Bohr inequalities for free holomorphic functions on polyballs

TL;DR

This paper extends classical inequalities like Bohr, Caratheodory, and Fejer to the setting of free holomorphic functions with operator coefficients on polyballs, using multivariable operator theory.

Contribution

It introduces multivariable analogues of several classical inequalities for free holomorphic functions with operator coefficients on polyballs, expanding the theoretical framework.

Findings

Established Bohr inequalities for free holomorphic functions on polyballs.

Derived analogues of Caratheodory, Fejer, and Egervary-Szazs inequalities.

Provided multivariable versions of Landau's and Bohr's inequalities using numerical radius.

Abstract

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball. In addition, we obtain analogues of Caratheodory, Fejer, and Egervary-Szazs inequalities for free holomorhic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.