Publications of (and about) Paul Erdös
Autor: Erdös, Paul; Renyi, Alfréd
Title: On random entire functions. (In English)
Source: Zastosowania Mat. 10, 47-55 (1969).
Review: Let f(z) = sum oon = 0anzn be an arbitrary entire function, held fixed in all that follows. For r > 0 let M(r) = max (|f(z)|: |z| = r) be the maximum modulus function of f and \mu(r) = max (|an| rn: n \geq 0) the maximum term in the series expansion of f. The following extension of Wiman's theorem was proved by Rosenbloom: for every \delta > 0 there exists a subset E\delta of fintie logarithmic measure such that if r \not in E\delta, then M(r) < \mu (r)[ log \mu (r)] ½[ log log \mu (r)]1+\delta.
For 0 \leq t < 1 let Rn(t) = sign \sin (2n \pi t) denote the n-th Rademacher function, n \geq 0. The present paper considers the class of entire functions obtained by giving random signs to the terms in the series expansion of f above; explicitly, the entire functions which can be written f(z,t) = sum oon = 0 anRn(t)zn, 0 \leq t < 1. Keeping the notation above, let M(r,t) = max (|f(z,t)|: |z| = r). The main result is: for every \delta > 0 and almost all t in [0,1), there exists a subset E\delta (t) \subset R_+ of finite logarithmic measure (depending on t) such that for r \not in E\delta (t), M(r,t) < \mu (r)[ log \mu (r)]1/4[ log log \mu (r)]1+\delta.
Two related results are also given.
Classif.: * 30D20 General theory of entire functions
60-XX Probability theory and stochastic processes
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