Meromorphic functions that partially share values with their first derivative

TL;DR

This paper investigates the uniqueness of meromorphic functions that partially share certain values with their derivatives, providing new results for functions sharing four values.

Contribution

It introduces new uniqueness theorems for meromorphic functions sharing values with their derivatives, especially for four partially shared values.

Findings

Established conditions under which meromorphic functions sharing values with their derivatives are unique.

Proved a new result for functions sharing four values partially with their derivatives.

Abstract

We consider uniqueness results for meromorphic functions $f:{\mathbb C} \to \widehat{\mathbb C}$ such that for certain values $a\in {\mathbb C}$ the implication $f(z)=a \Rightarrow f'(z)=a$ holds, i.e. that $f$ and $f'$ share values {\it partially}. In particular, we give a result for four partially shared values.