Bohr inequalities for free holomorphic functions on polyballs

Abstract

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball Bn, n=(n1,…,nk)∈Nk, which is a noncommutative analogue of the scalar polyball (Cn1)1×⋯×(Cnk)1. The Bohr radius Kmh(Bn) (resp. Kh(Bn)) associated with the multi-homogeneous (resp. homogeneous) power series expansions of the free holomorphic functions are the main objects of study in this paper. We extend a theorem of Bombieri and Bourgain for the disc D:={z∈C:|z|<1} to the polyball, and obtain the estimations 13k<Kmh(Bn)<2log⁡kk if k>1, extending Boas-Khavinson result for the scalar polydisc Dk.With respect to the homogeneous power series expansion, we prove that Kh(Bn)=1/3, extending the classical result, and obtain analogues of Carathéodory, Fejér, and Egerváry-Szász inequalities for free holomorphic 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.When specialized to the regular polydisc Dk (which corresponds to the case n1=⋯=nk=1), we obtain new results concerning Bohr, Landau, Fejér, and Harnack inequalities for operator-valued holomorphic functions and k-pluriharmonic functions on the scalar polydisc Dk. The results of the paper can be used to obtain Bohr type inequalities for the noncommutative ball algebra An, the Hardy algebra Fn∞, and the C⁎-algebra C⁎(S), generated by the universal model S of the polyball Bn.