  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.

Complex Variables