AI 中文总结
本文证明了K-拟共形调和映射相关的系数猜想成立，确定了其所属Hardy空间与加权Bergman空间的$p$范围，改进了相关估计并讨论了奇性映射的平行结果。
AI 中文摘要
近来，Li和Ponnusamy证明了Wang等人提出的关于单位圆盘内保定向K-拟共形单叶调和映射类$\boldsymbol{\text{S}}^0_H(K)$中几个重要几何子类的系数猜想。本文中，我们证明该猜想对一类通过拟从属定义的K-拟共形调和映射仍然成立。此外，我们确定了$p>0$的范围，使得这类映射属于Hardy空间$\boldsymbol{\text{h}}^p$和加权Bergman空间$\boldsymbol{\text{a}}^{\boldsymbol{p}}_{\boldsymbol{\beta}}$（其中$\beta>-1$）。我们的Hardy空间结果对Pavlović提出的问题取得了重要进展，而Bergman空间结果则改进了Das和Rasila得到的范围，将先前已知的边界值翻倍。此外，我们还得到了该子类的精确增长和积分均值估计，改进了早期结果，并为Das等人提出的一个开放问题提供了进一步证据。我们还讨论了奇性K-拟共形调和映射的平行结果。
英文摘要
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 $\mathbf{a}^{\mathbf{p}}_{\boldsymbolβ}$, for $β>-1$. Our Hardy space result makes significant progress toward a problem posed by Pavlović, 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.
