On quasiconformal extension of harmonic mappings with nonzero pole - Bhowmik, Bappaditya; Satpati, Goutam (Comptes Rendus. Mathématique)
Let $Σ _H^k(p)$ be the class of sense-preserving univalent harmonic mappings defined on the open unit disk $\mathbb{D}$ of the complex plane with a simple pole at $z=p \in (0,1)$ that have $k$-quasiconformal extensions ($0\le k<1$) to the extended complex plane. We first derive a sufficient condition for harmonic mappings defined on $\mathbb{D}$ with pole at $z=p\in (0,1)$ to belong in the class $Σ _H^k(p)$. As a consequence of this, we derive a convolution result involving functions in $\smash{Σ _H^{k_i}(p)}$, $0\le k_i<1$ for $i=1,2$. We also consider harmonic mappings with a nonzero pole defined on a linearly connected domain $Ω \subset \mathbb{D}$ and prove criteria for univalence and quasiconformal extensions for such mappings.