Quasiconformal Extension of Meromorphic Functions with High-Order Poles

TL;DR

This work studies meromorphic univalent functions with a pole of order $m$ at $p\in(0,1)$ that admit a $k$-quasiconformal extension, defining the generalized class $\Sigma^{(m)}_k(p)$. Using the Area Theorem and convolution techniques, it derives a generalized area-type inequality and explicit coefficient bounds that extend the prior $m=1$ results; it also proves a convolution (Hadamard-product) theorem showing $f\star g \in \Sigma^{(m)}_{\alpha_m}(p)$ with $\alpha_m=|a_{-1}||b_{-1}|k_1k_2(1-p)^{-2m}$. Additional contributions include a sharp Schwarzian-norm bound $\|S_f\| \le \frac{6k}{(1-p^2)^2}$ for $f\in\Sigma_k(p)$, a sufficient harmonic-mapping extension criterion on convex domains, and a generalization tying these results to the classical $m=1$ case of prior work. The paper also extends the area-method framework to a Hilbert-space viewpoint and discusses alternative proofs, highlighting the broader impact on quasiconformal extension theory and harmonic mappings.

Abstract

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].

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Remark 2.1
• Theorem 2.1
• Remark 2.2
• Corollary 2.1
• Remark 2.3
• proof : Proof of Lemma \ref{['Lem-2.1']}
• proof : Proof of Theorem \ref{['Th-1.2']}
• Theorem 2.2