Bicheng Yang

By means of the weight functions, Hermite-Hadamard's inequality and the technique of real analysis, a new more accurate half-discrete Hilbert-type inequality involving one higher-order derivative function is given. The equivalent conditions of the best possible constant factor related to a few parameters, the equivalent forms, several particular inequalities and the operator expressions are considered.

By means of the techniques of the real analysis and the weight functions, a few equivalent conditions of a Hilbert-type integral inequality with the general nonhomogeneous kernel in the whole plane are obtained. The constant factor is proved to be the best possible. As applications, a few equivalent conditions of a Hilbert-type integral inequality with the general homogeneous kernel in the whole plane are deduced. We also consider the operator expressions, a few particular cases and some examples. Key words: Hilbert-type integral inequality; weight function; equivalent form; operator; norm 2000 Mathematics Subject Classification. 26D15

By the use of the way of real analysis, we estimate the weight functions and give some new Hilbert-type integral inequalities in the whole plane with nonhomogeneous kernels and multiparameters. The constant factors related to the hypergeometric function and the beta function are proved to be the best possible. We also consider the equivalent forms, the reverses, and some particular cases in the homogeneous kernels.