The Distortion in Conformal Mapping in Hyperbolic Geometry

Abstract

Nehari-type inequalities for normalized univalent functions are combined with elementary monotonicity arguments to give quick and simple proofs for numerous sharp two-point distortion theorems for conformal maps. These inequalities are used to prove two theorems of sharp two-point distortion theorems for conformal maps from the unit disk into the unit disk equipped with hyperbolic geometry.