Goluzin’s extension of the Schwarz-Pick inequality

Journal of Inequalities and Applications

关键词: Bounded holomorphic functions;    Schwarz’;    s inequality;    Poincaré;    density.;   

DOI  :  10.1155/S1025583497000246

来源: DOAJ

【 摘 要 】

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 equalitycondition. For a holomorphic map from a hyperbolic plane domain into another, one can provea similar result in terms of the Poincaré metric.

【 授权许可】

Unknown