On periodicity of entire functions

Abstract:

A sequence $S = \{ {s_n}\}$ is said to be a periodic set of period $\tau ( \ne 0)$ if and only if ${S^\ast } = \{ {s_n} + \tau \}$ can be rearranged to be a sequence to coincide with $S$. Let $F$ be the class of all entire functions $f$ satisfying the growth condition: \[ \lim \limits _{r \to \infty } \sup \log \log \log M(r,f)/\log r < 1.\] In this paper it is shown that if $f \in F$ and the zero sets of $f$ and $f’$ both are periodic sets with the same period $\tau$, then $f$ can be expressed as $f(z) = {e^{cz}}g(z)$, where $c$ is a constant and $g(z)$ is a periodic entire function with period $\tau$. A counterexample is exhibited to show that the above condition is a necessary one.