Normality concerning the sequence of multiple functions

Abstract

Let \(\{f_n\}\) be a sequence of meromorphic functions defined in a domain D, and let \(\{\psi _n\}\) be a sequence of holomorphic functions on D, whose zeros are multiple, such that \(\psi _n\rightarrow \psi \) converges locally uniformly in D, where \(\psi (\not \equiv 0)\) is holomorphic in D. If, (1) \(f_n\ne 0\) and \(f_n^{(k)}\ne 0\); (2) all zeros of \(f_n^{(k)}-\psi _n\) have multiplicities at least \((k+2)/k\), then \(\{f_n\}\) is normal in D.