Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions

Abstract

For a meromorphic function f in the unit disk U = {z : |z| < 1} and arbitrary points z 1 , z 2 in U distinct from the poles of f , a sharp upper bound on the product |f ′ (z 1 )f ′ (z 2 )| is established. Further, we prove a sharp distortion theorem involving the derivatives f ′ (z 1 ), f ′ (z 2 ) and the Schwarzian derivatives S f (z 1 ), S f (z 2 ) for z 1 , z 2 ∈ U . Both estimates hold true under some geometric restrictions on the image f (U ).