Blaschke products in B

4.9+

BLASCHKE PRODUCTS IN

The class ~

~o

~o

consists of those functions f that are holomorphic

~4-I~I) I~(~)I =0.

in ~

and satisfy

It can be described alternatively as the class of functions in D

that

are derivatives of holomorphic functions having boundary values in the Zygmund class %, (the class of uniformly smooth functions) [I, p. 263]. It is a subclass of the class ~ of Bloch functions

(those holomorphic f in

D

satisfying

contains VMOA, the class of holomorphic functions

mlS~i(~-l~1) l~1(~)i~oo) ; see, e.g., [2]. in ~

It

whose boundary values have vanishing

mean oscillations [3]. The class ~ o N ~ m has an interesting interpretation: those functions in H ~ that are constant on each Gleason part of H ~.

it consists of

It is not too hard to come up with an example to show that the inclusion V M 0 ~ is proper. Indeed, it is known that %, contains functions that are not of bounded variation [I, p. 48]. If u is the Poisson integral of such a function and V is its harmonic conjugate, then the derivative of u + iv will be such an example. In connection with a problem in prediction theory mentioned in [4], I was interested in having an example of a bounded function in ~o which is not in VMOA, and that seems somewhat more difficult to obtain. Eventually I realized one can produce such an example on the basis of a result of Shapiro [5] and Kahane [6]. They showed, by rather complicated constructions, that there exist positive singular measures on

~

whose indefinite integrals are in %,.

inner function associated with such a measure is in functions in VMOA are the finite Blaschke products. If f is an inner function in function in

~o

and

It is easy to check that the singular

~0

That does it, because the only inner

Icl < I, then (f -- c)/(I

~o , and it is a Blaschke product for "most" values of c.

infinite Blaschke products. B!aschke products in

~f) is also an inner Thus,

~o

contains

I should like to propose the problem of characterizing

~o by means of the distribution of their zeros.

that the zeros of a Blaschke product in

~

the

One has the feeling

must, in some sense, be "spread smoothly" in

A natural first step in trying to find the correct condition would be to try to give + a direct, construction of an infinite Blaschke product in ~o 9 offer on the problem is very meager: A Blaschke product in larity on

~

.

The proof, unfortunately, T

~o cannot have an isolated singu-

is too involved to indicate here.

tion one might ask whether a Blaschke product in ~o some subarc of

The only information I can

can have a singular

As a test ques-

set which

meets

in a nonempty set of measure zero.

Another question, admittedly vague, concerns the abundance of Blaschke products in ~o For instance, a Blaschke product should be in ~o

if its zeros are evenly spread throughout

One is led to suspect that, in some sense, a Blaschke product with random zeros will be almost

surely in ~o" LITERATURE CITED

] 9

2. 3.

4. 5.

A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, Cambridge (1959). J. M. Anderson, J. Clunie, and Ch. Pommerenke, "On Bloch functions and normal functions," J. Reine Angew. Math., 270, 12-37 (1974). Ch. Pommerenke, "On univalent functions, Bloch functions and VMOA" (to appear). D. Sarason, "Functions of vanishing mean oscillation," Trans. Am. Math. Soc., 207, 391405 (1975). H. S. Shapiro, Monotonic singular functions of high smoothness," Michigan Math. J., 15, 265-275 (1968).

tDONALD SARASON. 94720.

University of California, Department of Mathematics,

Berkeley, California,

2257

.

2258

J.-P. Kahane, "Trois notes sur les ensembles parfaits lin~aires," Enseignement Math., (2), 15, 185-192 (1969).