Shinji Yamashita

Copyright © 1997 Shinji Yamashita. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For a function f holomorphic and bounded, |f|<1, with the expansion f(z)=a0+∑k=n∞akzk in the disk D={|z|<1},n≥1, we set Γ(z,f)=(1−|z|2)|f′(z)|/(1−|f(z)|2)A=|an|/(1−|a0|2),  and  ϒ(z)=zn(z+A)/(1+Az). Goluzin’s extension of the Schwarz-Pick inequality is that Γ(z,f)≤Γ(|z|,ϒ),  z∈D. We shall further improve Goluzin’s inequality with a complete description on the equality condition. For a holomorphic map from a hyperbolic plane domain into another, one can prove a similar result in terms of the Poincaré metric.