A Sharp Inequality for Harmonic Diffeomorphisms of the Unit Disk
We extend the classical Schwarz–Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between the unit disk and a convex domain that solve the Beltrami equation of second order.
KeywordsHarmonic functions Convex domain Hardy spaces
Mathematics Subject ClassificationPrimary 31R05 Secondary 42B30
I am grateful to the referee for useful suggestions and corrections.