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Journal of the London Mathematical Society 1998 58(1):127-140; doi:10.1112/S0024610798006334
© 1998 by London Mathematical Society
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© The London Mathematical Society

Identity Theorems for Functions of Bounded Characteristic

W. K. Hayman

Department of Mathematics, Imperial College 180 Queen's Gate, London SW7 2BZ

Received 7 December 1994. Revision received 17 October 1995.

Suppose that f(z) is a meromorphic function of bounded characteristic in the unit disk {Delta}:|z|<1. Then we shall say that f(z)isinN. It follows (for example from [3, Lemma 6.7, p. 174 and the following]) that


Formula

where h1(z), h2(z) are holomorphic in {Delta} and have positive real part there, while {Pi}1(z), {Pi}2(z) are Blaschke products, that is,


Formula

where p is a positive integer or zero, 0<|aj|<1, c is a constant and

{sum}(1–|aj|)<{infty}.

We note in particular that, if c!=0, so that f(z)nequiv0,


Formula
(1.1)

so that f(z)=0 only at the points aj. Suppose now that zj is a sequence of distinct points in {Delta} such that

|zj|->1 as j->{infty} and {sum}(1–|zj|)={infty}. (1.2)

If f(zj)=0 for each j and fisinN, then f(z){equiv}0.

N. Danikas [1] has shown that the same conclusion obtains if f(zj)->0 sufficiently rapidly as j->{infty}. Let {varepsilon}j, {lambda}j be sequences of positive numbers such that

{sum}{varepsilon}j<{infty} and {lambda}j ->{infty} as j->{infty}.

Danikas then defines


Formula

and proves Theorem A.


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