇ ˇ R R ˇ N 1 g.s/s ds d! ˇˇ N 1 ˇ S 0 ˇ  R D ˇˇg.r0 /  R ˇ R N 1 ˇ ˇ s ds d! 0 S N 1 ˇ ˇ Z R ˇ ˇ N N 1 ˇ ˇ g.s/s ds ˇ D ˇg.r0 /  N R 0 ˇZ R ˇ ˇ ˇ N ˇ N 1 D N ˇ s .g.r0 /  g.s//ds ˇˇ R 0

Z r0 Z R N N 1 N 1 s s  N jg.r0 /  g.s/j ds C jg.r0 /  g.s/j ds R r0 0 Z r0  ˛  N D g   s N 1 .r0  s/˛ ds r0 1;Œ0;r0   .˛ C 1/R N 0

Z R  ˛  s N 1 .s  r0 /˛ ds (5.6) C Dr0 g 1;Œr ;R 0

D

N  .˛ C 1/RN



 ˛  D g  r0  1;Œ0;r

Z 0

0

r0

r0

.r0  s/.˛C1/1 .s  0/N 1 ds

 ˛  C Dr g 1;Œr ;R  .1/N 1 0 0

Z R N 1 ˛  ..R  s/  R/ .s  r0 / ds r0

54

5 Multivariate Radial Mixed Fractional Ostrowski Inequalities

 ˛  N  .˛ C 1/.N  1/Š ˛CN D g  D r0 r0  1;Œ0;r0   .˛ C N C 1/  .˛ C 1/RN  ˛  C Dr g 1;Œr 0 Z  D

R

r0

0 ;R

N 1

 .1/

N 1 X

k

.1/ R

kD0

.R  s/N k1  .s  r0 /.˛C1/1 ds

k

N 1 k

!

(5.7)

 ˛   .˛ C 1/.N  1/Š ˛CN N D g   r r0  1;Œ0;r  N 0  .˛ C 1/R  .˛ C N C 1/ 0  ˛  C Dr g 1;Œr 0 

NŠ D N R

"

0

 ;R

N 1 X

.1/N Ck1 Rk

kD0

.N  1/Š kŠ.N  k  1/Š

.N  k  1/Š .˛ C 1/ .R  r0 /.N kC˛/  .N  k C ˛ C 1/

 ˛  D g   r0  1;Œ0;r  0

 ˛  g 1;Œr C Dr 0

0

r0˛CN  .˛ C N C 1/  ;R

N 1 X kD0

.1/N Ck1 kŠ .N C 1  k C ˛/ !#

 Rk .R  r0 /.N kC˛/

:

(5.8)

Hence in the radial mixed case we proved ˇ ˇ R ˇ ˇ B.0;R/ f .y/dy ˇ ˇ ˇf .r0 !/  ˇ ˇ Vol.B.0; R// ˇ ˇ ˇ Z R ˇ ˇ N N 1 ˇ ˇ g.s/s ds ˇ D ˇg.r0 /  N R 0 "  r0˛CN NŠ   N Dr˛0  g 1;Œ0;r  0  .˛ C N C 1/ R  ˛  g 1;Œr C Dr 0

N 1 X 0 ;R

kD0

.1/.N Ck1/ Rk .R  r0 /.N kC˛/ kŠ .N C 1  k C ˛/

The last inequality (5.9) is attained by ( .s  r0 /˛ ; s 2 Œr0 ; R; g.s/ D .r0  s/˛ ; s 2 Œ0; r0 : Hence g 2 AC m .Œr0 ; R/ and g 2 AC m .Œ0; r0 /:

!# : (5.9)

(5.10)

5.2 Main Results

55 .k/

See that g .k/  .r0 / D g C .r0 / D 0; k D 0; 1; :::; m  1: Hence there exists g .m1/ at r0 ; and also g .m1/ 2 AC.Œ0; R/; therefore g 2 AC m .Œ0; R/: But we have that  ˛   ˛  D g  D  .˛ C 1/ D Dr g 1;Œr r0  0 1;Œ0;r  0

We observe that N Š .˛ C 1/ R:H:S: .5.9/ D RN

"

0 ;R

:

(5.11)

r0˛CN  .˛ C N C 1/ C

N 1 X kD0

# .1/.N Ck1/ Rk .R  r0 /.N kC˛/ : kŠ .N C 1  k C ˛/ (5.12)

And also it holds Z R N L:H:S: .5.9/ D N g.s/s N 1 ds R 0 Z r0

Z R N ˛ N 1 ˛ N 1 .r0  s/ s ds C .s  r0 / s ds (5.13) D N  R r0 0

Z R N  .˛ C 1/.N  1/Š ˛CN N 1 N 1 ˛ C .1/ ..R  s/  R/ .s  r0 / ds D N r R  .˛ C 1 C N / 0 r0 " N  .˛ C 1/.N  1/Š ˛CN D N C .1/N 1 r R  .˛ C 1 C N / 0 ! # Z R N 1 X N 1 .1/k Rk  .R  s/N k1 .s  r0 /.˛C1/1 ds k r 0 kD0 N  .˛ C 1/.N  1/Š ˛CN r C .1/N 1 D N R  .˛ C 1 C N / 0 # N 1 X .N  1/Š k k N kC˛ .N k 1/Š .˛ C1/  .1/ R .R  r0 / kŠ.N  k  1/Š  .N k C˛C1/ kD0 # " N 1 X .1/N Ck1 Rk .R  r0 /N kC˛ N Š .˛ C 1/ r0˛CN D ; C RN  .˛ C N C 1/ kŠ .N  k C ˛ C 1/ kD0

(5.14) proving (5.9) attained and sharp.

56

5 Multivariate Radial Mixed Fractional Ostrowski Inequalities

We have proved Theorem 5.4. Let f W B.0; R/ ! R which is radial; that is, there exists g such that f .x/ D g.r/; r D jxj ; 8x 2 B.0; R/: Assume that g 2 AC m .Œ0; R/; m D d˛e ; ˛ > 0; and g .k/ .r0 / D 0; k D 1; :::; m  1; r0 2 Œ0; R fixed, and  ˛   ˛  D g  ; Dr g 1;Œr r0  0 1;Œ0;r  0

0 ;R

< 1:

Then .8! 2 s N 1 / ˇ ˇ R ˇ ˇ B.0;R/ f .y/dy ˇ ˇ ˇ ˇf .r0 !/  ˇ Vol.B.0; R// ˇ ˇ ˇ Z R ˇ ˇ N ˇ N 1 ˇ D ˇg.r0 /  N g.s/s ds ˇ ˇ ˇ R 0 "  NŠ  r0˛CN  N Dr˛0  g 1;Œ0;r  0  .˛ C N C 1/ R  ˛  C Dr g 1;Œr 0

N 1 X 0 ;R

.1/.N Ck1/ Rk .R  r0 /.N C˛k/ kŠ .N C ˛ C 1  k/

kD0

(5.15)

!# : (5.16)

Inequality (5.16) is sharp, it is attained by ( g.s/ D

.s  r0 /˛ ; s 2 Œr0 ; R .r0  s/˛ ; s 2 Œ0; r0 ; ˛ > 0:

(5.17)

We make Remark 5.5. (continuing Remark 5.3) We have again (by [5], p. 40) 1 g.r/  g.r0 / D  .˛/

Z

r r0

˛ .r  J /˛1 Dr g.J /dJ; 0

(5.18)

.J  r/˛1 Dr˛0  g.J /dJ;

(5.19)

8r 2 Œr0 ; R: And (by [3]), it holds g.r/  g.r0 / D

1  .˛/

Z

r0 r

8r 2 Œ0; r0 : Hence for ˛  1 we get jg.r/  g.r0 /j 

 ˛  1 .r  r0 /˛1 Dr g L .Œr ;R/ ; 0 1 0  .˛/

(5.20)

5.2 Main Results

57

8r 2 Œr0 ; R: Also   1 .r0  r/˛1 Dr˛0  g L .Œ0;r / ; 1 0  .˛/

jg.r/  g.r0 /j 

(5.21)

8r 2 Œ0; r0 : Hence ˇ ˇ R ˇ ˇ B.0;R/ f .y/dy ˇ ˇ ˇf .r0 !/  ˇ ˇ Vol.B.0; R// ˇ ˇ ˇ Z R ˇ ˇ N N 1 ˇ ˇ g.s/s ds ˇ D ˇg.r0 /  N R 0 Z r0

Z R N N 1 N 1  N s s jg.r0 /  g.s/j ds C jg.r0 /  g.s/j ds R r0 0 Z r0  ˛  N   Dr0  g L .Œ0;r / s N 1 .r0  s/˛1 ds  N 1 0 R  .˛/ 0

Z R  ˛  N 1 N 1 ˛1   ..R  s/  R/ .s  r0 / ds C Dr0 g L .Œr ;R/ .1/ 1

D

"

NŠ RN  .˛/

0

(5.22)  ˛  D g  r0  L

1 .Œ0;r0 /

 ˛  C Dr g L 0 

D

NŠ RN

"

r0

N 1 X kD0

 ˛  D g  r0  L



 .˛/ r ˛CN 1  .˛ C N / 0

1 .Œr0 ;R/

 .1/N 1

.1/k Rk .N  k  1/Š .˛/.R  r0 /N C˛k1 kŠ.N  k  1/Š .N C ˛  k/

1 .Œ0;r0

 /

 ˛  g L C Dr 0

#

r0˛CN 1  .˛ C N /

1 .Œr0 ;R/



N 1 X kD0

# .1/N Ck1 Rk .R  r0 /N C˛k1 : (5.23) kŠ .N C ˛  k/

We have proved: Theorem 5.6. Let f W B.0; R/ ! R which is radial, that is, there exists such that f .x/ D g.r/; r D jxj ; 8x 2 B.0; R/: Assume that g 2 AC m .Œ0; R/; m D d˛e ; ˛  1; and g.k/ .r0 / D 0; k D 1; :::; m  1; r0 2 Œ0; R fixed, and  ˛  D g  r0  L

1 .Œ0;r0 /

 ˛  ; Dr g L 0

1 .Œr0 ;R/

< 1:

58

5 Multivariate Radial Mixed Fractional Ostrowski Inequalities

Then .8! 2 S N 1 / ˇ ˇ ˇ R ˇ Z R ˇ ˇ ˇ ˇ N B.0;R/ f .y/dy ˇ ˇ g.s/s N 1 ds ˇˇ ˇ D ˇˇg.r0 /  N ˇf .r0 !/  ˇ Vol.B.0; R// ˇ R 0 "   ˛  NŠ  r ˛CN 1  N Dr˛0  g L .Œ0;r /  0 g L .Œr ;R/ C Dr 0 1 0 1 0 R  .˛ C N / !# N 1 X .1/N Ck1 Rk .R  r0 /N C˛k1  : kŠ .N C ˛  k/

(5.24)

kD0

We also make Remark 5.7. Let p; q > 1 W We have

1 p

C

1 q

D 1; with ˛ > 1  p1 :

Z

r

ˇ ˇ ˛ .r  J /˛1 ˇDr g.J /ˇ dJ 0

1 jg.r/  g.r0 /j   .˛/ 1   .˛/

r0

Z

r

r0

.r  J /

p.˛1/

1=p dJ

 ˛  D g  r0 L

q .Œr0 ;R/

1

 1 .r  r0 /.˛1/C p  D ˛ g  D : r0 Lq .Œr0 ;R/  .˛/ .p.˛  1/ C 1/1=p

(5.25)

That is, 1

 ˛  .r  r0 /.˛1/C p D g  jg.r/  g.r0 /j  r0 Lq .Œr0 ;R/; 1=p  .˛/.p.˛  1/ C 1/

(5.26)

8r 2 Œr0 ; R: Also we have 1 jg.r/  g.r0 /j   .˛/

Z r

r0

ˇ ˇ .J  r/˛1 ˇDr˛0  g.J /ˇ dJ 1



 .r0  r/.˛1/C p  1 D ˛ g  ; r0  Lq .Œ0;r0 / 1=p  .˛/ .p.˛  1/ C 1/

(5.27)

8r 2 Œ0; r0 : Consequently we obtain .8! 2 S N 1 / ˇ ˇ R ˇ ˇ B.0;R/ f .y/dy ˇ ˇ ˇf .r0 !/  ˇ ˇ Vol.B.0; R// ˇ ˇ ˇ Z R ˇ ˇ N N 1 ˇ ˇ g.s/s ds ˇ D ˇg.r0 /  N R 0

(5.28)

5.2 Main Results

N  N R

Z

r0

59

s

N 1

0

Z jg.r0 /  g.s/j ds C

R

s

N 1

r0

jg.r0 /  g.s/j ds

Z r0  ˛  1 N   s N 1 .r0  s/.˛1/C p ds Dr0  g L .Œ0;r /   N 1=p q 0  .˛/R .p.˛  1/ C 1/ 0

Z R 1  ˛  .˛1/C p N 1 N 1   C Dr0 g L .Œr ;R/  .1/  ..R  s/  R/ .s  r0 / ds q

0

r0

N D  .˛/.p.˛  1/ C 1/1=p R N C˛C p1 1

 r0

2   4Dr˛  g  0 L N N 1 X

 ˛  C Dr g L 0

q .Œr0 ;R/





kD0



.N  1/Š ˛ C p1    q .Œ0;r0 /  N C ˛ C p1



(5.29)

.N  1/Š .1/N Ck1 Rk kŠ.N  k  1/Š

3 .N  k  1/Š ˛ C p1 1   .R  r0 /N kC˛C p 1 5  1  N kC˛C p   2 N C˛1=q N Š ˛ C p1   r0 4Dr˛  g     D 0 Lq .Œ0;r0 /  .˛/.p.˛  1/ C 1/1=p RN  N C ˛ C p1 3 N 1 N Ck1 k .N kC˛1=q/ X  ˛  .1/ R .R  r / 0 5:   C Dr0 g L .Œr ;R/  q 0 1 kŠ N  k C ˛ C kD0 p

(5.30)

We have established Theorem 5.8. Let p; q > 1W p1 C q1 D 1; ˛ > 1 p1 ; d˛e D m: Let f WB.0; R/ ! R which is radial, that is, there exists g such that f .x/ D g.r/; r D jxj ; 8x 2 B.0; R/: Suppose that g 2 AC m .Œ0; R/; and g .k/ .r0 / D 0; k D 1; :::; m  1; r0 2 Œ0; R fixed, and  ˛   ˛  D g  D g  ; < 1: r0  r0 L .Œ0;r / L .Œr ;R/ q

0

q

0

Then .8! 2 S N 1 / ˇ ˇ ˇ R ˇ Z R ˇ ˇ ˇ ˇ N B.0;R/ f .y/dy ˇ ˇ N 1 ˇ ˇ g.s/s ds ˇ ˇ D ˇg.r0 /  N ˇf .r0 !/  ˇ Vol.B.0; R// ˇ R 0   2 N Š ˛ C p1   4Dr˛  g    0 Lq .Œ0;r0 / 1=p N  .˛/.p.˛  1/ C 1/ R

60

5 Multivariate Radial Mixed Fractional Ostrowski Inequalities N C˛ 1

q  ˛  r0 C Dr  g L .Œr ;R/ 1 0 q 0  .N C ˛ C p /   13 0 N 1 N kC˛ q1 N Ck1 k X .1/ R .R  r0 / A5 :   @ kŠ N  k C ˛ C p1 kD0

(5.31)

We give Corollary 5.9. Let ˛ > 12 ; d˛e D m; f W B.0; R/ ! R is radial, with g W f .x/ D g.r/; r D jxj ; 8x 2 B.0; R/: Suppose g 2 AC m .Œ0; R/ W g.k/ .r0 / D 0; k D 1; :::; m  1; r0 2 Œ0; R fixed, and  ˛  D g  r0  L

2 .Œ0;r0

 ˛  D g  ; r0 / L

2 .Œr0 ;R/

< 1:

Then .8! 2 S N 1 / ˇ ˇ ˇ R ˇ Z R ˇ ˇ ˇ ˇ N B.0;R/ f .y/dy ˇ ˇ N 1 ˇ ˇ !/  /  g.s/s ds D f .r g.r ˇ ˇ 0 ˇ 0 ˇ N ˇ Vol.B.0; R// ˇ R 0 "  ˛  N Š .˛ C 12 / D g  p  r0  L2 .Œ0;r0 / N  .˛/ 2˛  1R 





N C˛ 12

r0

 N C˛C N 1 X kD0

1 2

 ˛   C Dr g L 0

2 .Œr0 ;R/

:

1 .1/N Ck1 Rk .R  r0 /.N kC˛ 2 /   kŠ N  k C ˛ C 12

!# :

(5.32)

References 1. G. A. Anastassiou, “Ostrowski type inequalities”, Proc. AMS 123 (1995), 3775-3781. 2. G. A. Anastassiou, “Fractional Differentiation Inequalities”, Research Monograph, Springer, New York, 2009. 3. G. A. Anastassiou, “On Right Fractional Calculus, ”Chaos, Solitons and Fractals”, 42(2009), 365-376. 4. G. A. Anastassiou, Multivariate radial mixed Caputo fractional Ostrowski inequalities, submitted, 2011. 5. Kai Diethelm, Fractional Differential Equations, online: http://www.tubs.de/diethelm/lehre/ f-dg102/fde-skript.ps.gz. 6. A. M. A. El-Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol.3, No.12(2006), 81-95.

