The norm of the Hilbert matrix operator on Bergman spaces

Abstract Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha $ is equal to $\pi /\sin ((2+\alpha )\pi /p)$ when $-1<\alpha <p-2$ . In this article, we provide a proof of this conjecture for $0\leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}$ , and this range of $\alpha $ improves the best known result when $\alpha>\frac {1}{47}$ and $\alpha \not =1$ .