Regular functions of restricted growth and their zeros in tangential regions

Author:

Linden C. N.

Abstract

For a given function k k , positive, continuous, nondecreasing and unbounded on [ 0 , 1 ) [0,1) , let A ( k ) {A^{(k)}} denote the class of functions regular in the unit disc for which log | f ( z ) | > k ( | z | ) |f(z)| > k(|z|) when | z | > 1 |z| > 1 . Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in A ( k ) {A^{(k)}} to be Blaschke is that \[ 0 1 ( k ( t ) / ( 1 t ) ) d t \int _0^1 {\sqrt {(k(t)/(1 - t))\,dt} } \] is finite. It is shown that the imposition of a further regularity condition on the growth of k k ensures that in some tangential region the zero set of each function in A ( k ) {A^{(k)}} is also Blaschke.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference3 articles.

1. A critical growth rate for functions regular in a disk;Hayman, W. K.;Michigan Math. J.,1980

2. Functions regular in the unit circle;Linden, C. N.;Proc. Cambridge Philos. Soc.,1956

3. E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, London, 1939.

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Subharmonic functions with unilateral growth conditions;Lecture Notes in Mathematics;1987

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