Refining the constant in a maximum principle for the Bergman space

Abstract:

Let $A^2(\mathbb {D})$ be the Bergman space over the open unit disk $\mathbb {D}$ in the complex plane. Korenblum conjectured that there is an absolute constant $c,~0<c<1$, such that whenever $|f(z)|\leq |g(z)|$ ($f,g\in A^2(\mathbb {D})$) in the annulus $c<|z|<1$, then $\|f\|\leq \|g\|$. In this note we give an example to show that $c<0.69472.$