Behavior of the constant in Korenblum's maximum principle

Let A p () ( p ≥ 1) be the Bergman space over the open unit disk 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 ()) in the annulus c < | z | < 1, then ∥ f   A   p≤ ∥ g ∥   A   p. 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)