Length Distortion Of Curves Under Meromorphic Univalent Mappings
| Title: | Length Distortion Of Curves Under Meromorphic Univalent Mappings |
|---|---|
| Authors: | Bhowmik, Bappaditya, Maity, Deblina |
| Publication Year: | 2024 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Complex Variables, 30C35, 30C20, 30C55 |
| More Details: | 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 $-1$. Comment: 13 pages, submitted to a journal |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2412.19075 |
| Accession Number: | edsarx.2412.19075 |
| Database: | arXiv |
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