Normal Families of Zero-Free Meromorphic Functions

Abstract

Let $a(\ne \mathrm{0}),\hspace{0.17em}\hspace{0.17em}b\in ℂ$, and $n$ and $k$ be two positive integers such that $n\ge 2$. Let $ℱ$ be a family of zero-free meromorphic functions defined in a domain $𝒟$ such that for each $f\in ℱ$, $f+a{(f^{(k)})}^n-b$ has at most $nk$ zeros, ignoring multiplicity. Then $ℱ$ is normal in $𝒟$.