A generalization of Lappan’s five point theorem

In this paper, we prove the following result: Let \(\mathcal {F}\) be a family of meromorphic functions on a domain D and let \(S=\left\{ \varphi _i:1\le i \le 5\right\} \) be a set of five distinct meromorphic functions on D. If for each \(f \in \mathcal {F}\) and \(z_0 \in D\), there is a constant \(M>0\) such that \(f^{\#}(z_0) \le M\) whenever \(f(z_0)= \varphi (z_0)\) for some \(\varphi \in S\) and if \(f(z_0) \ne \varphi (z_0)\) for all \(\varphi \in S\) whenever \(\varphi _i(z_0) = \varphi _j(z_0) \) for some \(i,j \in \left\{ 1,2,3,4,5\right\} \) with \(i \ne j\), then \(\mathcal {F}\) is normal on D. Further we extend this result to the case where the set S contains fewer functions. In particular, our result generalizes the most significant theorem of Lappan (i.e. Lappan’s five point theorem).