On Littlewood's Constants

Abstract

In two papers, Littlewood studied seemingly unrelated constants: (i) the best α such that for any polynomial f, of degree n, the areal integral of its spherical derivative is at most ·nα, and (ii) the extremal growth rate rβ of the length of Green's equipotentials for simply connected domains. These two constants are shown to coincide, thus greatly improving known estimates on α. 2000 Mathematics Subject Classification 30C50 (primary), 30C85, 30D35 (secondary