On the growth of nonvanishing analytic functions

In this article by a new and simple method we derive new and old coefficient bounds and distortion theorems for functions G(z) analytic in the unit disk and satisfying \(\sup\limits _{|z|\le 1}(|G(z)|(1-|z|^2) ^\alpha )\leqq 1\) for some \(\alpha \geqq 0\). These results enable us to prove analogous theorems for the subclasses consisting of nonvanishing functions.