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
Yongxing Gu
  Department of Mathematics
  Chongqing University
  Chongqing 400044

In this paper, we prove that for a transcendental meromorphic function $f(z)$ on the complex plane, the inequality $T(r,f)<6N (r,1/(f^2f^{(k)}-1))+S(r,f)$ holds, where $k$ is a positive integer. Moreover, we prove the following normality criterion: Let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and let $k$ be a positive integer. If for each $f\in \mathcal{F}$, all zeros of $f$ are of multiplicity at least $k$, and $f^2f^{(k)}\ne 1$ for $z\in D$, then $\mathcal{F}$ is normal in the domain $D$. At the same time we also show that the condition on multiple zeros of $f$ in the normality criterion is necessary.
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