A note on the Hardy-Hilbert inequality (Q1353526)

Let \(\{a_n\}\) and \(\{b_n\}\) denote two nonnegative sequences in \(\ell^p\). The author proves that \[ \sum^\infty_{m=1} \sum^\infty_{n=1} (a_mb_n)/(m+ n)\leq \] \[ \Biggl\{\sum^\infty_{n=1} \Biggl(\pi/\sin{\pi\over p}- O_q(n)/n^{1/p}\Biggr)a^{1/p}_n\Biggr\}^{1/p}\times \Biggl\{\sum^\infty_{m= 1}\Biggl(\pi/\sin{\pi\over p}- O_p(n)/n^{1/q}\Biggr)b^{1/q}_n\Biggr\}^{1/q}. \] He obtains an explicit representation for \(O_r(n)\) for any \(r>1\), and shows that \(O_r(n)>0\). Thus the standard Hardy-Hilbert inequality becomes a corollary of this result.