A Reverse Hardy-Hilbert’s Inequality Involving One Partial Sum as the Terms of Double Series

Abstract

In this paper, by constructing proper weight coefficients and utilizing the Euler-Maclaurin summation formula and the Abel partial summation formula, we establish reverse Hardy-Hilbert’s inequality involving one partial sum as the terms of double series. On the basis of the obtained inequality, the equivalent conditions of the best possible constant factor associated with several parameters are discussed. Finally, we illustrate that more reverse inequalities of Hardy-Hilbert type can be generated from the special cases of the present results.

A Reverse Hardy–Hilbert’s Inequality Containing Multiple Parameters and One Partial Sum

“…). Yang, Wu and Huang [6] established a reverse Hardy-Hilbert's inequality with one partial sum B n = ∑ n k=1 b k as the term of the double series, as follows:…”

“…Inspired by the work of [4][5][6][7][8][9][10], in this paper, we construct a reverse Hardy-Hilbert's inequality which contains one partial sum and some extra parameters inside the weight coefficients, the reverse Hardy-Hilbert's inequality has different structural forms by comparing with existing results mentioned above. Our method is mainly based on some skillful applications of the Euler-Maclaurin summation formula and Abel's partial summation formula.…”