On half-discrete Hilbert’s inequality

摘要

In this paper we provide an application of the Euler–Maclaurin summation formula with the Bernoulli function for the proof of a strengthened version of the half-discrete Hilbert inequality with the best constant factor in terms of the Euler–Mascheroni constant. Some equivalent numerical representations, operator representations, two kinds of reverses as well as an extension in terms of parameters and the Beta function are also studied.