  The present article concerns the Bohr radius for $K$-quasiconformal
sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk
$\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic
function $\varphi$, and the purpose is to look into two cases: when $\varphi$
is convex, or a general univalent function in $\ID$. The results state that if
$h(z) =\sum_{n=0}^{\infty}a_n z^n$ and $g(z)=\sum_{n=1}^{\infty}b_n z^n$, then
$$\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist
(\varphi(0),\partial\varphi(\ID)) ~\mbox{ for $r\leq r^*$} $$ and give
estimates for the largest possible $r^*$ depending only on the geometric
property of $\varphi (\ID)$ 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 $K\rightarrow \infty$.
