Quasiconformal Extension of Meromorphic Functions with High-Order Poles
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arXiv
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| Formato: | Preprint |
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2025
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| author | Ahamed, Molla Basir Roy, Partha Pratim |
| author_facet | Ahamed, Molla Basir Roy, Partha Pratim |
| contents | In this paper, we study the class ${Σ^{(m)}(p)}$ of meromorphic univalent functions $f$ in $\mathbb{D}$ with a pole of order ${m \geq 1}$ at $p \in (0,1)$, admitting a $k$-quasiconformal extension ($0 \leq k < 1$) to $\widehat{\mathbb{C}}$. Using the Area Theorem and convolution methods, we establish a generalized area-type inequality and derive explicit analytic membership conditions for $Σ^{(m)}(p)$. We also extend the convolution theorem to a modified Hadamard product of $m$ functions, $f_j \in Σ^{(m)}_{k_j}(p)$, determining sufficient conditions for the product to be in ${Σ^{(m)}_α(p)}$, with $α$ defined by $k_j$ and $p$. Further results include a sufficient criterion for sense-preserving harmonic mappings on convex domains to admit quasiconformal extensions, and the sharp Schwarzian norm for $f \in Σ_k(p)$ (the $m=1$ case). These findings improve upon existing results of [{\em Proc. Amer. Math. Soc.}, {144}(6) (2016), 2593--2601]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasiconformal Extension of Meromorphic Functions with High-Order Poles Ahamed, Molla Basir Roy, Partha Pratim Complex Variables Primary 30A10, 30B10, 30C35, 30C62, 30H05, 31A05, Secondary 30C45 In this paper, we study the class ${Σ^{(m)}(p)}$ of meromorphic univalent functions $f$ in $\mathbb{D}$ with a pole of order ${m \geq 1}$ at $p \in (0,1)$, admitting a $k$-quasiconformal extension ($0 \leq k < 1$) to $\widehat{\mathbb{C}}$. Using the Area Theorem and convolution methods, we establish a generalized area-type inequality and derive explicit analytic membership conditions for $Σ^{(m)}(p)$. We also extend the convolution theorem to a modified Hadamard product of $m$ functions, $f_j \in Σ^{(m)}_{k_j}(p)$, determining sufficient conditions for the product to be in ${Σ^{(m)}_α(p)}$, with $α$ defined by $k_j$ and $p$. Further results include a sufficient criterion for sense-preserving harmonic mappings on convex domains to admit quasiconformal extensions, and the sharp Schwarzian norm for $f \in Σ_k(p)$ (the $m=1$ case). These findings improve upon existing results of [{\em Proc. Amer. Math. Soc.}, {144}(6) (2016), 2593--2601]. |
| title | Quasiconformal Extension of Meromorphic Functions with High-Order Poles |
| topic | Complex Variables Primary 30A10, 30B10, 30C35, 30C62, 30H05, 31A05, Secondary 30C45 |
| url | https://arxiv.org/abs/2511.09121 |