On an extension of the Hardy-Hilbert theorem

A weighted Hardy-Hilbert's inequality with the parameter lambda of form Sigma(infinity)(m=1)Sigma(infinity)(n=1) a(m)b(n)/(m+n)(lambda)< B*(lambda)(Sigma(infinity)(n=1)n(1-lambda)a(n)(p))(1/p)(Sigma(infinity)(n=1)n(1-lambda)b(n)(q))(1/q) is established by introducing two parameters s and lambda, where 1/p + 1/q = 1, p >= q > 1, 1 - q/p < lambda <= 2, B*(A) = B(lambda - (1 - 2-lambda/p), 1 - 2-lambda/p) is the beta function. B* (lambda) is proved to be best possible. A stronger form of this inequality is obtained by means of the Euler-Maclaurin summation formula.