References

61

7. G. A. Frederico and D. F. M. Torres, Fractional Optimal Control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol.3, No. 10 (2008), 479-493. 8. R. Gorenflo and F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http:// www.maphysto.dk/oldpages/events/LevyCAC2000/MainardiNotes/fm2k0a.ps 9. W. Rudin, Real and Complex Analysis, International Student edition, Mc Graw Hill, London, New York, 1970. 10. D. Stroock, A Concise Introduction to the Theory of Integration, Third Edition, Birkha¨user, Boston, Basel, Berlin, 1999. ¨ 11. A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10, (1938), 226-227. 12. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka i Technika, Minsk, 1987)].

Chapter 6

Shell Mixed Caputo Fractional Ostrowski Inequalities

Here we present general shell mixed Caputo fractional Ostrowski inequalities, radial and nonradial cases. One of them is proved to be sharp and attained. Estimates are with respect to kkp , 1  p  1: This chapter is based on [4].

6.1 Introduction In 1938, A. Ostrowski [11] proved the following important inequality. Theorem 6.1. Let f W Œa; b ! R be continuous on Œa; b and differentiable on .a; b/ whose derivative f 0 W .a; b/ ! R is bounded on .a; b/, i.e., kf 0 k1 WD sup jf 0 .t/j < C1: Then t 2.a;b/

ˇ ˇ "  2 # ˇ 1 Z b ˇ   x  aCb 1 ˇ ˇ 2  .b  a/ f 0 1 ; C f .t/ dt  f .x/ˇ  ˇ 2 ˇb  a a ˇ 4 .b  a/ for any x 2 Œa; b. The constant

1 4

(6.1)

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications to numerical analysis and probability. This chapter is greatly motivated and inspired by the following result. Theorem 6.2 (see [1]). Let f 2 C nC1 .Œa; b/, n 2 N and x 2 Œa; b be fixed, such that f .k/ .x/ D 0, k D 1; :::; n: Then it holds ˇ ˇ  ˇ 1 Z b ˇ f .nC1/   ˇ ˇ 1  f .y/ dy  f .x/ˇ  ˇ ˇb  a a ˇ .n C 2/Š

.x  a/nC2 C .b  x/nC2 ba

! : (6.2)

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 6, © George A. Anastassiou 2011

63

64

6 Shell Mixed Caputo Fractional Ostrowski Inequalities

Inequality (6.2) is sharp. In particular, when n is odd is attained by f  .y/ WD .y  x/nC1  .b  a/, while when n is even the optimal function is f .y/ WD jy  xjnC˛  .b  a/ ,

˛ > 1.

Clearly inequality (6.2) generalizes inequality (6.1) for higher order derivatives of f . Also in [2], see Chaps. 24–26, we gave a complete theory of left fractional Ostrowski inequalities. Here we combine both right and left Caputo fractional derivatives and produce Ostrowski inequalities in the multivariate setting of a shell for radial and nonradial functions. A nonradial case ball result is given at the end of the chapter. For the nonradial case results we use the left and right fractional radial derivatives. For the basic concepts of fractional calculus used here, we refer to [3, 6–9, 12].

6.2 Main Results We make Remark 6.3. Let the spherical shell A WD B .0; R2 /  B .0; R1 /, 0 < R1 < R2 , A  RN , N  2, x 2 A. Consider that f W A ! R is radial, that is, there exists g such that f .x/ D g .r/, r D jxj, r 2 ŒR1 ; R2 , 8 x 2 A: Here x can be written uniquely as x D r!, where r D jxj > 0 and ! D xr 2 S N 1 ; j!j D 1, see ([10], pp. 149–150 and [2], p. 421), furthermore for F W A ! R a Lebesgue integrable function we have that  Z R2 Z Z F .x/ dx D F .r!/ r N 1 dr d!: (6.3) S N 1

A

R1

Let d! be the element of surface measure on S N 1 and let Z

!N

N

2 2 D d! D  N  : N 1  2 S

! .RN RN / Here Vol .A/ D N 2N 1 , and we suppose that g 2 AC m .ŒR1 ; R2 / (i.e., g .m1/ 2 AC .ŒR1 ; R2 /); m D d˛e (de ceilling of number), ˛ > 0, and g.k/ .r0 / D 0, k D 1; :::; m  1; r0 2 ŒR1 ; R2  : We get by [6], p. 40, that

1 g .r/  g .r0 / D  .˛/

Z

r

r0

˛ .r  J /˛1 Dr g .J / dJ; 0

(6.4)

6.2 Main Results

65

˛ 8 r 2 Œr0 ; R2 , where Dr g is the left Caputo fractional derivative of order ˛, see 0 [6], p. 38. And, by [3], Z r0 1 g .r/  g .r0 / D .J  r/˛1 Dr˛0  g .J / dJ; (6.5)  .˛/ r

(where Dr˛0  g is the right Caputo fractional derivative of order ˛, see [7–9]), ˛ 8r 2 ŒR1 ; r0 . Here assume Dr g 2 L1 .Œr0 ; R2 / and Dr˛0  g 2 L1 .ŒR1 ; r0 /. 0 We obtain  .r  r0 /˛  D ˛ g  ; (6.6) jg .r/  g .r0 /j  r0 1;Œr0 ;R2   .˛ C 1/ 8 r 2 Œr0 ; R2  and jg .r/  g .r0 /j 

 .r0  r/˛  D ˛ g  ; r0  1;ŒR1 ;r0   .˛ C 1/

(6.7)

8 r 2 ŒR1 ; r0  : Next we observe that R ˇ ˇ ˇ ˇ   Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .r0 !/  A f .y/ dy ˇ D ˇg .r0 /  g .s/ s ds ˇ ˇ ˇ ˇ N N Vol .A/ R2  R 1 R1 ˇ   ˇZ R2 ˇ ˇ N ˇ D .g .r0 /  g .s// s N 1 ds ˇˇ ˇ N N R2  R1 R1   Z R2 N  (6.8) jg .r0 /  g .s/j s N 1 ds R2N  R1N R1     Z r0 Z R2 N N 1 N 1 D ds C ds jg .r0 /  g .s/j s jg .r0 /  g .s/j s R2N  R1N R1 r0    Z r0  ˛  1 N   Dr0  g 1;ŒR ;r   .r0  s/˛ s N 1 ds 1 0 R2N  R1N  .˛ C 1/ R1  Z R2  ˛  ˛ N 1   .sr0 / s ds DW ./: C Dr0 g 1;Œr ;R  0

2

r0

(6.9) Here we calculate Z r0 Z I1 WD .r0  s/˛ s N 1 ds D R1

D

kD0

D

R1

N 1  X

N 1 X kD0

r0

.r0  s/˛ ..s  R1 / C R1 /N 1 ds

 Z r0 N 1 k R1 .r0  s/.˛C1/1 .s  R1 /N k1 ds k R1

.N  1/Š  .˛ C 1/ .N  k  1/Š Rk .r0 R1 /˛CN k : kŠ .N  k  1/Š 1  .˛ C 1 C N  k/

66

6 Shell Mixed Caputo Fractional Ostrowski Inequalities

That is, I1 D .N  1/Š .˛ C 1/

N 1 X kD0

Also

Z

I2 WD

R2

r0

D .1/

s N 1 .s  r0 /˛ ds D .1/N 1 N 1

N 1 X kD0

D

N 1 X kD0

Z

R1k .r0 R1 /˛CN k : kŠ .˛ C 1 C N  k/ R2

..R2  s/  R2 /N 1 .s  r0 /˛ ds

r0

.N  1/Š .1/k R2k kŠ .N  k  1/Š

(6.10)

Z

R2 r0

.R2 s/.N k/1 .sr0 /.˛C1/1 ds

.N 1/Š .N  k  1/Š .˛ C 1/ .1/N Ck1 R2k .R2  r0 /N kC˛ : kŠ .N k1/Š  .N  k C ˛ C 1/

That is, I2 D .N  1/Š .˛ C 1/

N 1 X kD0

.1/N Ck1 R2k .R2  r0 /N kC˛ : kŠ .N  k C ˛ C 1/

(6.11)

Consequently we obtain (  N 1 X  ˛  NŠ R1k .r0 R1 /˛CN k D g  ./ D r0  N N  1;ŒR ;r 1 0 kŠ .˛ C 1 C N  k/ R2  R1 kD0  ˛  C Dr g 1;Œr 0

N 1 X 0 ;R2 

kD0

) .1/N Ck1 R2k .R2  r0 /N kC˛ : kŠ .N  k C ˛ C 1/ (6.12)

So far we have proved that (8 ! 2 S N 1 ) R ˇ ˇ ˇ ˇ   Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .r0 !/  A f .y/ dy ˇ D ˇg .r0 /  g .s/ s ds ˇ ˇ ˇ ˇ N N Vol .A/ R2  R 1 R1 !  ( N 1 X  ˛  R1k .r0 R1 /˛CN k NŠ    Dr0  g 1;ŒR ;r  1 0 kŠ .˛ C 1 C N  k/ R2N  R1N kD0

 ˛  g 1;Œr C Dr 0

N 1 X 0 ;R2 

kD0

.1/N Ck1 R2k .R2 r0 /N kC˛ kŠ .N kC˛C1/

!) :

(6.13) Inequality (6.13) (˛ > 0) is attained by ( .s  r0 /˛ ; s 2 Œr0 ; R2  ; g .s/ WD .r0  s/˛ ; s 2 ŒR1 ; r0  :

(6.14)

6.2 Main Results

67

Here g fulfills all assumptions and  ˛   ˛  D g  D Dr g 1;Œr r0  0 1;ŒR ;r  1 0

0 ;R2 

D  .˛ C 1/ :

(6.15)

Hence L:H:S: .6.13/ D

N R2N  R1N

N D N R2  R1N N D N R2  R1N

Z

R2

g .s/ s N 1 ds

R1

Z "

r0 R1

˛ N 1

.r0  s/ s

Z ds C

.N  1/Š .˛ C 1/

N 1 X kD0

R2 r0

˛ N 1

.s  r0 / s

  D

NŠ N R2  R1N

kD0

"

.1/N Ck1 R2k .R2  r0 /N kC˛ kŠ .N  k C ˛ C 1/

. .˛ C 1//

N 1 X kD0

ds

R1k .r0 R1 /˛CN k kŠ .˛ C 1 C N  k/

C .N  1/Š .˛ C 1/ N 1 X



!#

R1k .r0 R1 /˛CN k kŠ .˛ C 1 C N  k/

C . .˛ C 1// 

N 1 X kD0

.1/N Ck1 R2k .R2  r0 /N kC˛ kŠ .N  k C ˛ C 1/

D R:H:S: .6.13/;

!

!

!#

(6.16)

proving it attained and sharp (6.13). We have established Theorem 6.4. Let f W A ! R be radial, that is, there exists g such that f .x/ D g .r/, r D jxj, 8 x 2 A; ! 2 S N 1 . Assume that g 2 AC m .ŒR1 ; R2 /, m D d˛e, ˛ > 0, and g.k/ .r0 / D 0 (r0 2 ŒR1 ; R2  fixed), k D 1; :::; m  1; and Dr˛0  g 2 ˛ g 2 L1 .Œr0 ; R2 /. Then .8 ! 2 S N 1 / L1 .ŒR1 ; r0 /, Dr 0 R ˇ ˇ ˇ ˇ  Z R2  ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .r0 !/  A f .y/ dy ˇ D ˇg .r0 /  g .s/ s ds ˇ ˇ ˇ ˇ Vol .A/ R2N  R1N R1 ! (  N 1 ˛CN k k X  ˛  R .r R / NŠ 0 1 1 D g   r0  1;ŒR1 ;r0  kŠ .˛ C 1 C N  k/ R2N  R1N kD0

 ˛  g 1;Œr CDr 0

N 1 X 0 ;R2 

kD0

.1/N Ck1 R2k .R2  r0 /N kC˛ kŠ .N  k C ˛ C 1/

!) :

(6.17)

68

6 Shell Mixed Caputo Fractional Ostrowski Inequalities

The last inequality (6.17) is sharp, that is attained by 

.s  r0 /˛ ; s 2 Œr0 ; R2  ; .r0  s/˛ ; s 2 ŒR1 ; r0  :

g .s/ WD

(6.18)

We need to make Remark 6.5. Let ˛  1. We get easily by (6.4) that jg .r/  g .r0 /j 

 .r  r0 /˛1  D ˛ g  ; r0 L1 .Œr0 ;R2 /  .˛/

(6.19)

8 r 2 Œr0 ; R2  : Also by (6.5) we find that jg .r/  g .r0 /j 

 .r0  r/˛1  D ˛ g  ; r0  L1 .ŒR1 ;r0 /  .˛/

(6.20)

8 r 2 ŒR1 ; r0  :  ˛    Here we suppose Dr g L .Œr ;R / ; Dr˛0  g L .ŒR ;r / < 1: 0 1 0 2 1 1 0 Then it holds R ˇ ˇ ˇ ˇ Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .r0 !/  A f .y/ dy ˇ D ˇg .r0 /  g .s/ s ds ˇ ˇ ˇ ˇ N N Vol .A/ R2  R1 R1  

N R2N  R1N

Z

R2 R1

jg .r0 /  g .s/j s N 1 ds

(6.21)

  Z r0  Z R2 N N 1 N 1 D ds C ds jg .r0 / g .s/j s jg .r0 / g .s/j s R2N R1N R1 r0    Z r0   ˛  1 N ˛1 N 1   .r0  s/ s ds Dr0  g L .ŒR ;r /  1 1 0 R2N  R1N  .˛/ R1 

Z

 ˛  C Dr g L 0

1 .Œr0 ;R2 /



NŠ D R2N  R1N

(

 ˛  g L C Dr 0

 ˛  D g  r0  L

1 .Œr0 ;R2 /



N 1 X 1 .ŒR1 ;r0 /

N 1 X kD0

kD0

R2 r0

˛1 N 1

.s  r0 /

s

R1k .r0 R1 /˛CN k1 kŠ .˛ C N  k/

 ds

!

