Class of meromorphic functions partially shared values with their differences or shifts

TL;DR

This paper investigates the uniqueness of meromorphic functions sharing values partially with their shifts or differences, correcting previous results, providing new generalizations, and illustrating findings with examples and open questions.

Contribution

It introduces the concept of partially shared values, corrects and generalizes existing uniqueness theorems, and explores their applications to meromorphic functions and their shifts or differences.

Findings

Established new uniqueness results for meromorphic functions with partially shared values.

Corrected and generalized previous theorems in the literature.

Provided examples demonstrating the sharpness of conditions and posed open questions.

Abstract

Two meromorphic functions $ f $ and $ g $ are said to share a value $ s\in\mathbb{C}\cup\{\infty\} $ $ CM $ $ (IM) $ provided that $ f(z)-s $ and $ g(z)-s $ have the same set of zeros counting multiplicities (ignoring multiplicities). We say that a meromorphic function $ f $ share $ s\in\mathbb{C} $ partially $ CM $ with a meromorphic function $ g $ if $ E(s,f)\subseteq E(s,g)$. It is easy to see that the condition ``partially shared values $ CM $" is more general than the condition ``shared value $ CM $". With the idea of partially shared values, in this paper, we prove some uniqueness results between non-constant meromorphic functions and their shifts or generalized differences. We exhibit some examples to show that the result of {Charak \emph{et al.}} \cite{Cha & Kor & Kum-2016} is not true for $k=2 $ or $ k=3 $. We find some gaps in proof of the result of {Lin} \emph{et al.} \cite{Lin & Lin & Wu}, and we not only correct them but also generalize their result in a more convenient way. A number of examples have been exhibited to validate a certain claim of the main results of this paper and also to show that some of the conditions are sharp. In the end, we have posed some open questions for further investigation of the main result of the paper.