Normality through sharing of pairs of functions with derivatives

TL;DR

This paper improves existing criteria for the normality of families of holomorphic functions by analyzing shared pairs of functions with derivatives, leading to broader conditions ensuring normality.

Contribution

It introduces new normality criteria based on sharing pairs of functions with derivatives, extending Schwick's results and providing sharper conditions with examples.

Findings

Established new normality criteria for families of holomorphic functions.

Improved upon Schwick's 1992 result with broader conditions.

Provided examples demonstrating the sharpness of the criteria.

Abstract

Let $\mathcal{F}\subset\mathcal{M}(D)$ and let $a, b$ and $c$ be three distinct complex numbers. If, there exist a holomorphic function $h$ on $D$ and a positive constant $\rho$ such that for each $f\in\mathcal{F},$ $f$ and $f^{'}$ partially share three pairs of functions $(a,h), \ (b, c_f)$ and $(c,d_f)$ on $D,$ where $c_f$ and $d_f$ are some values in some punctured disk $D^*_{\rho}(0),$ then $\mathcal{F}$ is normal in $D$. This is an improvement of Schwick's result[Arch. Math. (Basel), \textbf{59} (1992), 50-54]. We also obtain several normality criteria which significantly improve the existing results and examples are given to establish the sharpness of results.