On the value distribution of $f^2f^{(k)}$

J. Aust. Math. Soc. 78 (2005), 17-26

On the value distribution of $f^2f^{(k)}$

Xiaojun Huang
Mathematics College
Sichuan University
Chengdu, Sichuan 610064
China
hx_jun@163.com

and

Yongxing Gu
Department of Mathematics
Chongqing University
Chongqing 400044
China
yxgu@cqu.edu.cn

Abstract

In this paper, we prove that for a transcendental meromorphic function

on the complex plane, the inequality $T(r,f)<6N (r,1/(f^2f^{(k)}-1))+S(r,f)$ holds, where

is a positive integer. Moreover, we prove the following normality criterion: Let $\mathcal{F}$ be a family of meromorphic functions on a domain

and let

be a positive integer. If for each $f\in \mathcal{F}$ , all zeros of

are of multiplicity at least

, and $f^2f^{(k)}\ne 1$ for $z\in D$ , then $\mathcal{F}$ is normal in the domain

. At the same time we also show that the condition on multiple zeros of

in the normality criterion is necessary.