Uniqueness of p(f) and P[f]

TL;DR

This paper investigates the uniqueness of certain differential polynomials and compositions of meromorphic functions sharing a value, generalizing previous results and answering open questions in complex analysis.

Contribution

It establishes new conditions under which p(f) and P[f] are unique when sharing a value, extending prior work in the field.

Findings

Proves uniqueness of p(f) and P[f] under shared value conditions.

Generalizes previous theorems by Zang, Lu, Banerjee, Majumder, Bhoosnurmath, and Kabbur.

Answers an open question posed by Zang and Lu.

Abstract

Let f be a non constant meromorphic function and a(not identically zero or infinity) be a meromorphic function satisfying T(r,a) = o(T(r,f)) as r tends to infinity, and p(z) be a polynomial of degree n greater than or equal to 1 with p(0) = 0. Let P[f] be a non constant differential polynomial of f. Under certain essential conditions, we prove the uniqueness of p(f) and P[f] when p(f) and P[f] share a with weight l greater than or equal to zero. Our result generalizes the results due to Zang and Lu, Banerjee and Majumder, Bhoosnurmath and Kabbur and answers a question of Zang and Lu.