A slight improvement to Korenblum's constant
Journal
Journal of Mathematical Analysis and Applications
Journal Volume
337
Journal Issue
1
Pages
464-465
Date Issued
2008
Author(s)
Abstract
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) | in the annulus c < | z | < 1, then {norm of matrix} f (z) {norm of matrix} ≤ {norm of matrix} g (z) {norm of matrix}. This conjecture had been solved by Hayman [W.K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19 (1999) 195-205. [1]], but the constant c in that paper is not optimal. Since then, there are many papers dealing with improving the upper and lower bounds for the best constant c. For example, in 2004 C. Wang gave an upper bound on c, that is, c < 0.67795, and in 2006 A. Schuster gave a lower bound, c > 0.21. In this paper we slightly improve the upper bound for c. © 2007 Elsevier Inc. All rights reserved.
Type
journal article