Szegő’s theorem on a bidisc

Abstract:

G. Szegö showed that \[ \inf \;\int _0^{2\pi } {|1 - f{|^2}w d\theta /2\pi = \exp \;\int _0^{2\pi } {\log w d\theta /2\pi } } \] where $f$ ranges over analytic polynomials with mean value zeros. We study extensions of the Szegö’s theorem on the disc to the bidisc. We show that the quantity is a mixed form of an arithmetic mean and a geometric one of $w$ in some special cases.