Three radii associated to Schur functions on the polydisk

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.