The norm of the Hilbert matrix operator on Bergman spaces

Karapetrovi\'c 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 paper, 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 $\alpha \not=1$.