.1/N Ck1 R2k .R2 r0 /N kC˛1 kŠ .N kC˛/

!) : (6.22)

6.2 Main Results

69

We have established Theorem 6.6. Let f W A ! R be radial, that is, there exists g such that f .x/ D g .r/, r D jxj, 8 x 2 A; ! 2 S N 1 . Assume that g 2 AC m .ŒR1 ; R2 / ; m D d˛e, ˛  1, and g .k/ .r0 / D 0, k D 1; :::; m  1I r0 2 ŒR1 ; R2  be fixed. Assume Dr˛0  g 2 ˛ L1 .ŒR1 ; r0 / and Dr g 2 L1 .Œr0 ; R2 /. Then (8 ! 2 S N 1 ) we get 0 R ˇ ˇ ˇ  Z ˇ f .y/ dy ˇ ˇ N ˇDˇg .r0 /  ˇf .r0 !/  A ˇ Vol .A/ ˇ ˇ RN RN 2



NŠ  N R2 R1N

(

 ˛  D g  r0  L

1

N 1 X 1 .ŒR1 ;r0 /

kD0

1 .Œr0 ;R2 /

R1

g .s/ s

N 1

ˇ ˇ ds ˇˇ !

R1k .r0 R1 /˛CN k1 kŠ .˛CN k/

N 1 X

 ˛  g L C Dr 0

R2

kD0

!) .1/N Ck1 R2k .R2 r0 /N kC˛1 : kŠ .N kC˛/ (6.23)

We continue with Remark 6.7. Let p; q > 1 W 1 jg .r/  g .r0 /j   .˛/

1 p

1 q

C

Z

r r0

D 1, ˛ > q1 . Then by (6.4) we derive

.r  J /

p.˛1/

 p1 dJ

 ˛  D g  r0 L

q .Œr0 ;R2 /

;

(6.24)

8 r 2 Œr0 R2  : That is, 1

.˛1/C p  ˛  .r  r0 / 1 D g  ; jg .r/  g .r0 /j  r0 1 Lq .Œr0 ;R2 /  .˛/ .p .˛  1/ C 1/ p

(6.25)

8 r 2 Œr0 ; R2 . Similarly by (6.5) we obtain 1

jg .r/  g .r0 /j 

 1 .r0  r/˛1C p  D ˛ g  ; r0  1 Lq .ŒR1 ;r0 /  .˛/ .p .˛  1/ C 1/ p

(6.26)

˛ g 2 Lq .Œr0 ; R2 /, and Dr˛0  g 2 8 r 2 ŒR1 ; r0  : Here we assume that Dr 0 Lq .ŒR1 ; r0 / : Hence we derive R ˇ ˇ ˇ ˇ   Z R2 ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .r0 !/  A f .y/ dy ˇ D ˇg .r0 /  g .s/ s ds ˇ (6.27) ˇ ˇ ˇ N N Vol .A/ R2  R1 R1 ˇ  ˇZ R2  ˇ ˇ N N 1 ˇ ˇ .g .r0 /  g .s// s ds ˇ D ˇ N N R2  R1 R1

70

6 Shell Mixed Caputo Fractional Ostrowski Inequalities

   D  

N N R2  R1N N N R2  R1N N  R1N

Z

R2

R1

Z

jg .r0 /  g .s/j s N 1 ds

r0

R1



jg .r0 /  g .s/j s N 1 ds C

Z

R2

r0

jg .r0 /  g .s/j s N 1 ds



1

 1  .˛/ .p .˛  1/ C 1/ p  Z r0   ˛  ˛ q1 N 1   .r0  s/ s ds  Dr0  g L .ŒR ;r / R2N

q

1 0

 ˛  C Dr g L 0

q .Œr0 ;R2 /

D

R2N



R1N



R1

Z

R2 r0

˛ q1

.s  r0 /

N 1

 .˛/ .p .˛  1/ C 1/ p

s

N 1

 ds

(6.28)



8   <  1 ˛   Dr0  g L .ŒR ;r / .N  1/Š ˛  C 1  q 1 0 : q 0

1 ˛ q1 CN k k R .r R / 0 1 1 A @ 1 kŠ ˛  C 1 C N  k kD0 q    ˛  1  g .N  1/Š ˛  C 1  C Dr 0 Lq .Œr0 ;R2 / q 19 0 N 1 N kC˛ q1 = N Ck1 k X .1/ R2 .R2  r0 /  A : @ ; kŠ N  k C ˛  q1 C 1 kD0 N 1 X

We have established Theorem 6.8. Let p; q > 1 W

1 p

C

1 q

D 1: Let f W A ! R be radial, that is,

there exists g such that f .x/ D g .r/, r D jxj, 8 x 2 A; ! 2 S N 1 . Assume that g 2 AC m .ŒR1 ; R2 /, m D d˛e, ˛ > q1 and g.k/ .r0 / D 0, k D 1; :::; m  1I r0 2 ˛ ŒR1 ; R2  be fixed. Suppose also Dr˛0  g 2 Lq .ŒR1 ; r0 / and Dr g 2 Lq .Œr0 ; R2 /. 0 N 1 Then .8! 2 S / we obtain R ˇ ˇ ˇ f .y/ dy ˇ ˇf .r0 !/  A ˇ ˇ Vol .A/ ˇ ˇ ˇ   Z R2 ˇ ˇ N N 1 ˇ g .s/ s ds D ˇˇg .r0 /  ˇ N N R2  R1 R1

(6.29)

6.2 Main Results

71

  ˛ C p1 N Š     1  .˛/ R2N  R1N .p .˛  1/ C 1/ p 8 1 0 N 1 ˛ q1 CN k < k X  R .r R / 0 1 1  A  Dr˛0  g L .ŒR ;r / @ q 1 0 : 1 kŠ ˛ C C N  k kD0

0

 ˛  C Dr g L 0

q .Œr0 ;R2 /

@

p

N 1 X kD0

19 1 .1/N Ck1 R2k .R2  r0 /N kC˛ q A=  : ; kŠ N  k C ˛ C p1 (6.30)

We mention Corollary 6.9 (p D q D 2 case). Let f W A ! R be radial, that is, there exists g such that f .x/ D g .r/, r D jxj, 8 x 2 A; ! 2 S N 1 . Assume that g 2 AC m .ŒR1 ; R2 / ; m D d˛e, ˛ > 12 and g .k/ .r0 / D 0, k D 1; :::; m  1I ˛ r0 2 ŒR1 ; R2 . Suppose Dr˛0  g 2 L2 .ŒR1 ; r0 / ; Dr g 2 L2 .Œr0 ; R2 /. Then 0 N 1 .8! 2 S / R ˇ ˇ ˇ ˇ  Z R2  ˇ ˇ ˇ ˇ N N 1 ˇ ˇf .r0 !/  A f .y/ dy ˇ D ˇg .r0 /  g .s/ s ds ˇ ˇ ˇ ˇ N N Vol .A/ R2  R1 R1    ˛ C 12 N Š   N  1  .˛/ R2  R1N .2˛  1/ 2 ( ! 1 N 1 X  ˛  R1k .r0 R1 /˛ 2 CN k      Dr0  g L .ŒR ;r / 2 1 0 kŠ ˛ C 12 C N  k kD0  ˛  g L C Dr 0

2 .Œr0 ;R2 /

N 1 X kD0

1

.1/N Ck1 R2k .R2  r0 /N kC˛ 2   kŠ N  k C ˛ C 12

!) : (6.31)

Finally we deal with nonradial L1 Ostrowski inequalities on the shell. We need (see also [2], p. 421) Definition 6.10. Let F W A ! R, ˛ > 0, m D d˛e such that F .!/ 2 AC m .ŒR1 ; R2 /, 8 ! 2 S N 1 . We call the Caputo left radial fractional derivative the following function @˛r0 F .x/ 1 WD @r ˛  .m  ˛/

Z

r

r0

.r  t/m˛1

@m F .t!/ dt; @r m

(6.32)

where x 2 A; that is, x D r!; r; r0 2 ŒR1 ; R2 , r0 is fixed, ! 2 S N 1 , 8 r  r0 :

72

6 Shell Mixed Caputo Fractional Ostrowski Inequalities

Clearly @0r0 F .x/ D F .x/ ; @r 0 @˛r0 F .x/ @˛ F .x/ D ; if ˛ 2 N: @r ˛ @r ˛

(6.33) (6.34)

The above function exists almost everywhere for x 2 A. We also need Definition 6.11. Let F W A ! R, ˛ > 0, m D d˛e such that F .!/ 2 AC m .ŒR1 ; R2 /, 8 ! 2 S N 1 . We call the Caputo right radial fractional derivative the following function Z r0 m @˛r0  F .x/ .1/m m˛1 @ F .J !/ WD .J  r/ dJ; (6.35) @r ˛  .m  ˛/ r @J m where x 2 A; that is, x D r!; r; r0 2 ŒR1 ; R2 , r0 is fixed, ! 2 S N 1 , 8 r  r0 : Clearly @0r0  F .x/ D F .x/ ; (6.36) @r 0 ˛ @˛r0  F .x/ ˛ @ F .x/ D .1/ ; if ˛ 2 N: (6.37) @r ˛ @r ˛ The above function exists almost everywhere for x 2 A. We need to make Remark 6.12. We treat here the general, not necessarily radial, case of f: We apply Theorem 6.4 to f .r!/, ! fixed, r 2 ŒR1 ; R2 , under the following assumptions: f .!/ 2 AC m .ŒR1 ; R2 / ; 8 ! 2 S N 1 , ˛ > 0, m D d˛e, where f W A ! R k is Lebesgue integrable; @@rfk , k D 1; :::; m  1; vanish on @B .0; r0 /, r0 is fixed   @˛ 0 f in ŒR1 ; R2  I and r 2 B A1 (bounded functions), where A1 WD B .0; R2 /  @r ˛   @˛0  f B .0; r0 /, and r@r 2 B A2 , where A2 WD B .0; r0 /  B .0; R1 /; along with ˛ ˛ Dr˛0  f .!/ 2 L1 .ŒR1 ; r0 /, Dr f .!/ 2 L1 .Œr0 ; R2 / ; 8 ! 2 S N 1 : 0 N 1 ) Then (8 ! 2 S ˇ  ˇ ˇf .r0 !/  ˇ

ˇ  Z R2 ˇ N N 1 ˇ f .s!/ s ds ˇ N N R2  R1 R1 !   ( ˛  N 1 X  @˛r0 f  @r0  f  NŠ R1k .r0 R1 /˛CN k    C   @r ˛  @r ˛  R2N  R1N 1;A2 kD0 kŠ .˛ C 1 C N  k/ !) N 1 X .1/N Ck1 R2k .R2  r0 /N kC˛ DW 1 :  kŠ .N  k C ˛ C 1/ kD0

   

1;A1

(6.38)

6.2 Main Results

73

Therefore ˇR Z R2  ˇˇ Z ˇ N ˇ S N 1 f .r0 !/ d! ˇ N 1   N f .s!/ s ds d! ˇ  1 : ˇ ˇ ˇ !N R2  R1N ! N S N 1 R1 (6.39) That is, ˇ  Z ˇ R ˇ N ˇ ˇ 2 A f .x/ dx ˇ f .r0 !/ d!  (6.40) ˇ ˇ  1 : N ˇ 2 2 S N 1 Vol .A/ ˇ Therefore it holds for x 2 A that ˇ  Z R ˇ ˇ ˇ ˇ ˇ  N2 f .x/ dx ˇ ˇˇ ˇ A ˇf .x/  ˇ  ˇf .x/  f .r !/ d! ˇ C 1 : 0 N ˇ ˇ N 1 Vol .A/ ˇ ˇ 2 S 2

(6.41)

We have proved the following. Theorem 6.13. Let f W A ! R be Lebesgue integrable with f .!/ 2AC m k .ŒR1 ; R2 /, ˛ > 0, m D d˛e, 8 ! 2 S N 1 I @@rfk , k D 1; :::; m  1; vanish on ˛     @˛0  f @ 0f @B .0; r0 / ; r0 fixed in ŒR1 ; R2  I and r 2 B A1 ; r@r 2 B A2 along with ˛ @r ˛ ˛ f .!/ 2 L1 .Œr0 ; R2 / ; 8 ! 2 S N 1 . Then for Dr˛0  f .!/ 2 L1 .ŒR1 ; r0 / ; Dr 0 x 2 A we have ˇ N  Z R ˇ ˇ ˇˇ ˇ ˇ ˇ  f .x/ dx ˇ ˇ 2 ˇf .x/  A ˇ  ˇf .x/  f .r0 !/ d! ˇ N ˇ ˇ ˇ ˇ N 1 Vol .A/ 2 2 S  

NŠ N R2  R1N

( ˛  @r0  f   @r ˛

   

N 1 X

1;A2

 ˛  @r0 f C  @r ˛

   

kD0

1;A1

R1k .r0 R1 /˛CN k kŠ .˛ C 1 C N  k/

N 1 X kD0

!

.1/N Ck1 R2k .R2  r0 /N kC˛ kŠ .N  k C ˛ C 1/

!) :

(6.42) We also make Remark 6.14. Let f W B .0; R/ ! R be a Lebesgue integrable function, that is, not necessarily a radial function. Suppose f .!/ 2 AC .Œ0; R/, 8 ! 2 S N 1 ; 0 < ˛ ˛  1I r0 2 Œ0; R fixed, Dr˛0  f .!/ 2 L1 .Œ0; r0 /, Dr f .!/ 2 L1 .Œr0 ; R/, 8 0 N 1 : We further assume that !2S  ˛  D f .t!/ r0  1;.t 2Œ0;r where K > 0:

0 /

 ˛  ; Dr f .t!/1;.t 2Œr 0

0 ;R/

 K; 8 ! 2 S N 1 ; (6.43)

74

6 Shell Mixed Caputo Fractional Ostrowski Inequalities

By inequality (16) of [5], applied on f .!/ ; 8 ! 2 S N 1 , we get ˇ ˇ Z R ˇ ˇ N 1 ˇ ˇf .r0 !/  N f .s!/ s ds ˇ ˇ N R 0 !# " N 1 X .1/.N Ck1/ Rk .R  r0 /.N C˛k/ N ŠK r0˛CN  N DW 2 : C R  .˛ C N C 1/ kŠ .N C ˛ C 1  k/ kD0

(6.44) Therefore it holds ˇR  ˇ Z R Z ˇ ˇ S N 1 f .r0 !/ d! N N 1 ˇ  f .s!/ s ds d! ˇˇ  2 : ˇ N !N R ! N S N 1 0

(6.45)

That is, we find ˇ  Z ˇ R ˇ N ˇ ˇ B.0;R/ f .x/ dx ˇ 2 f .r0 !/ d!  ˇ ˇ  2 : N ˇ 2 2 S N 1 Vol .B .0; R// ˇ

(6.46)

Consequently we derive ˇ ˇ ˇ ˇ R  Z ˇ ˇ ˇ ˇ  N2 ˇ ˇ ˇ B.0;R/ f .x/ dx ˇ f .r0 !/ d! ˇ C 2 : (6.47) ˇ  ˇf .x/  ˇf .x/  N ˇ ˇ N 1 Vol .B .0; R// ˇ ˇ 2 2 S We have proved Theorem 6.15. Let f W B .0; R/ ! R be a Lebesgue integrable function, that is, not necessarily a radial function. Assume f .!/ 2 AC .Œ0; R/, R > 0; 8 ! 2 S N 1 ; 0 < ˛  1; r0 2 Œ0; R fixed, Dr˛0  f .!/ 2 L1 .Œ0; r0 /, ˛ f .!/ 2 L1 .Œr0 ; R/ ; 8 ! 2 S N 1 :  Dr 0  ˛ f .t!/1;.t 2Œr ;R/  K; 8 ! 2 Suppose also Dr˛0  f .t!/1;.t 2Œ0;r / ; Dr 0 0 0 S N 1 ; where K > 0. Then ˇ ˇ ˇ ˇ R N  Z ˇ ˇ ˇ ˇ f .x/ dx  ˇ ˇ ˇ ˇ B.0;R/ 2 f .r !/ d! ˇf .x/  ˇ ˇ  ˇf .x/  0 N ˇ ˇ ˇ ˇ Vol .B .0; R// 2 2 S N 1 N ŠK C N R

"

N 1 X r0˛CN .1/.N Ck1/ Rk .R  r0 /.N C˛k/ C  .˛ C N C 1/ kŠ .N C ˛ C 1  k/

!# :

kD0

(6.48)

References

75

References 1. G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781. 2. G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009. 3. G.A. Anastassiou, On Right Fractional Calculus, Chaos, Solitons and Fractals, 42 (2009), 365-376. 4. G.A. Anastassiou, Shell mixed fractional Ostrowski inequalities, Communications in Applied Analysis, 15, No 1(2011), 101-112. 5. G.A. Anastassiou, Multivariate radial mixed Caputo fractional Ostrowski inequalities, submitted, 2011. 6. Kai Diethelm, Fractional Differential Equations, online: http://www.tu-bs.de/˜diethelm/ lehre/ f-dgl02/fde-skript.ps.gz. 7. A.M.A. El-Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12(2006), 81-95. 8. G.S. Frederico and D.F.M. Torres, Fractional Optimal Control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol. 3, No. 10(2008), 479-493. 9. R. Gorenflo and F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http:// www.maphysto.dk/oldpages/events/LevyCAC2000/MainardiNotes/fm2k0a.ps. 10. W. Rudin, Real and Complex Analysis, International Student edition, Mc Graw Hill, London, New York, 1970. ¨ 11. A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227. 12. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].

