On Some Improvement And Extension Of Hilbert Type Inequalities

Hilbert’s inequalities(including integral and discrete) are important inequalities in analysis. In this paper, some improvements and generalizations of the integral and half-discrete Hilbert inequalities are given by introducing the appropriate weight function method. It is proved that the constant factor is the best, and their equivalent formula and some special results are obtained. An application of a sharpening of H(?)lder’s inequality in improvement of Hardy- Hilbert-type inequality is considered. The layout of this dissertation is as follows:Chapter 1:The aim, background, methods and results of the dissertation are introduced.Chapter 2:By the use of the transfer formula, the methods of weight functions and technique of Real Analysis, a multidimensional Hilbert-type integral inequality with parameters and a best possible constant factor related to the kernel of logarithm function is given. The equivalent form and some reverses are obtained. The operator expressions and a few particular results related to the kernel of non-homogeneous and homogeneous are considered.Chapter 3:By using the way of weight functions and the Hermite-Hadamard inequality, a half-discrete reverse Mulholland-type inequality with a best constant factor is given. The extension with multi-parameters, the equality forms as well as the relating homogeneous inequalities are also considered.Chapter 4:By introducing the weight function,using the way of real analysis, some improvements of a Hardy-Hilbert type integral inequality with a quasi-homogeneous are given, a number of new inequalities are established.