On generalizations and reinforcements of a Hilbert's type inequality (Q422028)

Two Hilbert-type inequalities NEWLINE\[NEWLINE \begin{gathered} \sum_{n=1}^\infty \sum_{m=1}^\infty \frac{a_mb_n}{\max\{m^\lambda,n^\lambda\}} < \left\{ \sum_{n=1}^\infty \left[ \kappa(\lambda)-\frac 1{3qn^{(q+\lambda-2)/q}}\right] n^{1-\lambda} a_n^p\right\}^{1/p} \\ \times \left\{\sum_{n=1}^\infty \left[\kappa(\lambda)-\frac 1{3pn^{(p+\lambda-2)/p}}\right ] n^{1-\lambda} b_n^q\right\}^{1/q} \end{gathered} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \begin{aligned} & \sum_{m=1}^\infty m^{(p-1)(\lambda-1)} \left(\sum_{n=1}^\infty \frac{a_n}{\max\{m^\lambda,n^\lambda\}}\right)^p < \kappa(\lambda) \sum_{n=1}^\infty \left[\kappa(\lambda) - \frac 1{3qn^{(q+\lambda-2)/q}}\right] n^{1-\lambda} a_n^p \end{aligned} NEWLINE\]NEWLINE are established. Here, \(p,q>1\) is a conjugate pair, \(2-\min\{p,q\}<\lambda\leq 2\), \(a_n,b_n\geq 0\;\forall\;n\geq 1\), \(0<\sum\limits_{n=1}^\infty a_n^p\), \(\sum\limits_{n=1}^\infty b_n^q < \infty\), and NEWLINE\[NEWLINE \kappa(\lambda) = \frac{p\,q \lambda}{(p+\lambda-2)(q+\lambda-2)} > 0\;. NEWLINE\]