  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.
