Spherical Derivatives and Normal Functions

Abstract Note: Please see pdf for full abstract with equations. Let $k$ be a positive integer and $C>0$ an constant, and let $\varphi:[0,1)\to (0,\infty)$ be smoothly increasing. A meromorphic function $f$ in the unit disk is called a $\varphi$-normal function if $\sup_{z\in \Delta}\frac{f^{\#}(z)}{\varphi(|z|)}<\infty$. In this paper, we prove that $f$ is a $\varphi$-normal function if $|f^{(k)}(z)|/(\varphi(|z|)^{k}(1+|f(z)|^{k+1}))\geq C$ ($z\in \Delta)$. Also, the similar results for normal functions and $\alpha$-normal functions are given. 2000Mathematics subject classification:30D45