Normal families of meromorphic functions (Q2779051)

In 1967, Walter Hayman conjectured that if a family \({\mathcal F}\) of meromorphic functions has the property that there exist two finite complex numbers \(a\neq 0\) and \(b\) and an integer \(n\geq 3\) such that \(f'-af^n\neq b\) for each \(f\in {\mathcal F}\), then \({\mathcal F}\) is a normal family. This conjecture has been proved and has stimulated a number of generalizations, and the author gives a result that contains many of these generalizations. He proves that if \(n,k,p\), and \(t\) are four positive integers satisfying \(n-1>{k+1 \over p}+ {1\over t}\), if \(a\neq 0\) and \(b\) are two finite complex numbers, and if \({\mathcal F}\) is a family of meromorphic functions such that each function \(f\in{\mathcal F}\) has all its poles of order at least \(p\) and all its zeros of order \(t\), and also satisfies the equation \(f^{(k)}-a f^n\neq b\), then \({\mathcal F}\) is a normal family. In particular this result improves a result of \textit{H. H. Chen} and \textit{X. Hua} [J. Aust. Math. Soc., Ser A 59, No. 1, 112-117 (1995; Zbl 0843.30032)] which states that if \({\mathcal F}\) is a family of holomorphic functions for which there exist integers \(k\geq 1\) and \(n\geq 2\) and finite complex numbers \(a\neq 0\) and \(b\) such that \(f^{(k)}-a f^n\neq b\) for each \(f\in{\mathcal F}\), then \({\mathcal F}\) is a normal family. The author also proves that if \(f\) is a transcendental meromorphic function all of whose poles are of order at least \(p\) and all of whose zeros are of order at least \(t\), then \(f^{(k)}-af^n\) assumes every complex value infinitely often, where \(a\neq 0\) is a finite complex number and \(n,k,p\), and \(t\) are all positive integers satisfying the inequality \(n-1>{k+1\over p}+{1 \over t}\).