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 $f\in \mathcal{F}$, $f(z_0)\ne {\psi }_i(z_0)$ whenever ${\psi }_i(z_0)={\psi }_j(z_0)$ for $i,j\in \left\{1,2,3\right\}(i\ne j)$ 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.

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