Uniqueness of Meromorphic Functions With Respect To Their Shifts Concerning Derivatives

TL;DR

This paper investigates the uniqueness of meromorphic functions sharing small functions with their shifts concerning derivatives, extending previous results to more general cases and providing new conditions for function equality or specific exponential relations.

Contribution

It generalizes existing uniqueness results to meromorphic functions with derivatives, incorporating small functions and shifts, and introduces new conditions for function equivalence or exponential relations.

Findings

Established conditions under which a meromorphic function equals its shift or relates via an exponential function.

Extended previous results from entire to meromorphic functions and from derivatives to higher derivatives.

Provided explicit forms and conditions involving small functions and periodicity for function sharing scenarios.

Abstract

An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $\pi i$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We use a different method from Qi and Yang \cite {qy} to improves entire function to meromorphic function, the first derivative to the $k-th$ derivatives, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $\rho_{2}(f)<1$, let $c$ be a nonzero finite value, and let $a(z)\not\equiv\infty, b(z)\not\equiv\infty\in \hat{S}(f)$ be two distinct small functions of $f(z)$ such that $a(z)$ is a periodic function with period $c$ and $b(z)$ is any small function of $f(z)$. If $f(z)$ and $f(z+c)$ share $a(z),\infty$ CM, and share $b(z)$ IM, then either $f(z)\equiv f(z+c)$ or $$e^{p(z)}\equiv \frac{f(z+c)-a(z+c)}{f(z)-a(z)}\equiv \frac{b(z+c)-a(z+c)}{b(z)-a(z)},$$ where $p(z)$ is a non-constant entire function of $\rho(p)<1$ such that $e^{p(z+c)}\equiv e^{p(z)}$.