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 z1,z2 in U distinct from the poles of f, a sharp upper bound on the product |f′(z1)f′(z2)| is established. Further, we prove a sharp distortion theorem involving the derivatives f′(z1), f′(z2) and the Schwarzian derivatives Sf(z1), Sf(z2) for z1,z2∈U. Both estimates hold true under some geometric restrictions on the image f(U).