On the Generalizations of Br\"{u}ck Conjecture

TL;DR

This paper extends and generalizes the Brück conjecture to differential polynomials, providing new results, examples, and posing open questions for future research in complex analysis.

Contribution

It introduces broader conditions for the conjecture's generalizations and improves upon existing results with new theorems and illustrative examples.

Findings

Generalized Brück conjecture for differential polynomials

Provided examples demonstrating sharpness of conditions

Posed open problem for further exploration

Abstract

We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the necessity or sharpness of some conditions used in the paper. At last, we pose an open problem for future research.