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

Abstract

In this paper, we prove the following 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 ${\psi_j(z), \ j=1,2,3}$ are 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 Fang and Hong (Bull Malays Math Sci Soc 23(2), 143–151, 2000) and in particular, it generalizes the most celebrated theorem of Montel i.e. Montel’s theorem.