Normal family of compositions of holomorphic functions and their high order derivatives (Q1418271)

Let \(F=\{f\}\) be a family of entire functions and \(f^{(k)}\) the \(k-\)th derivative of \(f\). The author discusses the relationship between the normality of \(F\) and that of the family \(\{f^{(k)}\circ f:\; f\in F\}\) or the family \(\{ f\circ f^{(k)}:\; f\in F\}\). Three results are obtained. One of them is: if for each \(f\in F\), the order of zeroes of \(f\) is not smaller than \(k\) and \(| f^k(0)| <1\), \(| f(0)| <1\), and if both \(\{f^{(k)}\circ f:\; f\in F\}\) and \(\{ f\circ f^{(k)}:\; f\in F\}\) are normal in a domain \(D\), then \(F\) is normal in \(D\). This result generalizes a result obtained by Jianhua Zheng etc in 1996.