Bohr-type inequalities for classes of analytic maps and K-quasiconformal harmonic mappings

In this paper, a significant improvement has been achieved in the classical Bohr's inequality for the class $\mathcal{B}$ of analytic self maps defined on the unit disk $\mathbb{D}$. More precisely, we generalize and improve several Bohr-type inequalities by combining appropriate improved and refined versions of the classical Bohr's inequality with some methods concerning the area measure of bounded analytic functions in $\mathcal{B}$. In addition, we obtain Bohr-type and Bohr-Rogosinski-type inequalities for the subordination class and also for the class of $K$-quasiconformal harmonic mappings. All the results are proved to be sharp.