Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for:

(i)

a class of noncommutative holomorphic functions on the open unit ball of $B(\mathcal{H})^n$, generalizing the analytic functions on the open unit disc;

(ii)

the noncommutative disc algebra $\mathcal{A}_n$ and the noncommutative analytic Toeplitz algebra $F_n^\infty$;

(iii)

a class of noncommutative selfadjoint harmonic functions on the open unit ball of $B(\mathcal{H})^n$, generalizing the real-valued harmonic functions on the open unit disc;

(iv)

the Cuntz-Toeplitz algebra $C^*(S_1,\ldots, S_n)$, the reduced (resp. full) group $C^*$-algebra $C_{red}^*(\mathbb{F}_n)$ (resp.  $C^*(\mathbb{F}_n)$) of the free group with $n$ generators;

(v)

a class of analytic functions on the open unit ball of $\mathbb{C}^n$.

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.