On the coefficients estimate of K-quasiconformal harmonic mappings

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 $h(z)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$ and $g(z)={\sum }_{k=1}^{\infty }{b}_k{z}^k$ are analytic in the unit disk $|z|<1$, and the dilatation $\omega =g^{\prime }/h^{\prime }$ satisfies the condition $|\omega (z)|\le k<1$ for $\ID$, with $K=\frac{1+k}{1-k}\ge 1$. 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 ${\mathcal{S}}_H^0(K)$. 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..

20 pages; Revised for a journal