THE DIFFERENCE ANALOGUE OF THE TUMURA–HAYMAN–CLUNIE THEOREM

Abstract We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and $\Delta _c^n f$ omit zero in the whole complex plane, then either $f(z)=\exp (h_1(z)+C_1 z)$ , where $h_1$ is an entire function of period c and $\exp (C_1 c)\neq 1$ , or $f(z)=\exp (h_2(z)+C_2 z)$ , where $h_2$ is an entire function of period $2c$ and $C_2$ satisfies $$ \begin{align*} \bigg(\frac{1+\exp(C_2c)}{1-\exp(C_2 c)}\bigg)^{2n}=1. \end{align*} $$