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 $\partial \D^{{+}={z\in\IC:|z|=1,} {\rm{Im}}\,z&gt;0}$. 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 $\partial \D<sup>{+}$ 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 $I={z: {\rm{Re}}\,z=0; ~|{\rm{Im}}\,z|&lt;1}$ to the length of the image of the semi-circle $C&#39;={z: |z|=1;~ {\rm{Re}}\,z&lt;0}$ 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 $C&#39;$ with the corresponding arc of the unit circle passing through the point $-1$.