Moment duality and an improved lower bound for Korenblum's constant

Abstract

We introduce a moment-duality method for Korenblum's maximum principle in the Bergman space $A^2(\mathbb{D})$. Starting from an annular coefficient estimate of Wang, we show that admissibility of a constant~$c$ follows from the existence of a probability measure on $[c^2,1]$ whose ordinary and weighted moments lie on opposite sides of the Bergman moments $1/(k+1)$. This converts the norm comparison into a positive moment problem. We then give an explicit measure, consisting of eight atoms with rational data and Lebesgue measure on a terminal interval, for which the required inequalities admit a rigorous ball-arithmetic certificate. Consequently, \[ c_2\geq 0.4263, \] improving Wang's recent lower bound $c_2\ge 0.3554$.

8 pages