Uniqueness of shift and derivatives of meromorphic functions

Abstract

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function $f(z)$ and the k-th derivative of its shift, $f(z+c),$ where $j,k$ are integers with $0\leq j<k.$ In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by $E(a,f)$ the set of all zeros of $f-a,$ where each zero with multiplicity $m$ is counted $m$ times. In the paper proved, in particular, the following statement:\\ Let $f(z)$ be a non-constant meromorphic function of finite order, $c$ be a non-zero finite complex number and $j,k$ be integers such that $0\leq j<k.$ If $f^{(j)}(z)$ and $f^{(k)}(z+c)$ have the same $a-$points for a finite value $a(\neq 0)$ and satisfy conditions $$E(0,f^{(j)}(z))\subset E(0,f^{(k)}(z+c))\quad\text{and}\quad E(\infty,f^{(k)}(z+c))\subset E(\infty,f^{(j)}(z)),$$ then $f^{(j)}(z)\equiv f^{(k)}(z+c)$ (Theorem 6).