Behavior of the constant in Korenblum's maximum principle

✍ Scribed by Chunije Wang

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 125 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200510616

✦ Synopsis

Abstract

Let A^p^ (𝔻) (p ≥ 1) be the Bergman space over the open unit disk 𝔻 in the complex plane. Korenblum's maximum principle states that there is an absolute constant c ∈ (0, 1) (may depend on p), such that whenever |f (z)| ≤ |g (z)| (f, g ∈ A^p^ (𝔻)) in the annulus c < |z | < 1, then ∥f ≤ ∥g ∥. For p ≥ 1, let c~p~ be the largest value of c for which Korenblum's maximum principle holds. In this note we prove that c~p~ → 1 as p → ∞. Thus we give a positive answer of a question of Hinkkanen. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)