Normality and Shared Values of Meromorphic Functions with Differential Polynomial

Abstract

In this paper, we discuss the normality and shared values of meromorphic functions with differential polynomial. We obtain the main result: Let \(\mathcal {F}\) be a family of meromorphic functions in a domain D and k, q be two positive integers. Let \(P(z,w)=w^{q}+a_{q-1}(z)w^{q-1}+\cdots +a_{1}(z)w \) and \(H(f,f',\ldots ,f^{(k)})\) be a differential polynomial with \(\frac{\varGamma }{\gamma }|_H<k+1\). If \(P(z,f^{(k)})+H(f, f',\cdots ,f^{(k)})-1~\)has at most \(q(k+1)-1\) distinct zeros (ignoring multiplicity) for each function \(f\in \mathcal {F}, f(z)\ne 0\), then \(\mathcal {F}\) is normal in D. This result generalizes that of Chang [1].