Bohr radius for subordination and $K$-quasiconformal harmonic mappings

The present article concerns the Bohr radius for $K$-quasiconformal sense-preserving harmonic mappings $f=h+\bar{g}$ in the unit disk $\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic function $\phi$, and the purpose is to look into two cases: when $\phi$ is convex, or a general univalent function in $\text{ID}$. The results state that if $h(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ and $g(z)={\sum }_{n=1}^{\mathrm{\infty }}{b}_n{z}^n$, then \sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist (\varphi(0),\partial\varphi(\ID)) ~ for $r\leq r^*$ and give estimates for the largest possible $r^{\ast }$ depending only on the geometric property of $\phi (\text{ID})$ and the parameter $K$. Improved versions of the theorems are given for the case when $b_1=0$ and corollaries are drawn for the case when $K\to \mathrm{\infty }$.