Variation on a Theorem by Mues and Steinmetz

TL;DR

This paper proposes a generalization of meromorphic functions sharing a value with their derivatives without restrictions on ramification, and investigates whether classical theorems extend to this broader context.

Contribution

It introduces a new generalized sharing concept for meromorphic functions and explores its implications on established theorems about value sharing with derivatives.

Findings

Generalization of value sharing without ramification restrictions

Discussion on extending classical theorems to the generalized setting

Insights into the behavior of meromorphic functions sharing values with derivatives

Abstract

Let $f$ be a meromorphic function. We suggest a generalization of $f$ and its derivative $f'$ sharing a nonzero value $a$ IM that does not impose any a priori restrictions on the ramification of $f$. Then we discuss some results around the question whether the famous theorem on entire functions $f$ that share two values IM with $f'$ still holds for this weaker notion.