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

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.