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