On new strengthened Hardy-Hilbert's inequality (Q1384265)

Summary: A new inequality for the weight coefficient \(\omega(q,n)\) in the form \[ \omega(q,n):= \sum^\infty_{m= 1}{1\over m+ n} \Biggl({n\over m}\Biggr)^{1/q}< {\pi\over\sin(\pi/p)}- {1\over 2n^{1/p}+ n^{-1/q}} \quad \Biggl(q>1, {1\over p}+{1\over q}= 1, n\in\mathbb{N}\Biggr) \] is proved. This is followed by a strengthened version of the Hardy-Hilbert inequality.