Harmonic K-quasiconformal Koebe functions: construction and application to Pavlovic's problem

We first construct the harmonic K-quasiconformal Koebe functions, filling a long-standing foundational gap in geometric function theory. This construction provides a unified parametric candidate extremal function framework for conformal mappings, quasiconformal mappings, and harmonic mappings, and we formulate related conjectures for the extremal theory of harmonic K-quasiconformal mappings. By combining this construction with Astala and Koskela's Hp-theory for quasiconformal mappings, we establish a sharp result concerning the optimal order of harmonic K-quasiconformal mappings with bounded Schwarzian norm in harmonic Hardy spaces. Motivated by the work of Chuaqui, Hernandez, and Martin [Math. Ann. 367, 1099-1122, 2017], this result gives a partial solution to Pavlovic's 2014 open problem on the embeddings of harmonic quasiconformal mappings into Hardy spaces, and outlines a path toward its complete solution.