Length Distortion Of Curves Under Meromorphic Univalent Mappings

Let $f$ be a conformal (analytic and univalent) map defined on the open unit disk $\text{D}$ of the complex plane $\text{IC}$ that is continuous on the semi-circle $\mathrm{\partial }{\text{D}}^{+}=\{z\in \text{IC}:\mathrm{\mid }z\mathrm{\mid }=1,\mathrm{I}\mathrm{m}z>0\}$. 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 $\mathrm{\partial }{\text{D}}^{+}$ 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 $\text{D}$ and considering the ratio of the length of the image of the vertical diameter $I=\{z:\mathrm{R}\mathrm{e}z=0;\mathrm{\mid }\mathrm{I}\mathrm{m}z\mathrm{\mid }<1\}$ to the length of the image of the semi-circle $C^{\mathrm{\prime }}=\{z:\mathrm{\mid }z\mathrm{\mid }=1;\mathrm{R}\mathrm{e}z<0\}$ 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 $C^{\mathrm{\prime }}$ with the corresponding arc of the unit circle passing through the point $-1$.