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.