Recently, the Wang et al. [15] 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 \(\omega =g'/h'\) satisfies the condition \(|\omega (z)| \le k<1\) for \({\mathbb D}\) , with \(K=\frac{1+k}{1-k}\ge 1\) . The main aim of this article is to 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..
