  In this article, we establish the Bohr inequalities for the sense-preserving
$K$-quasiconformal harmonic mappings defined in the unit disk $\mathbb{D}$
involving classes of Ma-Minda starlike and convex univalent functions, usually
denoted by $\mathcal{S}^*(\psi)$ and $\mathcal{C}(\psi)$ respectively, and for
$\log (f(z)/z)$ where $f$ belongs to the Ma-Minda classes or satisfies certain
differential subordination. We also estimate Logarithmic coefficient's bounds
for the functions in $\mathcal{C}(\psi)$ for the case $\psi(\mathbb{D})$ be
convex.
