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arXiv · 2606.03306

Area Theorems and Quasiconformal Extensions of Harmonic Mappings with a Pole

Zhijun Chen·Limei Wang
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Abstract

In this paper, we study the class Σ_{H}^{k}(p) of sense-preserving univalent harmonic mappings in the unit disk \mathbb{D} that possess a simple pole at p\in[0,1) and admit a k-quasiconformal extension to the extended complex plane for k\in[0,1). In 2024, Bhowmik and Satpati established an area theorem and derived a sufficient condition for the k-quasiconformal extension of harmonic mappings belonging to Σ_{H}^{k}(p) without logarithmic terms. Motivated by their work, we investigate the corresponding problem when a logarithmic singularity is present. Our main contributions are two-fold: we first prove a generalized area theorem for all mappings in Σ_{H}^{k}(p); we then obtain a sufficient condition for sense-preserving univalent harmonic mappings in \mathbb{D} to admit explicit k-quasiconformal extensions. These results extend the aforementioned work to the setting where logarithmic singularities are allowed.

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Zhijun Chen, Limei Wang. 2026-06-10. Area Theorems and Quasiconformal Extensions of Harmonic Mappings with a Pole. https://arxiv.org/abs/2606.03306

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