Improved Bohr's phenomenon in quasi-subordination classes

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. Here, we consider the image of the origin and the boundary of the image of the unit disk under the function in question and let the distance between both play a central role in our theorems. 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 prove Bohr type theorems for functions subordinate to convex or univalent functions.