On Korenblum’s maximum principle

Let A 2 ( D ) A^2(\mathbb {D}) be the Bergman space over the open unit disk D \mathbb {D} in the complex plane. Korenblum’s maximum principle states that there is an absolute constant c ∈ ( 0 , 1 ) c\in (0,1) , such that whenever | f ( z ) | ≤ | g ( z ) | |f(z)|\leq |g(z)| ( f , g ∈ A 2 ( D ) f,g\in A^2(\mathbb {D}) ) in the annulus c > | z | > 1 c>|z|>1 , then ‖ f ‖ A 2 ≤ ‖ g ‖ A 2 \|f\|_{A^2}\leq \|g\|_{A^2} . In this paper we prove that Korenblum’s maximum principle holds with c = 0.25018 c=0.25018 .