Abstract
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
$$ \begin{align*} \bigg(\frac{1+\exp(C_2c)}{1-\exp(C_2 c)}\bigg)^{2n}=1. \end{align*} $$
Publisher
Cambridge University Press (CUP)