Coefficients and Integral Mean Estimates for $K$-Quasiconformal Harmonic Mappings

Abstract

Recently, Li and Ponnusamy~\cite{LiPonnusamy2025} established the coefficient conjecture proposed by Wang et al.~\cite{Wang2024} for several prominent geometric subclasses of $\mathcal{S}^0_H(K)$, the class of sense-preserving $K$-quasiconformal univalent harmonic mappings in the unit disk. In this paper, we show that the conjecture continues to hold for a class of $K$-quasiconformal harmonic mappings defined via quasi-subordination. Further, we determine the range of $p>0$ for which such mappings belong to the Hardy space ${\bf h}^p$ and the weighted Bergman space $\mathbf{a}^{\mathbf{p}}_{\boldsymbol{\beta}}$, for $\beta>-1$. Our Hardy space result makes significant progress toward a problem posed by Pavlovi\'{c}, while the Bergman space result sharpens the range obtained by Das and Rasila~\cite{DasRasila}, doubling the previously known bounds. In addition, we obtain refined growth and integral mean estimates for the subclass, improving earlier results and providing further evidence toward an open problem raised by Das et al.~\cite{DasRasila2025}. Parallel results are also discussed for odd $K$-quasiconformal harmonic mappings.