View article

Let be the algebra of all bounded linear operators on a complex Hilbert space . In this paper, we first establish several sharp improved and refined versions of the Bohr’s inequality for the functions in the class of bounded analytic functions from the unit disk into . For the complete circular domain , we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002–1053] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr’s inequalities for operator valued functions in Q.