Normality through sharing of pairs of functions with derivatives

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.