Chapter 7

Left Caputo Fractional Uniform Landau Inequalities

Here we present left Caputo fractional uniform Landau-type inequalities. We give applications and we recover the original Landau inequality on RC . This chapter relies on [3].

7.1 Introduction Let p 2 Œ1; 1; I D RC or I D R and f W I ! R is twice differentiable with f; f 00 2 Lp .I /, then f 0 2 Lp .I /. Moreover, there exists a constant Cp .I / > 0 independent of f , such that 1=2

1=2

kf 0 kp;I  Cp .I / kf kp;I kf 00 kp;I ;

(7.1)

where k  kp;I is the p-norm on the interval I , see [1, 4]. The research on these inequalities started by E. Landau [10] in 1914. For the case of p D 1 he proved that C1 .RC / D 2 and C1 .R/ D

p 2

(7.2)

are the best constants in (7.1). In 1932, G. H. Hardy and J. E. Littlewood [7] proved (7.1) for p D 2, with the best constants p C2 .RC / D 2 and C2 .R/ D 1: (7.3) In 1935, G. H. Hardy, E. Landau, and J. E. Littlewood [8] showed that the best constant Cp .RC / in (7.1) satisfies the estimate Cp .RC /  2 for p 2 Œ1; 1/;

(7.4)

which yields Cp .R/  2 for p 2 Œ1; 1/. G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 7, © George A. Anastassiou 2011

77

78

7 Left Caputo Fractional Uniform Landau Inequalities

p In fact in [6] and [9], it was shown that Cp .R/  2. In this chapter we prove fractional Landau inequalities with respect to k  k1 involving the left Caputo fractional derivative. We need Definition 7.1 ([5], p. 38). Let   0; n D de (de the ceiling of the number), f 2 AC n .ŒA; B/ (i.e., f .n1/ 2 AC.ŒA; B/; A; B 2 R). We define the left Caputo fractional derivative  f .x/ WD DA

1  .n  /

Z

x A

.x  t/n1 f .n/ .t/dt; .8/ x 2 ŒA; B;

(7.5)

and  DA f .x/ WD 0; if x < A:

Here  is the gamma function. Definition 7.2. Let 0 <   1, f 2 AC.ŒA; B/ we define  f .x/ DA

1 WD  .1  /

Z

x A

.x  t/ f 0 .t/dt; .8/ x 2 ŒA; B:

(7.6)

Notice n f WD f .n/ ; for n 2 N: DA

We make Remark 7.3. Let f 0 2 AC.ŒA; B/, 0 <   1, then d C 1e D 2, and  DA f 0 .x/ D

1  .1  /

Z

x

A

.x  t/ f 00 .t/dt

1 D  .2  . C 1//

Z

x

A

C1 .x  t/2.C1/1 f 00 .t/dt D DA f .x/:

Hence it holds  C1 f 0 .x/ D DA f .x/: DA

(7.7)

7.2 Main Results From [2], p. 618, Theorem 26.8 there we obtain Theorem 7.4. Let 0 <   1, A; B 2 R; A < B; f 2 AC.ŒA; B/. Suppose  DA f 2 L1 .ŒA; B/.

7.2 Main Results

79

Then ˇ ˇ ˇ

1 B A

Z

B

A

ˇ kD  f k 1;ŒA;B/ ˇ A f .x/dx  f .A/ˇ  .B  A/ :  . C 2/

(7.8)

We make Remark 7.5. Let 0 <   1, A; B 2 R; A < B; f 0 2 AC.ŒA; B/. Suppose  DA f 0 2 L1 .ŒA; B/. Then by (7.8), ˇ ˇ ˇ

1 B A

Z

B A

ˇ kD  f 0 k 1;ŒA;B ˇ A f 0 .x/dx  f 0 .A/ˇ  .B  A/ :  . C 2/

(7.9)

The above are equivalent to next statements. C1 f 2 Let 0 <   1, A; B 2 R; A < B; f 2 AC 2 .ŒA; B/. Assume DA L1 .ŒA; B/. Then ˇ 1 ˇ kD C1 f k 1;ŒA;B ˇ ˇ A (7.10) .f .B/  f .A//  f 0 .A/ˇ  .B  A/ : ˇ B A  . C 2/ Hence jf 0 .A/j 

C1 f k1;ŒA;B kDA 1 jf .B/  f .A/j  .B  A/ : B A  . C 2/

(7.11)

We make Remark 7.6. Let 0 <   1, A; b 2 R with f 2 AC 2 .ŒA; b/; .8/ b > A. C1 C1 We fix A. Assume DA f 2 L1 .ŒA; C1//, thus DA f 2 L1 .ŒA; b/. C1 2 Let A < a < b. Then f 2 AC .Œa; b/ and DA f 2 L1 .Œa; C1// and in C1 particular DA f 2 L1 .Œa; b/. Here C1 f .x/ D DA

1  .1  /

Z a

x

.x  t/ f 00 .t/dt:

(7.12)

If f 00 .t/  0 a.e., then 1  .1  /

Z

x A



.x  t/

1 f .t/dt   .  1/ 00

Z

x a

.x  t/ f 00 .t/dt;

i.e., C1 C1 DA f .x/  Da f .x/  0;

a.e., for x  a > A.

(7.13)

80

7 Left Caputo Fractional Uniform Landau Inequalities

Therefore it holds C1 C1 f k1;ŒA; C1/  kDa f k1;Œa; C1/ : 1 > kDA

(7.14)

So it is not strange to suppose that C1 C1 kDa f k1;Œa; C1/  kDA f k1;ŒA; C1/ ;

(7.15)

.8/ a  A (it is obvious when  D 1). We also make Remark 7.7. Let a; b 2 ŒA; C1/; a < b. Then as before we obtain jf 0 .a/j 

C1 f k1;Œa;b kDa 1 jf .b/  f .a/j C .b  a/ : ba  . C 2/

(7.16)

We also assume that kf k1; ŒA;C1/ < 1: Therefore jf 0 .a/j 

C1 f k1; ŒA;C1/ 2kf k1; ŒA;C1/ kDA C .b  a/ ; ba  . C 2/

(7.17)

.8/ a; b 2 ŒA; C1/; a < b. R.H.S. of (7.17) depends only on b  a. Consequently, it holds kf 0 k1; ŒA;C1/ 

C1 f k1; ŒA;C1/ 2kf k1; ŒA;C1/ kDA C .b  a/ : ba  . C 2/

(7.18)

We may call t D b  a > 0. Thus by (7.18), kf 0 k1; ŒA;C1/ 

C1 2kf k1; ŒA;C1/ kDA f k1; ŒA;C1/  C t ; .8/ t > 0: t  . C 2/

(7.19)

Call  WD 2kf k1; ŒA;C1/ ;  WD

C1 f k1; ŒA;C1/ kDA ;  . C 2/

both are greater than 0. We consider the function y.t/ D

 C  t  ; 0 <   1; t > 0: t

(7.20)

7.2 Main Results

81

We have

y 0 .t/ D 

then

 C t 1 D 0; t2

t 1 D

and

 ; t2

t C1 D ;

that is,

 ; 

t C1 D

(7.21)

with a unique solution   1=.C1/ : 

(7.22)

y 0 .t/ D t 2 C  t 1 and

(7.23)

t0 WD tcrit:no D We have

y 00 .t/ D 2t 3 C .  1/t 2 : We see that   3=.C1/    .C1/3 .C1/ C .  1/   !  2 C .  1/ 

y 00 .t0 / D 2t03 C .  1/t02 D 2 D

  3=.C1/ 

D

  3=.C1/ .2C1 C   1/ > 0: 

(7.24)

Therefore y has a global minimum at   1=.C1/ ; t0 D  which is     C1  y.t0 / D  1=.C1/ C    



D ./

1=.C1/

D ./

1=.C1/



 1=.C1/ 1 1 C1



C 

 C1





C



 C1   C1 

 1 C1  C1 

 C1

(7.25)

82

7 Left Caputo Fractional Uniform Landau Inequalities

 1  1 1 1 1      D  C1  C1  C1 C  C1  C1  C1 D  C1  C1  C1 C  C1 1

D . / C1 1

D . / C1

 1  C1 C . C 1/ 

 C1

1





 C1

1

D . / C1 1

 1    C1  C1 C 1 

 C1 

D . / C1 . C 1/  C1 :

(7.26)

    C1

(7.27)

That is, 

y.t0 / D . /

1 C1

. C 1/ 

:

Consequently y.t0 / D

C1 f k1; ŒA;C1/ kDa  . C 2/

1 ! .C1/



     2kf k1; ŒA;C1/ .C1/ . C 1/

 C1



:

(7.28) We have proved that kf 0 k1; ŒA;C1/  . C 1/ 

kf k1; ŒA;C1/

  .C1/ 

   .C1/ 1 2 . . C 2// .C1/ 

C1 kDA f k1; ŒA;C1/

1  .C1/

:

(7.29)

We have established the following result, left Caputo fractional Landau inequality for k  k1 : Theorem 7.8. Let 0 <   1, A; b 2 R with f 2 AC 2 .ŒA; b/; .8/ b > A, where C1 A is fixed. Assume kf k1; ŒA;C1/ < 1, DA f 2 L1 .ŒA; C1//, and C1 C1 f k1; Œa;C1/  kDA f k1; ŒA;C1/ ; .8/ a  A: kDa

Then

   .C1/ 1 2 . . C 2// .C1/ kf k1; ŒA;C1/  . C 1/     .C1/  C1  1 kf k1; ŒA;C1/ kDA f k1; ŒA;C1/ .C1/ :

0

(7.30)

In the assumptions of last Theorem 7.8, for A D 0 we get Corollary 7.9. Let 0 <   1, f 2 AC 2 .Œ0; b/; .8/ b > 0. Suppose kf k1; RC < C1 f 2 L1 .RC /, and 1, D0 C1 C1 kDa f k1; Œa;C1/  kD0 f k1; RC ; .8/ a  0:

7.3 Addendum

83

Then 0

kf k1; RC 

   .C1/ 1 2  . C 1/ . . C 2// .C1/ 

kf k1; RC

  .C1/ 

C1 kD0 f k1; RC

1  .C1/

:

(7.31)

When  D 1 we get Corollary 7.10. Let A 2 R, f 2 AC 2 .ŒA; b/; .8/ b > A. Assume f; f 00 2 L1 .ŒA; C1//. Then 1

1

00 2 2 kf 0 k1; ŒA;C1/  2kf k1; ŒA;C1/ kf k1; ŒA;C1/ :

(7.32)

Also when  D 1 we get Corollary 7.11. Let f 2 AC 2 .Œ0; b/; .8/ b > 0. Suppose f; f 00 2 L1 .RC /. Then 1

1

00 2 2 kf 0 k1; RC  2kf k1; RC kf k1; RC ;

(7.33)

which is the Landau [10] inequality, where 2 is the best constant.

7.3 Addendum Let g 2 C 2 .Œ0; R/; R > 0. Using integration by parts we obtain Z 0

R

g 00 .s/sds D Rg 0 .R/  g.R/ C g.0/:

(7.34)

If g 2 C 3 .Œ0; R/; R > 0, then similarly we get Z 0

R

g .3/ .s/s 2 ds D R2 g 00 .R/  2Rg 0 .R/ C 2g.R/  2g.0/:

(7.35)

For g 2 C 4 .Œ0; R/; R > 0, we find Z

R 0

etc.

g .4/ .s/s 3 ds D R3 g .3/ .R/  3R2 g 00 .R/ C 6Rg 0 .R/  6g.R/ C 6g.0/; (7.36)

84

7 Left Caputo Fractional Uniform Landau Inequalities

References 1. A. Aglic AljinoviKc, Lj. MaranguniKc and J. PeLcariKc, On Landau type inequalities via Ostrowski inequalities, Nonlinear Functional Analysis and Applications, Vol. 10, No. 4 (2005), pp. 565–579. 2. G.A. Anastassiou, Fractional Differentiation Inequalities, Springer, New York, 2009. 3. G.A. Anastassiou, Left Caputo fractional kk1 - Landau inequalities, Applied Math. Letters, accepted, 2011. 4. N. S. Barnett and S. S. Dragomir, Some Landau type inequalities for functions whose derivatives are of locally bounded variation, Tamkang Journal of Mathematics, Vol. 37, No.4, 301–308, winter 2006. 5. K. Diethelm, Fractional Differential Equations, On line: http://www.tu-bs.de/diethelm/lehre/ f-dgl02/fde-skript.ps.gz, 2003. 6. Z. Ditzian, Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities, Math. Inequal. Appl. 3(2000), 15–24. 7. G. H. Hardy and J. E. Littlewood, Some integral inequalities connected with the calculus of variations, Quarterly Journal of Mathematics, Oxford Ser. 3(1932), 241–252. 8. G. H. Hardy, E. Landau and J. E. Littlewood, Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Mathematische Zeitschrift 39(1935), 677–695. 9. R. R. Kallman and G. C. Rota, On the inequality kf 0 k2  4kf k  kf ”k in “Inequalities”, vol. II (O. Shisha, Ed), 187-192, Academic Press, New York, 1970. 10. E. Landau, Einige Ungleichungen fRur zweimal differentzierban Funktionen, Proc. Lond. Math. Soc. 13 (1913), 43–49.

Chapter 8

Left Caputo Fractional Lp -Landau-Type Inequalities

Here we present left Caputo fractional Lp -Landau-type inequalities and we give applications on RC . This chapter relies on [3].

8.1 Introduction Let p 2 Œ1; 1 ; I D RC or I D R, and f W I ! R is twice differentiable with f; f 00 2 Lp .I /, then f 0 2 Lp .I /. Moreover, there exists a constant Cp .I / > 0 independent of f , such that 1  1  0 f   Cp .I / kf k 2 f 00  2 ; p;I p;I p;I

(8.1)

where kkp;I is the p-norm on the interval I , see [1, 4]. The research on these inequalities started by E. Landau [10] in 1914. For the case of p D 1 he proved that p (8.2) C1 .RC / D 2 and C1 .R/ D 2 are the best constants in (8.1). In 1932, G.H. Hardy and J.E. Littlewood [7] proved (8.1) for p D 2, with the best constants p C2 .RC / D 2, and C2 .R/ D 1: (8.3) In 1935, G.H. Hardy, E. Landau, and J.E. Littlewood [8] showed that the best constant Cp .RC / in (8.1) satisfies the estimate Cp .RC /  2, for p 2 Œ1; 1/ ;

(8.4)

which yields Cp .R/  2 for p 2 Œ1; 1/. Infact in [6] and [9], it was shown that Cp .R/ 

p 2:

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 8, © George A. Anastassiou 2011

85

8 Left Caputo Fractional Lp -Landau-Type Inequalities

86

In this chapter we give fractional Landau inequalities with respect to kkp , p > 1, involving the left Caputo fractional derivative. We need Definition 8.1 ([5], p. 38). Let   0, n D de (de the ceiling of the number), f 2 AC n .ŒA; B/ (i.e., f .n1/ 2 AC .ŒA; B/ ; A; B 2 R). We define the left Caputo fractional derivative Z x 1  DA f .x/ WD .x  t/n1 f .n/ .t/ dt; (8.5)  .n  / A  8 x 2 ŒA; B, and DA f .x/ WD 0, if x < A. Here  is the gamma function.

