On quasiconformal extension of harmonic mappings with nonzero pole

Let Sigma(k)(H)(p) be the class of sense-preserving univalent harmonic mappings defined on the open unit disk D of the complex plane with a simple pole at z = p is an element of (0,1) that have k-quasiconformal extensions (0 <= k < 1) to the extended complex plane. We first derive a sufficient condition for harmonic mappings defined on D with pole at z = p is an element of (0,1) to belong in the class Sigma(k)(H)(p). As a consequence of this, we derive a convolution result involving functions in Sigma(k)(H)(i)(p), 0 <= k(i) < 1 for i=1,2. We also consider harmonic mappings with a nonzero pole defined on a linearly connected domain Omega subset of D and prove criteria for univalence and quasiconformal extensions for such mappings.