On a theorem of A. and C. R\'enyi and a conjecture of C. C. Yang concerning periodicity of entire functions

TL;DR

This paper extends a theorem on the periodicity of entire functions, providing new conditions under which such functions are periodic, and addresses open questions related to a conjecture by C. C. Yang.

Contribution

The paper generalizes a theorem on periodic entire functions and offers new criteria involving polynomial and differential expressions, advancing understanding of Yang's conjecture.

Findings

Proves entire functions are periodic if P(f(z)) or P(f(z))/f^{(k)}(z) is periodic with finite Picard value.

Extends the theorem to cases where f(z)^n + a_1 f'(z) + ... + a_k f^{(k)}(z) is periodic.

Determines the possible periods explicitly for the functions studied.

Abstract

A theorem of A. and C. R\'enyi on periodic entire functions states that an entire function $f(z) $ must be periodic if $ P(f(z)) $ is periodic, where $ P(z) $ is a non-constant polynomial. By extending this theorem, we can answer some open questions related to the conjecture of C. C. Yang concerning periodicity of entire functions. Moreover, we give more general forms for this conjecture and we prove, in particular, that $ f(z) $ is periodic if either $ P(f(z)) f^{(k)}(z) $ or $ P(f(z))/f^{(k)}(z) $ is periodic, provided that $ f(z) $ has a finite Picard exceptional value. We also investigate the periodicity of $ f(z) $ when $ f(z)^{n}+a_{1} f^{\prime}(z)+\cdots+a_{k} f^{(k)}(z) $ is periodic. In all our results, the possibilities for the period of $ f(z) $ are determined precisely.