A slight improvement to Korenblum's constant

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 $c$.