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 $|f^{\prime }(z_1)f^{\prime }(z_2)|$ is established. Further, we prove a sharp distortion theorem involving the derivatives $f^{\prime }(z_1)$, $f^{\prime }(z_2)$ 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 $f(U)$.

arXiv commentSubstantial style improvement; a number of points better explained; language editing; 8 pages