Some Refinements And Generalizations Of Hilbert-type Inequalities

In theory and applications , inequalities often play an important role. In some occasion, it is more important than that of equalities. Especially, it is impossible for a lot of equations to find their exact solutions via calculations. The estimation of the solutions of the equations can be given by inequalities. In particular, Hilbert' s inequality is widely used in analytic number theory, functional analysis, differential equation and approximation theory etc.On account of the importance of the Hilbert inequality in mathematical science, it has aroused the much interest of mathematicians. In recent years, Hilbert-type inequalities have been studied by mathematicians, and therefore various nice results have been obtained.In this thesis, we consider the following problems:How to apply Euler-Maclaurin summation formula to deal with the computation problem of double series; How to improve H(o|¨)lder' s inequality; How to choose appropriate unit vector to establish new inequalities.This thesis is divided into four chapters.In chapter 1, states briefly the course of development, researching status quo and the project work of this thesis.In chapter 2, a new refinement of the Hilbert inequality for double series is established by means of Euler-Maclaurin Summation formula and by introducing a proper weight function. A similar result for the Hilbert integral inequality is also proved. As applications, some strengthened results of the Hardy-Littlewood theorem and the Widder theorem are given.In chapter 3, firstly , applying the sharpening of Cauchy' s inequality to improve H(o|¨)lder' s inequality, and then select a unit vector h such that the expressions of R_λ(with R_λ≠0,R_λ＜1)are obtained. Some Hardy-Hilbert type inequalitiesfor double series are improved by using the following inequality (a,b)＜||a|| ||b|| (1 - R)~k. Some new inequalities of the form:andare built .In the end, let' s select a proper function h(x,y)and obtainan expression of R_λ.And then Hardy-Hilbert' s integral inequality with parameters is further studied by means of a sharpened of H(o|¨)lder' s inequality: (f,g)<||f||_p ||g||_q (1-R)~k The following new inequality:is obtained.