Three radii associated to Schur functions on the polydisk

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0.3006"> <mml:semantics> <mml:mn>0.3006</mml:mn> <mml:annotation encoding="application/x-tex">0.3006</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we improve some estimates of P. G. Dixon on Agler norms of homogeneous polynomials using some modern inequalities.