  Recently the present authors established refined versions of Bohr's
inequality in the case of bounded analytic functions. In this article, we state
and prove a generalization of these results in a reformulated "distance form"
version and thereby we extend the refined versions of the Bohr inequality for
the class of the quasi-subordinations which contains both the classes of
majorization and subordination as special cases. As a consequence, we obtain
several new results.
