Sharing of a set of meromorphic functions and Montel's theorem

TL;DR

This paper generalizes Montel's theorem by proving that a family of meromorphic functions sharing a specific set and satisfying certain conditions is normal, extending previous results in complex analysis.

Contribution

It introduces a new normality criterion for families of meromorphic functions sharing a set of three functions, broadening the scope of Montel's theorem.

Findings

Generalizes Montel's theorem for meromorphic functions

Establishes normality under shared set conditions

Extends previous results by Fang and Hong

Abstract

In this paper we prove the result: Let $\mathcal{F}$ be a family of meromorphic functions on a domain $\Omega$ such that every pair of members of $\mathcal{F}$ shares a set $S:=\left\{\psi_1(z), \psi_2(z), \psi_3(z) \right\}$ in $\Omega$, where $\psi_j(z), \ j=1,2,3$ is meromorphic in $\Omega.$ If for every $f\in \mathcal{F}$, $f(z_0)\neq \psi_i (z_0)$ whenever $\psi_i(z_0)=\psi_j(z_0)$ for $i,j\in \left\{1,2,3 \right\}(i\neq j)$ and $z_0\in \Omega ,$ then $\mathcal{F}$ is normal in $\Omega$. This result generalizes a result of M.Fang and W.Hong [Some results on normal family of meromorphic functions, Bull. Malays. Math. Sci. Soc. (2)23 (2000),143-151,] and in particular, it generalizes the most celebrated theorem of Montel-the Montel's theorem.