The norm of the Hilbert matrix operator on Bergman spaces

Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A_{\alpha }^p$ is equal to $\pi \mathrm{/}\sin ((2+\alpha )\pi \mathrm{/}p)$ when $-1<\alpha <p-2$. In this paper, we provide a proof of this conjecture for $0\le \alpha \le \frac{6p^3-29p^2+17p-2+2p\sqrt{6p^2-11p+4}}{(3p-1{)}^2}$, and this range of $\alpha$ improves the best known result when $\alpha >\frac{1}{47}$ and $\alpha ≠1$.