Normality and uniqueness property of meromorphic function in terms of some differential polynomials

TL;DR

This paper investigates the conditions under which families of meromorphic functions exhibit normality and uniqueness when their differential polynomials share a small function, relaxing previous zero-order restrictions.

Contribution

It establishes new criteria for normality and uniqueness of meromorphic functions based on shared differential polynomial values without requiring zeros of the polynomial to have large order.

Findings

Established normality criteria for meromorphic functions

Proved uniqueness results under shared differential polynomial conditions

Relaxed zero-order restrictions compared to previous studies

Abstract

In this paper, we will consider normality and uniqueness property of a family $\mathcal{F}$ of meromorphic functions when $[Q(f)]^{(k)}$ and $[Q(g)]^{(k)}$ share $\alpha$ ignoring multiplicities, for any $f,g\in \mathcal{F}$, where $Q$ is a polynomial and $\alpha $ is a small function. Our results do not need all of zeros of $Q$ have large order as other authors's results.