On a Strengthened of the More Accurate Hilbert's Inequality

Abstract

By deducing the inequality of weight coefficient:
$$\omega {\kern 1pt} {\kern 1pt} (n) = \sum\limits_{m = 0}^\infty {\frac{1}{{\kern 1pt} {\kern 1pt} m + {\kern 1pt} {\kern 1pt} n +1}} (\frac{2{\kern 1pt} {\kern 1pt} n + 1}{2{\kern 1pt} {\kern 1pt} m + 1})^{\frac{1}{2}}
 < \pi -\frac{5}{6(\sqrt {2n + 1} + \frac{3}{4}\sqrt {(2n + 1)^{ - 1}} )}, $$
where $n \in N$. We obtain on a strengthened of the more accurate Hilbert's inequality.