A slight improvement to Korenblum's constant

TL;DR

This paper improves the upper bound on Korenblum's constant in the context of Bergman spaces, refining previous bounds and contributing to the understanding of function dominance in complex analysis.

Contribution

The paper provides a slight improvement to the known upper bound of Korenblum's constant, narrowing the range of possible values for this conjectured universal constant.

Findings

Upper bound on c is reduced from 0.67795 to a lower value.

Refinement of Korenblum's conjecture bounds in Bergman spaces.

Enhanced understanding of function dominance in complex analysis.

Abstract

Let $A^2(D)$ be the Bergman space over the open unit disk $D$ in the complex plane. Korenblum conjectured that there is an absolute constant $c \in (0,1)$ such that whenever $|f(z)|\le |g(z)|$ in the annulus $c<|z|<1$ then $||f(z)|| \le ||g(z)||$.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 $c$.