On the norm of the Hilbert matrix operator on weighted Bergman spaces

It is known that the norm of the Hilbert matrix operator on weighted Bergman spaces $A_{\alpha }^p$ was conjectured by Karapetrović to be $\frac{\pi }{\sin \frac{(\alpha +2)\pi }{p}}$ when $\alpha >-1$ and $p>\alpha +2$. The conjecture has been confirmed by Božin and Karapetrović in the case $\alpha =0$. In this paper we prove the conjecture for the cases both $\alpha =1$ and $0<\alpha \le \frac{1}{47}$. Moreover, we also show that the conjecture is valid when $-1<\alpha <0$ and $p\ge 2(\alpha +2)$.