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

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.