Length Distortion Of Curves Under Meromorphic Univalent Mappings

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 $I=\{z:\mathrm{Re}z=0;|\mathrm{Im}z|<1\}$ to the length of the image of the semi-circle $C^{\prime }=\{z:|z|=1;\mathrm{Re}z<0\}$ 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 $C^{\prime }$ with the corresponding arc of the unit circle passing through the point $-1$.

13 pages, submitted to a journal