Three radii associated to Schur functions on the polydisk

TL;DR

The paper analyzes three radii associated to bounded analytic functions on the polydisk—the Bohr radius $K_d$, the Bohr-Agler radius $K(\\mathcal{A}_d)$, and the Schur-Agler radius $SA_d$—and compares their growth with respect to the dimension. Using Agler transfer-function realizations, operator-inequality criteria, and probabilistic constructions (Kahane–Salem–Zygmund), it derives explicit lower and upper bounds and asymptotic trends: $K(\\mathcal{A}_d) \\ge \\frac{1}{\\sqrt{d}+2}$, $SA_d \\ge \\frac{1}{\\sqrt{d-1}}$, and $K(\\mathcal{A}_d)$, $SA_d$ exhibit growth adjacent to the classical Bohr radius; it also shows $K(\\mathcal{A}_d)$ is bounded above by $e^{1/2}\\sqrt{(\\log d)/d}$ up to subleading factors. A corollary gives $K_2 \\ge 0.3006$ on the bidisk. Finally, the work improves Dixon’s bounds on Agler norms through refined Bohnenblust–Hille and polarization estimates, clarifying the landscape of these radii and highlighting remaining gaps between lower and upper bounds.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Theorem 3.1: grinshpan Theorem 1.3
• Lemma 3.2