  In this paper, we introduce the study of the Bohr phenomenon for a
quasi-subordination family of functions, and establish the classical Bohr's
inequality for the class of quasisubordinate functions. As a consequence, we
improve and obtain the exact version of the classical Bohr's inequality for
bounded analytic functions and also for $K$-quasiconformal harmonic mappings by
replacing the constant term by the absolute value of the analytic part of the
given function. We also obtain the Bohr radius for the subordination family of
odd analytic functions.
