Normality Criteria of Meromorphic Functions

Abstract

In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D, m, n, k$ and $d$ be four positive integers satisfying $m ≥ n + 2$ and $d ≥ \frac{k + 1}{m − n − 1}$, and $a(≠0), b$ be two finite constants. Suppose that every $f ∈ \mathcal{F}$ has all its zeros and poles of multiplicity at least $k$ and $d$, respectively. If $(f^n)^{(k)}−af^m$ and $(g^n)^{(k)}−ag^m$ share the value $b$ for every pair of functions $(f, g)$ of $\mathcal{F}$, then $\mathcal{F}$ is normal in $D$. Our results improve the related theorems of Schwick (Schwick W. Normality criteria for families of meromorphic function. $J$. $Anal$. $Math$., 1989, 52: 241–289), Li and Gu (Li Y T, Gu Y X. On normal families of meromorphic functions. $J$. $Math$. $Anal$. $Appl$., 2009, 354: 421–425).