On an extension of the Hardy-Hilbert theorem

A weighted Hardy-Hilbert’s inequality with the parameter λ of form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {\frac{{a_m b_n }}{{(m + n)^\lambda }}} < B^* (\lambda )\left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } a_{a_n }^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } b_n^q } } \right)^q }$$ \end{document} is established by introducing two parameters s and λ, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{1}{p} + \tfrac{1}{q} = 1,p \geqq q > 1,1 - \tfrac{q}{p} < \lambda \leqq 2,B^* (\lambda ) = B(\lambda - (1 - \tfrac{{2 - \lambda }}{p}),1 - \tfrac{{2 - \lambda }}{p})$$ \end{document} is the beta function. B *(λ) is proved to be best possible. A stronger form of this inequality is obtained by means of the Euler-Maclaurin summation formula.