Definition 8.2. Let 0 <   1, f 2 AC .ŒA; B/, we define  f .x/ D DA

1  .1  /

Z

x A

.x  t/ f 0 .t/ dt;

(8.6)

8 x 2 ŒA; B : n Notice DA f .x/ D f .n/ , for n 2 N. We make Remark 8.3. Let f 0 2 AC .ŒA; B/, 0 <   1, then d C 1e D 2, and  f0 DA

1 .x/ D  .1  / D

Z

x A

.x  t/ f 00 .t/ dt

1  .2  . C 1//

Z

x

A

C1 .x  t/2.C1/1 f 00 .t/ dt D DA f .x/ :

Hence it holds  C1 f 0 .x/ D DA f .x/ : DA

(8.7)

8.2 Main Results We need Theorem 8.4 ([2], p. 620). Let p; q > 1 W p1 C q1 D 1, 1  p1 <   1; f 2  AC .ŒA; B/, A; B 2 R, A < B. Assume DA f 2 Lq .ŒA; B/. Then   ˇ ˇ Z B D  f  ˇ ˇ 1 1 A Lq .ŒA;B/ ˇ   .BA/1C p : f .x/ dxf .A/ˇˇ  1 ˇB  A A  ./ .p .  1/ C 1/ p  C p1 (8.8)

8.2 Main Results

87

We make Remark 8.5. Let p; q > 1 W p1 C q1 D 1, 1  p1 <   1; f 0 2 AC .ŒA; B/,  f 0 2 Lq .ŒA; B/. Then A; B 2 R, A < B. Suppose DA   0 ˇ ˇ Z B D f  ˇ 1 ˇ 1 A Lq .ŒA;B/ 0 0 ˇ ˇ   .BA/1C p : f .x/ dxf .A/ˇ  1 ˇB  A 1 A  ./ .p .  1/ C 1/ p  C p

(8.9) Equivalently we have: For p; q > 1 W C1 f DA

1 C q1 p

D

1, 1 p1

2

<   1; f 2 AC .ŒA; B/,

2 Lq .ŒA; B/. Then A; B 2 R, A < B. Assume ˇ ˇ ˇ ˇ 1 0 ˇ ˇ ˇ B  A .f .B/  f .A//  f .A/ˇ  C1  D f  1 A Lq .ŒA;B/   .B  A/1C p :  1 1  ./ .p .  1/ C 1/ p  C p

(8.10)

Therefore it holds ˇ ˇ 0 ˇf .A/ˇ 

 C1  D f  1 A 1 Lq .ŒA;B/   .BA/1C p : jf .B/ f .A/j  1 BA  ./ .p .  1/ C 1/ p  C p1 (8.11)

We also make Remark 8.6. Let 1 p1 <   1I p; q > 1 W p1 C q1 D 1; f 2 AC 2 .ŒA; b/, 8b > A. C1 C1 Assume DA f 2 Lq .ŒA; C1// (thus DA 2 Lq .ŒA; b/). Let A < a < b. Then C1 C1 2 f 2 AC .Œa; b/ and DA f 2 Lq .Œa; C1// and DA 2 Lq .Œa; b/. Here Z x 1 C1 Da f .x/ D .x  t/ f 00 .t/ dt: (8.12)  .1  / a If f 00 .t/  0 a.e., then C1 C1 f .x/  Da f .x/  0, a.e., for x  a: DA

Thus  C1   C1   C1  1 > DA f q;ŒA;C1/  DA f q;Œa;C1/  Da f q;Œa;C1/ : So it is not strange to suppose that  C1  D f  a

q;Œa;C1/

8 a  A (it is obvious when  2 N).

 C1   DA f q;ŒA;C1/ ;

(8.13)

8 Left Caputo Fractional Lp -Landau-Type Inequalities

88

We need Remark 8.7. Let a; b 2 ŒA; C1/, a < b. Then we obtain  C1  D f  1 a 1 Lq .Œa;b/   .ba/1C p : jf .b/f .a/jC 1 1 ba  ./ .p .  1/ C 1/ p  C p (8.14) We also assume that kf k1;ŒA;C1/ < 1. Therefore  C1  D f  ˇ 0 ˇ 2 kf k1;ŒA;C1/ 1 A q;ŒA;C1/ ˇf .a/ˇ    .b  a/1C p ; C 1 1 ba  ./ .p .  1/ C 1/ p  C ˇ ˇ 0 ˇf .a/ˇ 

p

(8.15) 8 a; b 2 ŒA; C1/, a < b. The R:H:S: (8.15) depends only on b  a. Therefore  C1  D f   0 2 kf k1;ŒA;C1/ 1 A q;ŒA;C1/ f    .b  a/1C p :  C 1 1;ŒA;C1/ 1 ba  ./ .p .  1/ C 1/ p  C p

(8.16) We may call t D b  a > 0. Thus  C1  D f   0 2 kf k1;ŒA;C1/ 1 A q;ŒA;C1/ f    t 1C p ;  C 1 1;ŒA;C1/ t  ./ .p .  1/ C 1/ p  C p1 (8.17) 8 t 2 .0; 1/ : Notice that 0 <   1 C

1 p

< 1. Call

e  WD 2 kf k1;ŒA;C1/ ;  C1  D f  A q;ŒA;C1/ e  ;  WD 1  ./ .p .  1/ C 1/ p  C p1

(8.18)

both are positive, and 1 ; (e  2 .0; 1/ ). p

(8.19)

e  ee C t  ; t 2 .0; 1/ : t

(8.20)

e  WD   1 C We consider the function e y .t/ D

8.2 Main Results

89

The only critical number here is e t0 D



 1 e  e C1 ; e e 

(8.21)

and e y has a global minimum at e t0 , which is 



 1     e  e C1 C1 : .e  C 1/e  e e  e e y e t0 D e

(8.22)

Consequently, we derive 0   e y e t0 D @

1 11  C1  C p D f  A q;ŒA;C1/ A    1  ./ .p .  1/ C 1/ p  C p1

   2 kf k1;ŒA;C1/

1 1C p 1 C p

!

   1  1 1C C p p

1 1C p 1 C p

!

: (8.23)

We have proved that  0 f  1;ŒA;C1/

1 0  2  C p1 A @   1 C p1



1 1

.p .  1/ C 1/ .pC1/

 kf k1;ŒA;C1/



1 1C p 1 C p

1 1C p 1 C p

!

 !

1 1

. .// .C p / 1



(8.24)

  11  C1   DA f q;ŒA;C1/ .C p / :

We have established the following Lq result, left Caputo fractional Landau Lq inequality: Theorem 8.8. Let p; q > 1 W

1 p

C

1 q

D 1; 1 

1 p

<   1; f 2 AC 2 .ŒA; b/,

C1 f 2 Lq .ŒA; C1// ; kf k1;ŒA;C1/ < 1; and 8b > A. Suppose DA

 C1   C1  D f   DA f q;ŒA;C1/ ; a q;Œa;C1/ 8 a  A. Then  0 f 

1;ŒA;C1/

1 0  2  C p1 A @   1 C p1

1 1C p 1 C p

(8.25)

!



(8.26)

8 Left Caputo Fractional Lp -Landau-Type Inequalities

90

1 1 1 C p

. .// . 

kf k1;ŒA;C1/





1

/ .p .  1/ C 1/ .pC1/

1 1C p 1 C p

!

  11  C1   DA f q;ŒA;C1/ .C p / :

We give Corollary 8.9. Let p; q > 1 W p1 C q1 D 1; 1  p1 <   1; f 2 AC 2 .Œ0; b/, C1 8b > 0. Suppose D0 f 2 Lq .Œ0; C1// ; kf k1;RC < 1; and  C1   C1  D f   D0 f q;R ; a q;Œa;C1/

(8.27)

C

8 a  0. Then  0 f  1;R

1 0  2  C p1 A @   1 C p1

C

1 1C p 1 C p

!



1 1 1 C p

. .// .

1

/ .p .  1/ C 1/ .pC1/

1 1C p 1 C p

  kf k1;RC

!

(8.28)



  11  C1   D0 f q;R .C p / : C

We mention the case of p D q D 2: C1 Corollary 8.10. Let 12 <   1; f 2 AC 2 .Œ0; b/, 8b > 0. Assume D0 f 2 L2 .RC / ; f 2 L1 .RC / ; and

 C1  D f  a

 C1   D0 f 2;R ;

2;Œa;C1/

C

8 a  0. Then  0 f   1;R C

2 C 1   12

!



 12 C 21

1 . .// . 

  kf k1;RC

 21 C 12

1 C 21



1 / .2  1/ .2C1/



 

 11   C1   D0 f 2;R .C 2 / : C

(8.29)

(8.30)

References

91

When  D 1 we derive Corollary 8.11. Let p; q > 1 W p1 C f 00 2 Lq .RC / ; f 2 L1 .RC / : Then

1 q

D 1; f 2 AC 2 .Œ0; b/, 8b > 0. Assume

 0 1 f   .2 .p C 1// pC1  1;R

(8.31)

C

p 1    pC1  pC1  00    f q;R : kf k1;RC C

We finish chapter with Corollary 8.12. Let f 2 AC 2 .Œ0; b/, 8b > 0. Suppose f 00 2 L2 .RC / ; f 2 L1 .RC / : Then  13    23  p  0 3 f  f 00   6   : kf k 1;R 1;R C 2;R C

C

(8.32)

References 1. A. Agli´c Aljinovic, Lj. Marangunic and J. Pecari´c, On Landau type inequalities via Ostrowski inequalities, Nonlinear Funct. Anal. & Appl., Vol. 10, No. 4 (2005), pp. 565-579. 2. G.A. Anastassiou, Fractional Differentiation Inequalities, Springer, New York, 2009. 3. G.A. Anastassiou, Lp -left Caputo fractional Landau inequalities, J. Comp. Anal. and Appl., accepted, 2011. 4. N.S. Barnett and S.S. Dragomir, Some Landau type inequalities for functions whose derivatives are of locally bounded variation, Tamkang Journal of Mathematics, Vol. 37, No. 4, 301-308, winter 2006. 5. K. Diethelm, Fractional Differential Equations, On line: http://www.tu-bs.de/˜diethelm/ lehre/f-dgl02/fde-skript.ps.gz, 2003. 6. Z. Ditzian, Remarks, questions and conjections on Landau-Kolmogorov-type inequalities, Math. Ineq. Appl. 3 (2000), 15-24. 7. G.H. Hardy and J.E. Littlewood, Some integral inequalities connected with the calculus of variations, Quart. J. Math. Oxford Ser. 3 (1932), 241-252. 8. G.H. Hardy, E. Landau and J.E. Littlewood, Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math. Z. 39 (1935), 677-695. 9. R.R. Kallman and G.-C. Rota, On the inequality kf 0 k2  4 kf k  kf 00 k, in “Inequalities”, vol. II (O. Shisha, Ed), 187-192, Academic Press, New York, 1970. 10. E. Landau, Einige Ungleichungen f¨ur zweimal differentzierban funktionen, Proc. London Math. Soc. 13 (1913), 43-49.

Chapter 9

Right Caputo Fractional Lp -Landau-Type Inequalities

We present right Caputo fractional kkp -Landau type inequalities, p 2 .1; 1 with applications on R . This chapter is based on [4].

9.1 Introduction Let p 2 Œ1; 1 ; I D RC or I D R, and f W I ! R be twice differentiable with f; f 00 2 Lp .I /, then f 0 2 Lp .I /. Moreover, there exists a constant Cp .I / > 0 independent of f , such that 1  0  1 f   Cp .I / kf k 2 f 00  2 ; p;I p;I p;I

(9.1)

where kkp;I is the p-norm on the interval I , see [1, 5]. Development of these inequalities was initiated by E. Landau [13] in 1914. For the case of p D 1 he proved that C1 .RC / D 2 and C1 .R/ D

p 2

(9.2)

are the best constants in (9.1). In 1932, G.H. Hardy and J.E. Littlewood [10] proved (9.1) for p D 2, with the best constants p C2 .RC / D 2, and C2 .R/ D 1: (9.3) In 1935, G.H. Hardy, E. Landau, and J.E. Littlewood [11] showed that the best constant Cp .RC / in (9.1) satisfies Cp .RC /  2, for p 2 Œ1; 1/ ;

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 9, © George A. Anastassiou 2011

(9.4)

93

94

9 Right Caputo Fractional Lp -Landau-Type Inequalities

which yields Cp .R/  2 for p 2 Œ1; 1/. Infact in [6] and [12], it was shown that p Cp .R/  2: In this chapter we give fractional Landau inequalities with respect to kkp , p 2 .1; 1, involving the right Caputo fractional derivative.

9.2 Main Results We need Definition 9.1 ([7–9, 14]). Let f 2 L1 .Œa; b/, ˛ > 0. The right Riemann– Liouville fractional operator of order ˛ is given by ˛ f .x/ D Ib

1  .˛/

Z

b x

.J  x/˛1 f .J / dJ ,

(9.5)

0 8 x 2 Œa; b, where  is the gamma function. We set Ib WD I (the identity operator).

Definition 9.2 ([7–9, 14]). Let f 2 AC m .Œa; b/ (f .m1/ is in AC .Œa; b/), m 2 N, m D d˛e, ˛ > 0 (de the ceiling of the number). We define the right Caputo fractional derivative of order ˛ > 0 by Z b .1/m ˛ Db f .x/ D .J  x/m˛1 f .m/ .J / dJ; 8 x  b: (9.6)  .m  ˛/ x If ˛ D m 2 N, then m f .x/ D .1/m f .m/ .x/ , 8 x 2 Œa; b : Db ˛ If x > b we define Db f .x/ D 0.

