Operator-valued multivariable Bohr type inequalities are obtained for:
(i) a class of noncommutative holomorphic functions on the open unit ball of , generalizing the analytic functions on the open unit disc;
(ii) the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra ;
(iii) a class of noncommutative selfadjoint harmonic functions on the open unit ball of , generalizing the real-valued harmonic functions on the open unit disc;
(iv) the Cuntz-Toeplitz algebra , the reduced (resp. full) group -algebra (resp. ) of the free group with generators;
(v) a class of analytic functions on the open unit ball of .
The classical Bohr inequality is shown to be a consequence of Fejér's inequality for the coefficients of positive trigonometric polynomials and Haager- up-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.
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