Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for: a class of noncommutative holomorphic functions, generalizing the analytic functions on the open unit disc; the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra; a class of noncommutative selfadjoint harmonic functions, generalizing the real-valued harmonic functions on the open unit disc; Cuntz-Toeplitz algebras ; the reduced (resp.full) group C*-algebra of the free group with n generators; a class of analytic functions on the open unit ball of C^n. The classical Bohr inequality is shown to be a consequence of Fejer's inequality for the coefficients of positive trigonometric polynomials and Haagerup-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.