A uniqueness problem concerning entire functions and their derivatives

TL;DR

This paper characterizes entire functions satisfying specific value-sharing conditions between the function and its derivative, solving an open problem in the field of uniqueness theory and introducing a potentially useful normality criterion.

Contribution

It completely determines entire functions with certain derivative-value implications, addressing an open question in uniqueness theory and proposing a new normality criterion.

Findings

Identified all entire functions satisfying the given derivative-value implications.

Provided a new normality criterion relevant to complex analysis.

Solved an open problem in the theory of entire functions.

Abstract

We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.