Generalizations of Bohr inequality for Hilbert space operators

Let B ( H ) be the space of all bounded linear operators on a complex separable Hilbert space H . Bohr inequality for Hilbert space operators asserts that for A , B ∈ B ( H ) and p , q > 1 real numbers such that 1 / p + 1 / q = 1 , | A + B | 2 ⩽ p | A | 2 + q | B | 2 with equality if and only if B = ( p − 1 ) A . In this paper, a number of generalizations of Bohr inequality for operators in B ( H ) are established. Moreover, Bohr inequalities are extended to multiple operators and some related inequalities are obtained. The results in this paper generalize results known so far. The idea of transforming problems in operator theory to problems in matrix theory, which are easy to handle, is the key role.