On half-discrete Hilbert's inequality

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.