On the coefficients estimate of K-quasiconformal harmonic mappings
Peijin Li, Saminathan Ponnusamy
TL;DR
The paper addresses sharp coefficient bounds for $K$-quasiconformal harmonic mappings in the class ${S}_H^0(K)$ and its starlike, convex, and typically real subfamilies. By employing the shearing method and coefficient-dominance techniques, it derives unified, sharp bounds for $|a_n|$ and $|b_n|$ in terms of $A(n,k)$, $B(n,k)$ with a pivotal parameter $k_0= rac{2k}{1+k^2}$, and proves their monotonicity in $k$. The authors verify Conjecture B (disk containment) for several subfamilies and establish sharp coefficient estimates for convex $K$-quasiconformal harmonic mappings, including explicit $n=2$ bounds; they further extend these results to typically real and related mappings. Overall, the work provides substantial evidence supporting the Wang et al. conjecture for a broad class of $K$-quasiconformal harmonic mappings and supplies sharp, extremal-function-attained coefficients via the harmonic Koebe function family $P_k$.
Abstract
Recently, the Wang et al. \cite{wwrq} proposed a coefficient conjecture for the family ${\mathcal S}_H^0(K)$ of $K$-quasiconformal harmonic mappings $f = h + \overline{g}$ that are sense-preserving and univalent, where $h(z)=z+\sum_{k=2}^{\infty}a_kz^k$ and $g(z)=\sum_{k=1}^{\infty}b_kz^k$ are analytic in the unit disk $|z|<1$, and the dilatation $ω=g'/h'$ satisfies the condition $|ω(z)| \leq k<1$ for $\ID$, with $K=\frac{1+k}{1-k}\geq 1$. The main aim of this article is provide an affirmative answer in support of this conjecture by proving this conjecture for every starlike function (resp. close-to-convex function) from $\mathcal{S}^0_H(K)$. In addition, we verify this conjecture also for typically real $K$-quasiconformal harmonic mappings. Also, we establish sharp coefficients estimate of convex $K$-quasiconformal harmonic mappings. By doing so, our work provides a document in support of the main conjecture of Wang et al..