Journal of Inequalities and Applications
Volume 1 (1997), Issue 4, Pages 345-356

Goluzin’s extension of the Schwarz-Pick inequality

Shinji Yamashita

Department of Mathematics, Tokyo Metropolitan University Minami-Osawa, Hachioji, Tokyo 192-03, Japan

Received 23 December 1996

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.


For a function f holomorphic and bounded, |f|<1, with the expansion f(z)=a0+k=nakzk in the disk D={|z|<1},n1, we set Γ(z,f)=(1|z|2)|f(z)|/(1|f(z)|2)A=|an|/(1|a0|2),andϒ(z)=zn(z+A)/(1+Az). Goluzin’s extension of the Schwarz-Pick inequality is that Γ(z,f)Γ(|z|,ϒ),zD. 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.