A FUNDAMENTAL INEQUALITY AND ITS APPLICATION

Let f(z) be meromorphie in |z|k+4+2/k].In this note,a fundamental inequality is established such that thecharacreristic function T(r,f)can be limibd by N(r,1/f)and _(τ-1)(r,1/(f~(k)-1).As anapplication,the following criterion for normality is also proved:Let be a family ofmeromorphic functions in a region D.If for every f(z)∈ ,f(z)≠0 and all the zeros off~(k)(z)-1 are of multiplicity >k+4+2/k]in D,then is normal there.

A Fundamental Inequality and Its Application

Let f(z) be meromorphic in |z|k+4+\frac{2}{k}] In this note, a fundamental inequality is established such that the characreristic function T(r, f)can be limited by N (r,\frac{1}{f}) and ${\bar N_{\tau - 1}}(r,\frac{1}{{{f^{(k)}} - 1}})\]$. As an application, the following criterion for normality is also proved: Let F be a family of meromorphic functions in a region D. If for every f(z)\in F,f(z)\ne 0 and all the zeros of $f^(k)(z)-1$ are of multiplicity >k+4+\frac{2}{k}] in D, then F is normal there.