Area Theorems and Quasiconformal Extensions of Harmonic Mappings with a Pole

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.