On further strengthened Hardy-Hilbert's inequality (Q1774761)

Summary: We obtain an inequality for the weight coefficient \(\omega(q,n)\) (\(q>1\), \(1/p+1/q=1\), \(n\in \mathbb{N}\)) in the form \[ \omega(q,n)=:\sum^{\infty}_{m=1}(1/(m+n))(n/m)^{1/q} < \pi/\sin(\pi/p)-1/(2n^{1/p}+(2/a)n^{-1/q}) \] where \(0<a<147/45\), as \(n\geq 3\); \(0<a<(1-C)/(2C-1)\), as \(n=1,2\), and \(C\) is the Euler constant. We show a generalization and improvement of Hilbert's inequalities. The results of the paper by \textit{B. Yang} and \textit{L. Debnath} [Int. J. Math. Math. Sci. 21, No. 2, 403--408 (1998; Zbl 0897.26008)] are improved.