Multidimensional analogues of the refined Bohr type inequalities

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk Dⁿ. We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables: one by replacing the constant term with the absolute value of the function, and another by replacing it with the square of the absolute value of the function. Furthermore, we establish multidimensional analogues of known results concerning the modulus of the derivative of analytic functions in the unit disk D, replacing the derivative with the radial derivative of holomorphic functions in Dⁿ. All of the established results are shown to be sharp.