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

TL;DR

This paper establishes sharp bounds on the derivatives and Schwarzian derivatives of meromorphic functions in the unit disk, under geometric restrictions, extending classical distortion theorems with precise inequalities.

Contribution

It introduces new sharp upper bounds for the product of derivatives and a distortion theorem involving derivatives and Schwarzian derivatives of meromorphic functions.

Findings

Sharp upper bounds on |f'(z_1)f'(z_2)|

Distortion theorem involving derivatives and Schwarzian derivatives

Results hold under geometric restrictions on f(U)

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 $z_1,z_2\in U$. Both estimates hold true under some geometric restrictions on the image $f(U)$.