# 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.

## Key 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 $f(U)$.