In this paper, we first establish a version of multidimensional analogues of the refined Bohr’s inequality. Then we establish two versions of multidimensional analogues of improved Bohr’s inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr’s inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp.

In this paper, we investigate the Bohr-type radii for several different forms of Bohr-type inequalities of analytic functions in the unit disk, we also investigate the Bohr-type radius of the alternating series associated with the Taylor series of analytic functions. We will prove that most of the results are sharp.