An Extension of the Hilbert's Integral Inequality

It is shown that an extension of the Hilbert's integral inequality can be established by introducing two parameters Open image in new window and Open image in new window . The constant factors expressed by the Euler number and Open image in new window as well as by the Bernoulli number and Open image in new window , respectively, are proved to be the best possible. Some important and especial results are enumerated. As applications, some equivalent forms are given.