The Bohr's Phenomenon involving multiple Schwarz functions

The primary objective of this paper is to establish several sharp versions of Bohr inequalities for bounded analytic functions in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ involving multiple Schwarz functions. Moreover, we obtain an improved version of the classical Rogosinski inequality for analytic functions in $\mathbb{D}$.