Goluzin's extension of the Schwarz-Pick inequality

• Main Author: Yamashita Shinji
• Format: Article
• Language: English
• Published: SpringerOpen 1997-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Bounded holomorphic functions; Schwarz's inequality; Poincaré density
• Online Access: http://www.journalofinequalitiesandapplications.com/content/1/781962

<p/> <p>For a function <inline-formula><graphic file="1029-242X-1997-781962-i1.gif"/></inline-formula> holomorphic and bounded, <inline-formula><graphic file="1029-242X-1997-781962-i2.gif"/></inline-formula>, with the expansion <inline-formula><graphic file="1029-242X-1997-781962-i3.gif"/></inline-formula> in the disk <inline-formula><graphic file="1029-242X-1997-781962-i4.gif"/></inline-formula>, we set <inline-formula><graphic file="1029-242X-1997-781962-i5.gif"/></inline-formula> Goluzin's extension of the Schwarz-Pick inequality is that <inline-formula><graphic file="1029-242X-1997-781962-i6.gif"/></inline-formula> 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 Poincar&#233; metric.</p>