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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2512.09877 |
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| _version_ | 1866918242391425024 |
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| author | Bhowmik, Bappaditya Maity, Deblina Sugawa, Toshiyuki |
| author_facet | Bhowmik, Bappaditya Maity, Deblina Sugawa, Toshiyuki |
| contents | Let $f$ be a meromorphic univalent function on the open unit disk having a simple pole at $p\in (0,1)$ that extends continuously to the left half $\IT^{-}$ of the unit circle. In this article, we prove that the ratio of the length of the image of the vertical diameter $\IA$ of the unit disk to the length of the image of $\IT^{-}$ under the mapping $f$ is bounded by a constant depending only on $p.$ Next, we extend this result by considering any hyperbolic geodesic and any Jordan curve in $\D$ sharing the same endpoints. These results extend the classical Gehring-Hayman inequality to meromorphic univalent functions and also prove a conjecture posed by Bhowmik and Maity [Bull. Sci. Math. \textbf{199} (2025), \# 103583]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gehring-Hayman Inequality for Meromorphic Univalent Mappings Bhowmik, Bappaditya Maity, Deblina Sugawa, Toshiyuki Complex Variables Let $f$ be a meromorphic univalent function on the open unit disk having a simple pole at $p\in (0,1)$ that extends continuously to the left half $\IT^{-}$ of the unit circle. In this article, we prove that the ratio of the length of the image of the vertical diameter $\IA$ of the unit disk to the length of the image of $\IT^{-}$ under the mapping $f$ is bounded by a constant depending only on $p.$ Next, we extend this result by considering any hyperbolic geodesic and any Jordan curve in $\D$ sharing the same endpoints. These results extend the classical Gehring-Hayman inequality to meromorphic univalent functions and also prove a conjecture posed by Bhowmik and Maity [Bull. Sci. Math. \textbf{199} (2025), \# 103583]. |
| title | Gehring-Hayman Inequality for Meromorphic Univalent Mappings |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2512.09877 |