Quasiconformal Extension of Meromorphic Functions with High-Order Poles

TL;DR

This paper investigates quasiconformal extensions of meromorphic functions with high-order poles, providing new inequalities, extension criteria, and convolution theorems that advance the understanding of such functions' geometric properties.

Contribution

It introduces generalized area inequalities, extends convolution theorems for meromorphic functions with poles, and establishes criteria for quasiconformal extendability and harmonic mappings.

Findings

Derived a generalized area-type inequality for meromorphic functions.

Extended convolution theorem to a modified Hadamard product for functions with poles.

Provided sharp bounds for Schwarzian norms in the class.

Abstract

In this paper, we study the class ${\Sigma^{(m)}(p)}$ 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 $\Sigma^{(m)}(p)$. We also extend the convolution theorem to a modified Hadamard product of $m$ functions, $f_j \in \Sigma^{(m)}_{k_j}(p)$, determining sufficient conditions for the product to be in ${\Sigma^{(m)}_{\alpha}(p)}$, with $\alpha$ 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 $f \in \Sigma_k(p)$ (the $m=1$ case). These findings improve upon existing results of [{\em Proc. Amer. Math. Soc.}, {144}(6) (2016), 2593--2601].