In particular we have Definition 9.3. Let 0 < ˛  1, f 2 AC .ŒA; B/ and define Z B 1 ˛ f .x/ WD .J  x/˛ f 0 .J / dJ: DB  .1  ˛/ x

(9.7)

Remark 9.4. Let f 0 2 AC .ŒA; B/, 0 < ˛  1, then d˛ C 1e D 2 and ˛ f0 DB

1 .x/ D  .1  ˛/ D

Z x

B

.J  x/˛ f 00 .J / dJ

 .1/2  .2  .˛ C 1//

Z x

B

˛C1 .J  x/2.˛C1/1 f 00 .J / dJ D DB f .x/ ;

9.2 Main Results

95

i.e., and hence

˛C1 ˛ DB f 0 .x/ D DB f .x/

(9.8)

 ˛C1   ˛  D f 0  D DB f 1;ŒA;B : B 1;ŒA;B

(9.9)

The focus is now on .1; B, B 2 R. Let a < b < B, a; b 2 R. If f 00 .J /  0, a.e., then Z

1  .1  ˛/

x

B

.J  x/˛ f 00 .J / dJ

1   .1  ˛/

Z

b x

.J  x/˛ f 00 .J / dJ  0; a.e.;

i.e., ˛C1 ˛C1 f .x/  Db f .x/  0; a.e., for x  b: DB

(9.10)

It holds therefore that  ˛C1   ˛C1  1 > DB f 1;.1;B  Db f 1;.1;b :

(9.11)

It is reasonable then to suppose that  ˛C1  D f  b

1;.1;b

 ˛C1   DB f 1;.1;B :

(9.12)

(it is obvious when ˛ D 1), 8 b  B. Remark 9.5. Let 0 < ˛  1I B; a 2 R, with f 2 AC 2 .Œa; B/, 8a < B. Assume ˛C1 ˛C1 DB f 2 L1 ..1; B/ ; thus DB f 2 L1 .Œa; B/. Let a < b < B; then ˛C1 ˛C1 f 2 AC 2 .Œa; b/, and DB f 2 L1 ..1; b/ and DB f 2 L1 .Œa; b/ : By Theorem 6 of [2], applied on f 0 , we obtain that  ˛C1  D  ˇ 0 ˇ b f 1;Œa;b ˇf .b/ˇ  1 jf .b/  f .a/j C .b  a/˛ (9.13) ba  .˛ C 2/  ˛C1  ˛ D  2 kf k1;.1;B B f 1;.1;B .b  a/ C ;  ba  .˛ C 2/

(9.14)

where we also assume that kf k1;.1;B < 1: Setting t D b  a,  0 f 

1;.1;B

8 t 2 .0; 1/ :



2 kf k1;.1;B t

C

 ˛C1  ˛ D  B f 1;.1;B  t  .˛ C 2/

;

(9.15)

96

9 Right Caputo Fractional Lp -Landau-Type Inequalities

One now can prove easily that the R:H:S: (9.15) has a global minimum and find it (see also [3]) with a usual calculus method. We have established the following. Theorem 9.6. Let 0 < ˛  1I B; a 2 R, with f 2 AC 2 .Œa; B/, 8a < B, where ˛C1 B is fixed. Assume kf k1;.1;B < 1; DB f 2 L1 ..1; B/ ; and  ˛C1   ˛C1     D b f 1;.1;b  DB f 1;.1;B ;

(9.16)

8 b < B, then  0 f   .˛ C 1/  1;.1;B 

  kf k1;.1;B

˛ ˛C1



  2 ˛ 

˛ ˛C1

 1

 . .˛ C 2// .˛C1/ 

(9.17)

1   .˛C1/  ˛C1   DB f 1;.1;B :

The case of B D 0 is given by the following Corollary. Corollary 9.7. Let 0 < ˛  1I f 2 AC 2 .Œa; 0/, 8 a < 0. Suppose that ˛C1 f 2 L1 .R / ; and kf k1;R < 1; D0  ˛C1  D f b

1;.1;b

 ˛C1   D0 f 1;R ;

(9.18)

8 b < 0. Then  0 f   .˛ C 1/  1;R 



kf k1;R



˛ ˛C1



  2 ˛



˛ ˛C1

 1

 . .˛ C 2// .˛C1/ 

(9.19)

1   .˛C1/  ˛C1   D0 f 1;R :

The case of ˛ D 1 is given by the following corollary. Corollary 9.8. Let B; a 2 R, with f 2 AC 2 .Œa; B/, 8a < B, B is fixed. Assume kf k1;.1;B < 1; f 00 2 L1 ..1; B/ ; then  12  0   12  00  f  f   2   : kf k 1;.1;B 1;.1;B 1;.1;B

(9.20)

Corollary 9.9. Let f 2 AC 2 .Œa; 0/, 8a < 0. Suppose f; f 00 2 L1 .R / : Then  12  0   12  00  f  f   2   : kf k 1;R  1;R 1;R

(9.21)

9.2 Main Results

97

Remark 9.10. Let any a; b 2 .1; B, a < b. Let p; q > 1 W p1 C q1 D 1; ˛C1 1  p1 < ˛  1; and f 2 AC 2 .Œa; b/. Assume also Db f 2 Lq .Œa; b/, then by 0 [2], Theorem 8, for f we get that  ˛C1  D f  1 1 b Lq .Œa;b/   .ba/˛1C p : jf .b/f .a/jC 1 ba  .˛/ .p .˛  1/ C 1/ p ˛ C p1 (9.22) Suppose here  ˛C1   ˛C1  D    (9.23) b f q;.1;b  DB f q;.1;B < 1; ˇ ˇ 0 ˇf .b/ˇ 

8 b  B: Therefore,  ˛C1  D  ˇ 0 ˇ 2 kf k1;.1;B 1 B f q;.1;B ˇf .b/ˇ    .b  a/˛1C p ; C 1 1 ba  .˛/ .p .˛  1/ C 1/ p ˛ C p (9.24) 8 a; b W a < b  B, where B is fixed. Let t WD b  a > 0, then  ˛C1  D   0 2 kf k1;.1;B 1 B f q;.1;B f    t ˛1C p ;  C 1 1;.1;B t  .˛/ .p .˛  1/ C 1/ p ˛ C p1 (9.25) 8 t 2 .0; 1/ ; where it is assumed that kf k1;.1;B < 1: As before we have proved 1 p

Theorem 9.11. Let p; q > 1 W

C

1 q

D 1; 1 

1 p

< ˛  1; f 2 AC 2 .Œa; B/,

˛C1 f 2 Lq ..1; B/ ; kf k1;.1;B < 1; and 8a < B. Assume DB

 ˛C1   ˛C1     D b f q;.1;b  DB f q;.1;B ; 8 b  B, then  0 f 

1;.1;B

1 0  2 ˛ C p1 A @ ˛  1 C p1

1 ˛1C p 1 ˛C p

. .˛// . 

kf k1;.1;B



1

/ .p .˛  1/ C 1/ .p˛C1/

1 a1C p 1 ˛C p

!

!



1 1 1 ˛C p

(9.26)



 11   ˛C1   DB f q;.1;B .˛C p / :

(9.27)

98

9 Right Caputo Fractional Lp -Landau-Type Inequalities

The case of B D 0 is given by Corollary 9.12. Corollary 9.12. Let p; q > 1 W p1 C q1 D 1; 1  p1 < ˛  1; f 2 AC 2 .Œa; 0/, ˛C1 8a < 0. Suppose D0 f 2 Lq .R / ; kf k1;R < 1; and  ˛C1  D f b

q;.1;b

 ˛C1   D0 f q;R ;

8 b  0. Then  0 f  1;R

1 0  2 ˛ C p1 A @ ˛  1 C p1

1 ˛1C p 1 ˛C p

!



1 1 1 ˛C p

. .˛// . 

kf k1;R

1

/ .p .˛  1/ C 1/ .p˛C1/

1 ˛1C p 1 ˛C p



!

(9.28)

(9.29)



  11  ˛C1   D0 f q;R .˛C p / :

The case for p D q D 2 follows. ˛C1 Corollary 9.13. Let 12 < ˛  1; f 2 AC 2 .Œa; 0/, 8a < 0. Suppose D0 f 2 L2 .R / ; kf k1;R < 1; and

 ˛C1   ˛C1     D b f 2;.1;b  D0 f 2;R ; 8 b  0, then



 0 f 

2˛ C 1 ˛  12



1;R

!

˛ 21 ˛C 12

1 1 ˛C 21

. .˛// .   kf k1;R



˛ 12 ˛C 21



1 / .2˛  1/ .2˛C1/

(9.30)





(9.31)



  11  ˛C1   D0 f 2;R .˛C 2 / :

The case ˛ D 1 gives: Corollary 9.14. Let p; q > 1 W p1 C f 00 2 Lq .R / ; f 2 L1 .R / ; then  0 f 

1;R

1 q

D 1; f 2 AC 2 .Œa; 0/, 8a < 0. Assume

p  pC1  1    1  .2 .p C 1// pC1  kf k1;R pC1  f 00 q;R :

(9.32)

References

99

Finally, for p D q D 2 we have Corollary 9.15. Let f 2 AC 2 .Œa; 0/, 8a < 0. Assume f 00 2 L2 .R / ; f 2 L1 .R / ; then  23 p  0  13  00   3 f  f   6   : kf k 1;R 1;R 2;R

(9.33)

References 1. A. Agli´c Aljinovic, Lj. Marangunic and J. Pecari´c, On Landau type inequalities via Ostrowski inequalities, Nonlinear Funct. Anal. & Appl., Vol. 10, No. 4 (2005), pp. 565-579. 2. G.A. Anastassiou, Univariate right fractional Ostrowski inequalities, CUBO, accepted, 2011. 3. G.A. Anastassiou, Left Caputo fractional kk1 -Landau inequalities, Applied Math. Letters, accepted, 2011. 4. G.A. Anastassiou, Right Caputo fractional Landau inequalities, Sarajevo J. of Math., accepted, 2011. 5. N.S. Barnett and S.S. Dragomir, Some Landau type inequalities for functions whose derivatives are of locally bounded variation, Tamkang Journal of Mathematics, Vol. 37, No. 4, 301-308, winter 2006. 6. Z. Ditzian, Remarks, questions and conjections on Landau-Kolmogorov-type inequalities, Math. Ineq. Appl. 3 (2000), 15-24. 7. A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95. 8. G.S. Frederico, D.F.M. Torres, Fractional Optimal Control in the sense of Caputo and the fractional Noether’s theorem, International Mathematical Forum, Vol. 3, No. 10 (2008), 479-493. 9. R. Gorenflo, F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center, http://www.maphysto.dk/oldpages/events/LevyCAC2000/ MainardiNotes/fm2k0a.ps. 10. G.H. Hardy and J.E. Littlewood, Some integral inequalities connected with the calculus of variations, Quart. J. Math. Oxford Ser. 3 (1932), 241-252. 11. G.H. Hardy, E. Landau and J.E. Littlewood, Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math. Z. 39 (1935), 677-695. 12. R.R. Kallman and G.-C. Rota, On the inequality kf 0 k2  4 kf k  kf 00 k, in “Inequalities”, vol. II (O. Shisha, Ed), 187-192, Academic Press, New York, 1970. 13. E. Landau, Einige Ungleichungen f¨ur zweimal differentzierban funktionen, Proc. London Math. Soc. 13 (1913), 43-49. 14. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].

Chapter 10

Mixed Caputo Fractional Lp -Landau-Type Inequalities

Here we give mixed Caputo fractional jj  jjp -Landau type inequalities, p 2 .1; 1 with applications on R. This chapter relies on [5].

10.1 Introduction Let p 2 Œ1; 1, I D RC or I D R and f W I ! R is twice differentiable with f; f 00 2 Lp .I /; then f 0 2 Lp .I /. Moreover, there exists a constant Cp .I / > 0 independent of f , such that 1

1

2 2 jjf 00 jjp;I ; jjf 0 jjp;I  Cp .I /jjf jjp;I

(10.1)

where jj:jjp;I is the p-norm on the interval I , see [1, 6]. The research on these inequalities started by E. Landau [15] in 1914. For the case of p D 1 he proved that C1 .RC / D 2 and C1 .R/ D

p

2

(10.2)

are the best constants in (10.1). In 1932, G. H. Hardy and J. E. Littlewood [12] proved (10.1) for p D 2, with the best constants p C2 .RC / D 2 and C2 .R/ D 1: (10.3) In 1935, G. H. Hardy, E. Landau, and J. E. Littlewood [13] showed that the best constant Cp .RC / in (10.1) satisfies the estimate Cp .RC /  2;

for p 2 Œ1; 1/;

(10.4)

which yields Cp .R/  2 for p 2 Œ1; 1/. p In fact, in [8] and [14] it was shown that Cp .R/  2. G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 10, © George A. Anastassiou 2011

101

10 Mixed Caputo Fractional Lp -Landau-Type Inequalities

102

In this chapter we prove mixed fractional Landau inequalities with respect to jj  jjp ; p 2 .1; 1, involving the right and left Caputo fractional derivatives.

10.2 Main Results We need (see [9–11, 16]) Definition 10.1. Let f 2 AC m .Œa; b/ (space of functions from Œa; b into R with m  1 derivative absolutely continuous function on Œa; b), m 2 N, where m D d˛e; ˛ > 0 (d:e the ceiling of the number). We define the right Caputo fractional derivative of order ˛ > 0 by ˛ f .x/ D Db

.1/m  .m  ˛/

Z

b x

.  x/m˛1 f .m/ ./d:

(10.5)

0 We set Db f .x/ D f .x/; 8x 2 Œa; b.

Remark 10.2. Let f 2 AC m .Œa; b/; m D d˛e, with ˛ > 0 then f .m1/ 2 AC.Œa; b/, which implies that f m exists a.e. on Œa; b and that f m 2 L1 .Œa; b/. ˛ f .x/ exists a.e. on Œa; b and Consequently, if f 2 AC m .Œa; b/, then Db 2 L1 .Œa; b/, see [7], p. 13. Observe that when ˛ D m 2 N, then

˛ Db f

m Db f .x/ D .1/m f .m/ .x/; 8x 2 Œa; b:

(10.6)

˛ f .x/ D 0. If x > b we define Db We also need

Definition 10.3 ([7], p. 38). Let f 2 AC m .Œa; b/; m 2 N; where m D d˛e; ˛ > 0. We define the left Caputo fractional derivative of order ˛ > 0, by ˛ f .x/ D Da

1  .m  ˛/

Z a

x

.x  t/m˛1 f .m/ .t/dt;

(10.7)

0 8x 2 Œa; b. We set Da f .x/ D f .x/, 8x 2 Œa; b. ˛ ˛ f exists a.e. on Œa; b and Da f 2 L1 .Œa; b/, see [7], pp. 13 and Again here Da 37–38. When ˛ D m 2 N then m Da f .x/ D f .m/ .x/; ˛ If x < ˛ we define Da f .x/ D 0.

8x 2 Œa; b:

(10.8)

10.2 Main Results

103

We make Remark 10.4. Let ˛ > 0; m D d˛e; ˛ … N and f 2 AC m .Œa; b/ with x0 2 Œa; b. Then Z x0 .1/m .  x/m˛1 f .m/ ./d; 8x  x0 ; (10.9) Dx˛0  f .x/ D  .m  ˛/ x Dx˛0  f .x/ D 0;

8x > x0 I

and Dx˛0  f .x/ D

1  .m  ˛/

Dx˛0  f .x/ D 0;

Z

x

x0

.x  t/m˛1 f .m/ .t/dt;

8x  x0 ;

(10.10)

8x < x0 :

Suppose first f 2 C 1 .R/ and of compact support ŒA; BI A < B. Then f .m/ is of support in ŒA; B. Hence possibly Dx˛0  f .x/ ¤ 0 only for x0 ; x 2 ŒA; B, i.e., only ŒA; B counts. We proved earlier that Dx˛0  f .x/ is jointly continuous on ŒA; B2 , see [3], so that sup jjDx˛0  f jj1;R < 1. In particular sup jjDx˛0  f jj1;Œx0 ;C1/ < 1. xo 2R

xo 2R

Totally similar we have for this case that sup jjDx˛0  f jj1;.1;x0  < C1. xo 2R

We also make Remark 10.5. Let x  x0 ; x; x0 2 R. We want to prove that Z

x x0

p

eit dt xt

(10.11)

is uniformly bounded for all x; x0 2 R; x  x0 . We call u WD x  t, then Z

x x0

Z

xx0

Z

xx0

eiu p du u 0 0   du 1 D calling w WD u 2 ; 2dw D p u Z pxx0 2 D 2eix eiw dw .the Fresnel integral/

eit dt D p xt

ei.xu/ p du D eix u

0

.assume first x  x0 > 1/ # "Z Z pxx0 1 ix iw2 iw2 e dw C e dw : D 2e 0

1

(10.12)

10 Mixed Caputo Fractional Lp -Landau-Type Inequalities

104

In the last eix and

R1

2

eiw dw are bounded.

0

p

WD

Call

x  x0 > 1:

So, it is enough to bound Z e

Z

iw2

dw D

1

We observe that

2

1

weiw dw D ./: w

(10.13)

2

deiw 2 D eiw .2iw/ dw

and

2

deiw 2 D eiw wdw: 2i



(10.14)

So that ˇ    2 Z 1 1 1 iw2 ˇˇw deiw iw2  e d D e ˇ w 2i w w 1 1 1 # "  Z 2 eiw 1 1 iw2 i e dw : C D e 2i w2 1

1 ./ D  2i

Z

 In the last expression We further see that ˇZ ˇ ˇ ˇ

1

ˇ Z 2 ˇ eiw dwˇˇ  2 w 1

(10.15)

  ei is bounded.

