Criteria for quasiconformal extensions of harmonic mappings

We obtain a criterion for quasiconformal extension of sense-preserving harmonic mappings defined on a bounded convex domain in the complex plane. Thereafter, we extend the harmonic analog of Noshiro–Warschawski univalence criterion to obtain a sufficient condition for quasiconformal extensions of harmonic mappings. Finally, as an application of one of our obtained results, we found neighborhood of a bi-Lipschitz holomorphic function defined on the open unit disk of the complex plane that consists of harmonic mappings having quasiconformal extensions to the whole complex plane.