An Upper Bound on Korenblum’s Constant

Abstract.

Let $A^2 (\mathbb{D})$ be the Bergman space over the open unit disk $\mathbb{D}$ in the complex plane. Korenblum conjectured that there is an absolute constant $c \in (0,1)$, such that whenever $|f(z)| \leq |g(z)| (f, g \in A^2 (\mathbb{D}))$ in the annulus $c < |z| < 1$, then $||f|| \leq ||g||$. In 1999 Hayman proved Korenblum’s conjecture. But the sharp value of c (we use γ to denote this sharp value and call it Korenblum’s constant) is still unknown. In this paper we give an upper bound on γ, that is, γ < 0.685086, which improves an earlier result of the author.