Three radii associated to Schur functions on the polydisk

TL;DR

This paper investigates three radii related to bounded analytic functions on the polydisk, providing bounds and asymptotic behaviors, and connecting these radii to classical results like the Bohr radius.

Contribution

It establishes explicit bounds and asymptotic growth rates for the Bohr-Agler and Schur-Agler radii, linking them to the classical Bohr radius and improving existing estimates.

Findings

Bound the Bohr-Agler radius explicitly.

Show Schur-Agler radius has similar growth to the Bohr radius.

Lower bound for the Bohr radius on the bidisk is 0.3006.

Abstract

This article examines three radii associated to bounded analytic functions on the polydisk: the well-known Bohr radius, the Bohr-Agler radius, and the Schur-Agler radius. We prove explicit upper and lower bounds for the Bohr-Agler radius, an explicit lower bound for the Schur-Agler radius, and an asymptotic upper bound for the Schur-Agler radius. The Bohr-Agler radius obeys the same (known) asymptotic as the Bohr radius while we show the Schur-Agler radius is roughly of the same growth as the Bohr radius. As a corollary, we bound the Bohr radius on the bidisk below by 0.3006. Finally, we improve some estimates of P.G. Dixon on Agler norms of homogeneous polynomials using some modern inequalities.