Length Distortion Of Curves Under Meromorphic Univalent Mappings

TL;DR

This paper extends known length distortion bounds for conformal maps on the unit disk to include meromorphic univalent functions with poles, analyzing the ratio of lengths of specific boundary and interior curves.

Contribution

It generalizes previous results by introducing poles in the conformal maps and considering more general boundary curves, including hyperbolic geodesics.

Findings

Established upper bounds for length ratios involving meromorphic univalent maps.

Extended classical results to functions with simple poles inside the disk.

Generalized length distortion inequalities to broader classes of boundary curves.

Abstract

Let $f$ be a conformal (analytic and univalent) map defined on the open unit disk $\D$ of the complex plane $\IC$ that is continuous on the semi-circle $\partial \D^{+}=\{z\in\IC:|z|=1, {\rm{Im}}\,z>0\}$. The existence of a uniform upper bound for the ratio of the length of the image of the horizontal diameter $(-1,1)$ to the length of the image of $\partial \D^{+}$ under $f$ was proved by Gehring and Hayman. In this article, at first, we generalize this result by introducing a simple pole for $f$ in $\D$ and considering the ratio of the length of the image of the vertical diameter $I=\{z: {\rm{Re}}\,z=0; ~|{\rm{Im}}\,z|<1\}$ to the length of the image of the semi-circle $C'=\{z: |z|=1;~ {\rm{Re}}\,z<0\}$ under such $f$. Finally, we further generalize this result by replacing the vertical diameter $I$ with a hyperbolic geodesic symmetric with respect to the real line, and by replacing $C'$ with the corresponding arc of the unit circle passing through the point $-1$.