Resumen de Multivariable Bohr inequalities

Gelu Popescu

• 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.