On Korenblum’s constant ✩

Let A2(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c ∈ (0, 1), such that whenever |f (z)| |g(z)| (f, g ∈ A2(D)) in the annulus c < |z| < 1, then f g . In 1999 Hayman proved Korenblum’s conjecture. But the sharp value of c (we use γ to denote this sharp value) is still unknown. In this paper we give an upper bound on γ , that is, γ <0.67795, which improves an earlier result of the author.  2004 Elsevier Inc. All rights reserved.