1 iw2

e

dw D w2

Z

ˇ ˇ 1 ˇˇ 1 ˇˇ1 1 w dw D  ˇ D ˇ D 1  < 1; w 1 w w (10.16) 2

1

 with 1  w1 > 0. Hence the last integral is bounded. p If x  x0  1, we observe that ˇZ ˇ ˇ ˇ

0

p xx0

iw2

e

ˇ Z ˇ dwˇˇ 

0

p xx0

ˇ iw2 ˇ ˇe ˇdw D px  x0  1;

a bounded integral. Hence integral (10.11) is uniformly bounded for all x; x0 2 R; x  x0 :

(10.17)

10.2 Main Results

105

Conclusion 10.6 The integrals Z

x x0

sin t dt; p xt

Z

x x0

cos t dt p xt

(10.18)

are uniformly bounded for all x; x0 2 R; x  x0 : Explanation: We observe that Z

x

x0

eit dt D p xt

Z

x x0

cos t C i sin t dt D p xt

Z

x

x0

 Z x  cos t sin t dt C i dt : p p xt xt x0 (10.19)

That is, ˇZ ˇ C1 > ˇˇ

x

x0

ˇ sZ x 2 Z x 2 ˇ eit cos t sin t dt ˇˇ D dt C dt p p p (10.20) xt xt xt x0 x0 ˇZ ˇ  ˇˇ

x x0

ˇ ˇ cos t dt ˇˇ; p xt

ˇZ ˇ ˇ ˇ

x x0

ˇ ˇ sin t dt ˇˇ: p xt

(10.21)

Similarly, for sin At; cos Bt, and their linear combinations, etc. Conclusion 10.7 We demonstrated one more reason that it makes sense to suppose for ˛ > 0 that sup jjDx˛0  f jj1;.1;x0  < C1;

xo 2R

˛ sup jjDx f jj1;Œx0 ;C1/ < C1: 0

(10.22)

xo 2R

We now make Remark 10.8. So we can suppose that there are big classes of functions f such that ˛ f jj1;Œx0 ;C1/ < C1; sup jjDx 0

xo 2R

sup jjDx˛0  f jj1;.1;x0  < C1:

(10.23)

xo 2R

So, for 0 < ˛  1; m D 1 we treat f 2 AC.Œa; b/; 8Œa; b  R. Then by [4], Theorem 6 and by [2], Theorem 26.8, p. 618 we derive ˇ ˇ 1 Z ˇ ˇb  a

b a

ˇ ˇ f .x/dx  f .x0 /ˇˇ

 1 ˛ ˛ (10.24) max jjDx0  f jj1;Œ˛;x0  ; jjDx0 f jj1;Œx0 ;b  .b  a/˛   .˛ C 2/ 

1 ˛ ˛  max sup jjDx0  f jj1;.1;x0  ; sup jjDx0  f jj1;Œx0 ;C1/  .ba/˛ :  .˛ C 2/ xo 2R xo 2R (10.25)

10 Mixed Caputo Fractional Lp -Landau-Type Inequalities

106

Next assume f 2 AC 2 .Œa; b/; 8Œa; b  R and assume jjf jj1;R < 1, along with ˛C1 sup jjDx f jj1;Œx0 ;C1/ < 1; 0

sup jjDx˛C1 f jj1;.1;x0  < 1: 0

xo 2R

xo 2R

Hence it holds ˇ ˇ Z b ˇ ˇ 1 0 0 ˇ ˇ f .x/dx  f .x / 0 ˇ ˇb  a a 

1  max sup jjDx˛0  f 0 jj1;.1;x0  sup jjDx˛0  f 0 jj1;Œx0 ;C1/  .b  a/˛ :  .˛ C 2/ xo 2R xo 2R (10.26) Equivalently, it holds

ˇ ˇ ˇ ˇ 1 1 0 ˇ jf .b/  f .a/j  ˇ .f .b/  f .a//  f .x0 /ˇˇ jf .x0 /j  ba ba 

1 ˛C1 ˛C1  max sup jjDx0  f jj1;.1;x0  sup jjDx0  f jj1;Œx0 ;C1/  .b  a/˛ :  .˛ C 2/ xo 2R xo 2R (10.27) Consequently, 0

jf 0 .x0 /j 

max

2 1 jjf jj1;R ba  .˛ C 2/

sup

xo 2R

jjDx˛C1 f 0

jj1;.1;x0  ; sup

xo 2R

jjDx˛C1 f 0



jj1;Œx0 ;C1/  .b  a/˛ : (10.28)

So that 2jjf jj1;R jjf 0 jj1;R  ba

 ˛C1 max sup jjDx˛C1 f jj ; sup jjD f jj 1;.1;x0  1;Œx0 ;C1/ x0  0 C

xo 2R

xo 2R

 .˛ C 2/

 .b  a/˛ : (10.29)

The R.H.S. of (10.29) depends only on t WD b  a > 0, that t can be anything positive. Thus 2jjf jj1;R jjf 0 jj1;R  t 

˛C1 ˛C1 max sup jjDx0  f jj1;.1;x0  ; sup jjDx0  f jj1;Œx0 ;C1/ C

xo 2R

xo 2R

 .˛ C 2/

 t˛;

8 t > 0:

(10.30)

10.2 Main Results

107

We call M WD 2jjf jj1;R > 0;

max

sup

xo 2R

 WD

jjDx˛C1 f 0

jj1;.1;x0  ; sup

xo 2R

jjDx˛C1 f 0

 jj1;Œx0 ;C1/ > 0: (10.31)

 .˛ C 2/

Hence jjf 0 jj1;R  Set y.t/ WD

M C t ˛ ; t

M C t ˛ ; t

8 t > 0:

(10.32)

8 t > 0:

(10.33)

Using basic calculus the function y.t/ has a global minimum which is 1

˛

y.t0 / WD .M ˛ / ˛C1  .˛ C 1/  ˛  ˛C1 :

(10.34)

Therefore, we have proved that jjf 0 jj1;R 1 2

3 ˛C1  ˛ ˛C1 ˛C1 ˛ 6 max sup jjDx0  f jj1;.1;x0  ; sup jjDx0  f jj1;Œx0 ;C1/ 2 jjf jj1;R 7 xo 2R xo 2R 6 7 6 7 4 5  .˛ C 2/ ˛

.˛ C 1/˛ ˛C1 :

(10.35)

We have established the following mixed fractional Landau inequality with respect to jj:jj1 norm. Theorem 10.9. Let 0 < ˛  1, f 2 AC 2 .Œa; b/; a < b; 8Œa; b  R. Assume ˛C1 jjf jj1;R < 1 and sup jjDx f jj1;Œx0 ;C1/ < 1; sup jjDx˛C1 f jj1;.1;x0  < 1. 0 0 xo 2R

Then

jjf 0 jj1;R  .˛ C 1/



max

xo 2R

˛ 1 ˛   ˛C1   ˛C1   ˛C1 2   .˛ C 2/  jjf jj1;R ˛ 1  ! ˛C1

sup jjDx˛C1 f jj1;.1;x0  ; sup jjDx˛C1 f jj1;Œx0 ;C1/ 0 0

xo 2R

xo 2R

: (10.36)

10 Mixed Caputo Fractional Lp -Landau-Type Inequalities

108

When ˛ D 1, we obtain Corollary 10.10. Let f 2 AC 2 .Œa; b/; 8Œa; b  R. Suppose jjf jj1;R < 1, jjf 00 jj1;R < 1. Then p (10.37) jjf 0 jj1;R  2  jjf jj1;R  jjf 00 jj1;R : We continue with Remark 10.11. Let p; q > 1 W p1 C q1 D 1. If f 2 C 1 .R/ and of compact support ŒA; B, then as we saw earlier, it only counts x0 ; x 2 ŒA; B for having ˛ f .x/jj1;ŒA;B2 < 1: Dx˛0  f .x/ ¤ 0 and   WD jjDx 0

Thus ˛ jjDx jj D 0 Lq .R/

Z

B A

ˇ ˇ ˛ ˇD f .x/ˇq dx x0

 q1

1

 .B  A/ q    < 1; 8x0 2 ŒA; B: (10.38)

Hence ˛ f jjLq .Œx0 ;C1// < 1: sup jjDx 0

(10.39)

jjDx˛0  f jjLq ..1;x0 / < 1;

(10.40)

xo 2R

Similarly, we obtain

for the above case; similarly for trigonometric polynomials, etc. So, we can assume f that ˛ sup jjDx f jjLq .Œx0 ;C1// < 1 0

xo 2R

and sup jjDx˛0  f jjLq ..1;x0 / < 1

(10.41)

xo 2R

is valid for large classes of functions f . Remark 10.12. Let p; q > 1 W p1 C q1 D 1; 1  p1 < ˛  1, f 2 AC 2 .Œa; b/; 8 Œa; b  R. Suppose jjf jj1;R < 1; x0 2 Œa; b and sup jjDx˛C1 f jjLq ..1;x0 / < 1I 0

xo 2R

˛C1 f jjLq .Œx0 ;C1// < 1: sup jjDx 0

xo 2R

10.2 Main Results

109

Then by [4], Theorem 8 and by [2] Theorem 26.10, p. 620 we obtain that 2jjf jj1;R 1    C 1 ba  .˛/ Œp.˛  1/ C 1 p ˛ C p1 

1 ˛C1  .b  a/˛ q : (10.42) max sup jjDx˛C1 f jj ; sup jjD f jj L ..1;x / L .Œx ;C1// q 0 q 0 x0 0 jf 0 .x0 /j 

xo 2R

xo 2R

Hence jjf 0 jj1;R 

2jjf jj1;R t

max C 1

 t ˛ q ;



˛C1 f jjLq ..1;x0 / ; sup jjDx f jjLq .Œx0 ;C1// sup jjDx˛C1 0 0 xo 2R xo 2R   1  .˛/ Œp.˛  1/ C 1 p ˛ C p1

8 t 2 .0; 1/:

(10.43)

Similarly as before we get 1

jjf 0 jj1;R

 1 ˛ q1 0  ˛C p 2 ˛ C p1 A @  ˛  q1

max

1

1

1

1

. .˛// ˛C p Œp.˛  1/ C 1 ˛pC1

˛C1 sup jjDx˛C1 f jjLq ..1;x0 / ; sup jjDx f jjLq .Œx0 ;C1// 0 0

xo 2R

  ˛ q1 ˛C p  jjf jj1;R 

1

 ! ˛C1 1

p

xo 2R

:

(10.44)

We have established the mixed fractional Lp -Landau inequality. Theorem 10.13. Let p; q > 1 W p1 C q1 D 1; 8 Œa; b  R. Assume jjf jj1;R < 1 and

1 q

< ˛  1, f 2 AC 2 .Œa; b/;

f jjLq ..1;x0 / < 1I sup jjDx˛C1 0

xo 2R

˛C1 sup jjDx f jjLq .Œx0 ;C1// < 1: 0

xo 2R

Then 1

0

jjf jj1;R

 1 ˛ q1 0  ˛C p 1 2 ˛C p A @   ˛  q1



max

1

1 1

1

1

. .˛// ˛C p Œp.˛  1/ C 1 ˛pC1

˛C1 f jjLq ..1;x0 / ; sup jjDx f jjLq .Œx0 ;C1// sup jjDx˛C1 0 0

xo 2R

  ˛ q1 ˛C p  jjf jj1;R 

xo 2R

 ! ˛C1 1

p

: (10.45)

10 Mixed Caputo Fractional Lp -Landau-Type Inequalities

110

Corollary 10.14. Let jjf jj1;R < 1 and

1 2

< ˛  1, f 2 AC 2 .Œa; b/; 8 Œa; b  R. Assume sup jjDx˛C1 f jjL2 ..1;x0 / < 1I 0

xo 2R

˛C1 f jjL2 .Œx0 ;C1// < 1: sup jjDx 0

xo 2R

Then 2˛ C 1 ˛  12

jjf 0 jj1;R 



max

! ˛ 211 ˛C 2



1 1

1

1

  ˛ 211 ˛C 2  jjf jj1;R 

. .˛// ˛C 2 .2˛  1/ 2˛C1

˛C1 f jjL2 ..1;x0 / ; sup jjDx f jjL2 .Œx0 ;C1// sup jjDx˛C1 0 0

xo 2R

 ! ˛C1 1

xo 2R

2

:

(10.46) Corollary 10.15. Let p; q > 1 W p1 C q1 D 1; f 2 AC 2 .Œa; b/; 8 Œa; b  R. Suppose that jjf jj1;R < 1; jjf 00 jjLq .R/ < 1. Then 0

jjf jj1;R

p 1 1   pC1    pC1  pC1 00  2.p C 1/  jjf jj1;R  jjf jjLq .R/ :

(10.47)

We finish chapter with Corollary 10.16. Let f 2 AC 2 .Œa; b/; 8 Œa; b  R. Assume jjf jj1;R < 1; jjf 00 jjL2 .R/ < 1. Then 0

jjf jj1;R 

p 3

  13   23 00 6  jjf jj1;R  jjf jjL2 .R/ :

(10.48)

References 1. A. Aglic Aljinovic, Lj. Marangunic and J. Pecaric, “On Landau type inequalities via Ostrowski inequalities”, Nonlinear Funct. Anal. Appl., Vol.10, 4(2005), pp. 565-579. 2. G. Anastassiou, “Fractional Differentiation inequalities”, Research monograph, Springer, New York, 2009. 3. G. Anastassiou, “Fractional Korovkin Theory”, Chaos Solitons and Fractals, 42(2009), 20802094. 4. G. Anastassiou, “Univariate right fractional Ostrowski inequalities”, CUBO, accepted, 2011. 5. G. Anastassiou, “Mixed Caputo fractional Landau inequalities”, submitted, 2011. 6. N.S. Barnett and S.S. Dragomir, “Some Landau type inequalities for functions whose derivatives are of locally bounded variation”, Tamkang Journal of Mathematics, Vol. 37, No. 4, 301-308, winter 2006.

References

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7. Kai Diethelm, Fractional Differential Equations, online: http://www.tubs.de/ ˜diethelm/lehre/ f-dgl02/fde-skript.ps.gz 8. Z. Ditzian, “Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities”, Math. Inequal. Appl., 3 (2000), 15-24. 9. A. M. A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic J. Theoretical Physics, Vol.3, 12(2006), 81—95. 10. G. S. Frederico and D.F.M. Torres, “Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem”, International Mathematical Forum, Vol.3, No. 10 (2008), 479-493. 11. R. Gorenflo, F. Mainardi, “Essentials of Fractional Calculus”, 2000, Maphysto Center, http://www.maphysto.dk/oldpages/events/LevyCAC2000/ MainardiNotes/fm2k0a.ps. 12. G. H. Hardy and J.E. Littlewood, “Some integral inequalities connected with the calculus of variations”, Quart. J. Math. Oxford Ser. 3(1932), 241-252. 13. G. H. Hardy, E. Landau and J. E. Littlewood, “Some inequalities satisfied by the integrals or derivatives of real or analytic functions”, Math. Z. 39(1935), 677-695. 14. R. R. Kallman and G. C. Rota, “On the inequality jjf 0 jj2  4jjf jj  jjf 00 jj”, in “Inequalities”, Vol. II, (O. Shisha, Ed.), 187-192, Academic Press, New York, 1970. 15. E. Landau, “Einige Ungleichungen f¨ur zweimal differentzierban funktionen”, Proc. London Math. Soc. 13(1913), 43-49. 16. S. G. Samko, A. A. Kilbas, O. I. Marichev, “Fractional Integrals and Derivatives, Theory and Applications”, (Gordon and Breach, Amsterdam, 1993). [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)].

