The norm of the Hilbert matrix operator on Bergman spaces

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