Meromorphic Functions Sharing One Value with Their Derivatives Concerning the Difference Operator

Abstract

Let f(z) be a non-constant meromorphic function of finite order, $c\in \mathbb {C}\setminus \{0\}$ and $k\in \mathbb {N}$. Suppose f(z) and $f^{(k)}(z+c)$ share 1 CM (IM), f(z) and $f(z+c)$ share $\infty $ CM. If $N(r,0;f)=S(r,f) \left( N\left( r,0;f(z)\right) +N\left( r,0;f^{(k)}(z+c)\right) =S(r,f)\right) $, then either $f(z)\equiv f^{(k)}(z+c)$ or f(z) is a solution of the following equation:

$\begin{aligned}&f'(z+c)-1=a(z)\left( f(z)-1\right) \left( f(z)+\frac{1}{a(z)}\right) ,\;\; \hbox {and}\\&\quad N\left( r,0;f(z)+\frac{1}{a(z)}\right) =S(r,f)\\&\quad \left( f'(z+c)-1=a(z)\left( f(z)-1\right) \left( f(z)+\frac{1}{a(z)}\right) \right) \end{aligned}$

where $a(z)\left( \not \equiv -\,1,0,\infty \right) \left( a(z)\left( \not \equiv 0,\infty \right) \right) $ is a meromorphic function satisfying $T(r,a)=S(r,f)$. Also we exhibit some examples to show that the conditions of our results are the best possible.

Keywords

Uniqueness / Meromorphic function / Shift operator / Difference polynomial / Sharing values