Chapter 11

Multivariate Caputo Fractional Landau Inequalities

Here we give multivariate left Caputo fractional Lp -Landau-type inequalities, p 2 .1; 1 with applications on RN , N  1. This chapter is based on [5].

11.1 Introduction Let p 2 Œ1; 1, I D RC or I D R and f W I ! R is twice differentiable with f; f 00 2 Lp .I /, then f 0 2 Lp .I /. Moreover, there exists a constant Cp .I / > 0 independent of f , such that 1

1

2 2 jjf 0 jjp;I  Cp .I /jjf jjp;I jjf 00 jjp;I

(11.1)

where jj:jjp;I is the p-norm on the interval I , see [1, 6]. The research on these inequalities started by E. Landau [12] in 1914. For the case of p D 1 he proved that p (11.2) C1 .RC / D 2 and C1 .R/ D 2 are the best constants in (11.1). In 1932, G. H. Hardy and J. E. Littlewood [9] proved (11.1) for p D 2, with the best constants p C2 .RC / D 2 and C2 .R/ D 1: (11.3) In 1935, G. H. Hardy, E. Landau, and J. E. Littlewood [10] showed that the best constant Cp .RC / in (11.1) satisfies the estimate Cp .RC /  2;

for p 2 Œ1; 1/;

(11.4)

which yields Cp .R/  2 for p 2 Œ1; 1/.

G.A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0703-4 11, © George A. Anastassiou 2011

113

114

11 Multivariate Caputo Fractional Landau Inequalities

p In fact, in [8] and [11], it was shown that Cp .R/  2. In this chapter we prove multivariate fractional Landau inequalities with respect to jj:jjp ; p 2 .1; 1, involving the left Caputo fractional radial derivative.

11.2 Main Results We need Definition 11.1. ([7], p. 38) Let f 2 AC m .Œa; b/, m 2 N (i.e., f .m1/ 2 AC.Œa; b/),where m D d˛e; ˛ > 0, (d:e the ceiling of the number). We define the left Caputo fractional derivative of order ˛ > 0 by Z x 1 ˛ Da f .x/ D .x  t/m˛1 f .m/ .t/dt; (11.5)  .m  ˛/ a 0 8 x 2 Œa; b: We set Da f .x/ D f .x/; 8x 2 Œa; b. ˛ ˛ Here Da f exists a.e. on Œa; b and Da f 2 L1 .Œa; b/, see [7], pp. 13 and 37–38. When ˛ D m 2 N, then m f .x/ D f .m/ .x/; 8x 2 Œa; b: Da ˛ f .x/ D 0. If x < a we define Da

We make Remark 11.2. Here we follow [13], pp. 149–150. For x 2 RN  f0g; N > 1, we can write uniquely x D rw, r D jxj > 0, w D x=r 2 S N 1 , jwj D 1. Clearly RN  f0g D .0; 1/  S N 1 ;

where S N 1 WD fx 2 RN W jxj D 1g:

Let A > 0, we consider B.0; A/ WD fx 2 RN W jxj < Ag  RN ; We consider

jxj  Euclidean norm of x 2 RN :

RN  B.0; A/ D ŒA; C1/  S N 1

(11.6)

on which we establish Landau fractional inequalities. We need to define the left Caputo radial fractional derivative for our case, see also [2], p. 421. Definition 11.3. Let f W RN  B.0; A/ ! R,   0, mWD de, such that f .w/ 2 AC m .ŒA; b/, 8b > A, for all w 2 S N 1 . We call the left Caputo radial fractional derivative the following function Z r m @A f .x/ 1 m1 @ f .tw/ WD .r  t/ dt; (11.7) @r   .m  / A @r m where x 2 RN  B.0; A/, that is, x D rw; r 2 ŒA; 1/; w 2 S N 1 .

11.2 Main Results

115

Clearly @0A f .x/ D f .x/; @r 0  @A f .x/ D @ @rf .x/ if  2 N:  ; @r 

(11.8) (11.9)

The above function (11.7) exists almost everywhere for x 2 RN  B.0; A/, see [2], p. 422. We make Remark 11.4. Let 0 <   1; A > 0 be fixed, with f .w/ 2 AC 2 .ŒA; b/; 8b > A, 8 w 2 S N 1 . Suppose jjf jj1;RN B.0;A/ < 1,  C1   @A f    <1  @r C1  N 1;R B.0;A/ and C1 C1 f .w/jj1;Œa;C1/  jjDA f .w/jj1;ŒA;C1/ ; 8 a  A; 8 w 2 S N 1 : jjDa (11.10)

Then, by Theorem 8 of [3], we obtain that  1   C1   C1 2 jjf .w/jj1;ŒA;C1/  . C 1/   . C 2/  1   C1    C1  jjf .w/jj1;ŒA;C1/ C1  jjDA f .w/jj1;ŒA;C1/

0

  1  C1  C1    2  . C 1/   . C 2/  jjf jj1;RN B.0;A/ C1  1 !C1  C1   @A f     DW :  @r C1  N 1;R B.0;A/

Hence

   @f     :  @r  N 1;R B.0;A/

(11.11)

(11.12)

We have proved the multivariate fractional Landau inequality. Theorem 11.5. Let 0 <   1; A > 0 be fixed, with f .w/ 2 AC 2 .ŒA; b/; 8b > A, 8 w 2 S N 1 . Suppose jjf jj1;RN B.0;A/ < 1,  C1  @A f   @r C1

   

1;RN B.0;A/

<1

116

11 Multivariate Caputo Fractional Landau Inequalities

and C1 C1 f .w/jj1;Œa;C1/  jjDA f .w/jj1;ŒA;C1/ ; 8 a  A; 8 w 2 S N 1 : jjDa (11.13) Then  1     C1   C1  @f  2    . C 1/   . C 2/  @r  N  1;R B.0;A/

 C1   @A f   C1  jjf jj1;RN B.0;A/    @r C1

1 ! C1

   

:

1;RN B.0;A/

(11.14) We give when  D 1, Corollary 11.6. Let A > 0 fixed, with f .w/ 2 AC 2 .ŒA; b/; 8b > A, 8 w 2 S N 1 . Suppose jjf jj1;RN B.0;A/ < 1,  2 @ f   @r 2

   

1;RN B.0;A/

< 1:

Then s    2  @f  @ f    2  jjf jj1;RN B.0;A/    @r  N  @r 2 1;R B.0;A/

   

:

(11.15)

1;RN B.0;A/

We make Remark 11.7. All entities here are assumed to make sense and to be well-defined. We see that .q > 1/:   C1 D f .w/q A

q;ŒA;C1/

Z D Z D

C1

ˇ C1 ˇ ˇD f .rw/ˇq dr A

A C1 A

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1  r 1N dr A

(11.16)

(see r  A implies r 1A  A1N ; N > 1). Hence it holds  C1  D f .w/q  A1N  A q;ŒA;C1/

Z

C1 A

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr: A

(11.17)

11.2 Main Results

117

R

Observe that

dw

S N 1

N

D 1;

wN

2 2  :  N2

where wN D

(11.18)

Therefore we find  1  1   C1 D f .w/ .C p /   A

q;ŒA;C1/



A

1N q





1 1 C p

.

/

Z

C1 A

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr A

 q1 

! 1 1 C p

:

(11.19)

1 :   .C1 1 / 1 p  ./  .p.  1/ C 1/ pC1

(11.20)

Set   c WD

2 C p1

  1   1C p C p1



1C p1

So here we have p; q > 1 W p1 C q1 D 1, with 1  p1 <   1, f .w/ 2 AC 2 .ŒA; b/; 8b > A, 8 w 2 S N 1 , A > 0 is fixed. C1 Assume DA f .w/ 2 Lq .ŒA; C1//, 8 w 2 S N 1 , jjf jj1;RN B.0;A/ < 1 and  C1   C1  D f .w/  DA f .w/q;ŒA;C1/ ; 8 a  A; 8 w 2 S N 1 : (11.21) a q;Œa;C1/ Hence by Theorem 8 of [4] and (11.19), we derive  0

jjf .w/jj1;ŒA;C1/  c  A    jjf .w/jj1;ŒA;C1/



1 1C p 1 C p



Z 

C1

A

1N q





1

.C p1 / 

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr A

We call



ˇ  WD c  A

1N q

 



1 1 q C p

.

/

: (11.22)

1

.C p1 / ;

and     WD ˇ   jjf jj1;RN B.0;A/



(11.23)  1

1C p 1 C p

:

(11.24)

Therefore jjf 0 .w/jj1;ŒA;C1/ 

(11.25)

118

11 Multivariate Caputo Fractional Landau Inequalities

  ˇ   jjf .w/jj1;ŒA;C1/ Z   

C1

A



1 1C p 1 C p



Z 

C1 A

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr A

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr A



1 1 q C p

.

/

:



1 1 q C p

.

/

(11.26)

That is, we proved Z

0

jf .rw/j    

C1

A

ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr A



1 1 q C p

.

/

;

(11.27)

8 w 2 S N 1 .

8r 2 ŒA; C1/; Call

1 q .C p / ı  WD   :

(11.28)

Then by (11.27) we obtain .

1 q C p

jf 0 .rw/j

0

jf .rw/j

Z  ı 

C1

A

 ˇ ˇ C1 ˇD f .rw/ˇq  r N 1 dr ; A

(11.29)

8 w 2 S N 1 .

8r 2 ŒA; C1/; Therefore Z

/

.

1 q C p

/

Z

Z

dw  ı  

S N 1

S N 1

C1 A

 ! ˇ C1 ˇq N 1 ˇD f .rw/ˇ  r dr dw A (11.30)

Z D ı  RN B.0;A/

ˇ ˇ ˇ @C1 f .x/ ˇq ˇ A ˇ ˇ ˇ dx: ˇ @r C1 ˇ

(11.31)

Consequently we derive 0 @

Z S N 1

0 1 q1   q 1 .C p1 / q B dwA  .ı  /  @ jf 0 .rw/j

Z RN B.0;A/

1 q1 ˇ ˇ ˇ @C1 f .x/ ˇq C ˇ A ˇ ˇ ˇ dx A ˇ @r C1 ˇ (11.32)

 C1 1  @A f q D .ı  /    @r C1

   

q;RN B.0;A/

:

(11.33)

11.2 Main Results

119

Therefore it holds    1   0   f .r/ C p   

q;S N 1

 C1 1  @A f q  .ı  /    @r C1

   

;

(11.34)

q;RN B.0;A/

8r 2 ŒA; C1/. Hence we get     C p1   @f    .rw/    @r 

.q;S N 1 ;w/

    

.1;ŒA;C1/;r/

 C1 1  @A f q  .ı  /    @r C1

   

:

q;RN B.0;A/

(11.35) Notice that 

1 q

.ı  / D .  /

C p1



D .ˇ  / DA DA 

1N q

1N q

C p1







   jjf jj1;RN B.0;A/ 

 .c /   

C p1

2 C p1





1C p1

  1C p1

   jjf jj1;RN B.0;A/

 1C 1  p

1C p1

1

  C p1 =.pC1/

 ./  .p.  1/ C 1/

   jjf jj1;RN B.0;A/

  1C p1

:

(11.36) We have proved the following multivariate Lp fractional Landau inequality. Theorem 11.8. Let p; q > 1 W p1 C q1 D 1, with q1 <   1, f .w/ 2 AC 2 .ŒA; b/; 8b > A, A > 0 fixed, 8 w 2 S N 1 . C1 Assume that DA f .w/ 2 Lq .ŒA; C1//, 8 w 2 S N 1 , jjf jj1;RN B.0;A/ < 1 and  C1  D f .w/ a

q;Œa;C1/

 C1   DA f .w/q;ŒA;C1/ ; 8 a  A; 8 w 2 S N 1 : (11.37)

120

11 Multivariate Caputo Fractional Landau Inequalities

Then     C p1    @f   .rw/     @r 1N



A q   ./

    

.q;S N 1 ;w/ .1;ŒA;C1/;r/  1C 1  1 p

 

2 C p



1C p1

  jjf jj1;RN B.0;A/

   1C p1

1

.C p1 /



.p.  1/ C 1/ .pC1/  C1   @A f    :  @r C1  N q;R B.0;A/

(11.38)

It follows the p D q D 2 case. Corollary 11.9. Let 12 <   1, f .w/ 2 AC 2 .ŒA; b/; 8b > A, A > 0 fixed, 8 w 2 S N 1 . C1 Suppose that DA f .w/ 2 L2 .ŒA; C1//, 8 w 2 S N 1 , jjf jj1;RN B.0;A/ < 1 and  C1   C1  D f .w/  DA f .w/2;ŒA;C1/ ; 8 a  A; 8 w 2 S N 1 : (11.39) a 2;Œa;C1/ Then

 .C 12 /    @f   .rw/     @r

    

.2;S N 1 ;w/ .1;ŒA;C1/;r/

1N



A 2   ./

 

2 C 12

 . 1 / 2

1 1   .C 2 / 2  1 .2C1/  C1   . 12 /  @A f    jjf jj1;RN B.0;A/ :  @r C1  N 2;R B.0;A/  12



(11.40)

We finish chapter with the p D q D 2;  D 1 case. Corollary 11.10. Let f .w/ 2 AC 2 .ŒA; b/; 8b > A, A > 0 fixed ,8 w 2 S N 1 . Assume that f 00 .w/ 2 L2 .ŒA; C1//, 8 w 2 S N 1 and jjf jj1;RN B.0;A/ < 1 . Then   1:5   @f      .rw/     @r  N 1  .2;S

;w/ .1;ŒA;C1/;r/

 2  1N  p  q @ f  A 2  6  jjf jj1;RN B.0;A/    @r 2

   

2;RN B.0;A/

:

(11.41)

References

121

References 1. A. Aglic Aljinovic, Lj. Marangunic and J. Pecaric, “On Landau type inequalities via Ostrowski inequalities”, Nonlinear Funct. Anal. Appl., Vol.10, 4(2005), pp. 565-579. 2. G. Anastassiou, “Fractional Differentiation inequalities”, Research monograph, Springer, New York, 2009. 3. G. Anastassiou, “Left Caputo fractional jj  jj1 - Landau inequalities”, Applied Math. Letters, accepted, 2011. 4. G. Anastassiou, “Lp - left Caputo fractional Landau inequalities”, J. Computational Anal. and Appl., accepted, 2011. 5. G. Anastassiou, “Multivariate Landau fractional inequalities”, submitted, 2011. 6. N.S. Barnett and S.S. Dragomir, “Some Landau type inequalities for functions whose derivatives are of locally bounded variation”, Tamkang Journal of Mathematics, Vol. 37, No. 4, 301-308, winter 2006. 7. K. Diethelm, Fractional Differential Equations, online: http://www.tubs.de/ ˜diethelm/lehre/ f-dg102/fde-skript.ps.gz, 2003. 8. Z. Ditzian, “Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities”, Math. Inequal. Appl., 3 (2000), 15-24. 9. G. H. Hardy and J.E. Littlewood, “Some integral inequalities connected with the calculus of variations”, Quart. J. Math. Oxford Ser. 3(1932), 241-252. 10. G. H. Hardy, E. Landau and J. E. Littlewood, “Some inequalities satisfied by the integrals or derivatives of real or analytic functions”, Math. Z. 39(1935), 677-695. 11. R. R. Kallman and G. C. Rota, “On the inequality jjf 0 jj2  4jjf jj  jjf 00 jj”, in “Inequalities”, Vol. II, (O. Shisha, Ed.), 187-192, Academic Press, New York, 1970. 12. E. Landau, “Einige Ungleichungen f¨ur zweimal differentzierban funktionen”, Proc. London Math. Soc. 13(1913), 43-49. 13. W. Rudin, “Real and Complex Analysis”, International Student Edition, McGraw Hill, London, New York, 1970.