关键词: 权函数, Euler-Maclaurin求和公式, Abel部分求和公式, 半离散Hilbert型不等式, 高阶导函数, 部分和

Abstract: Firstly, by using  the method of weight functions, Euler-Maclaurin summation formula, Abel’s summation by parts formula, and the technique of real analysis, we gave a new half-discrete Hilbert-type inequality involving higher-order derivative function and partial sums. Secondly, as applications, we discussed the equivalent conditions of the best constant factor to connect multiple parameters in a  inequality with particular  parameter and several particular inequalities.

Key words: weight function, Euler-Maclaurin summation formula, Abel’s summation by parts formula, half-discrete Hilbert-type inequality, higher-order derivative function, partial sums