New extensions of Hilbert's inequality with multiple parameters (Q955195)

The classical Hardy-Hilbert inequalities state that if \((a_n),(b_n)\in\ell_2({\mathbb R})\setminus\{(0)\}\), \(p>1\), \(\frac{1}{p}+\frac{1}{q}=1\), then \[ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{a_mb_n}{m+n+1}<\frac{\pi}{\sin\frac{\pi}{p}} \biggl(\sum_{n=0}^{\infty}a_n^p\biggr)^{\frac{1}{p}} \biggl(\sum_{n=0}^{\infty}b_n^q\biggr)^{\frac{1}{q}}. \] The parametrized versions with \((m+n+1)^{\lambda}\) and \((m+n)^{\lambda}\) in the denominator on the left hand side are known. The paper treats another parametric case of \((am+bn+c)^{\lambda}\) and \((m^{\alpha}+n^{\beta})^{\lambda}\), respectively (with the suitable assumptions).