On Weighted Hardy-Hilbert Type Inequality And Application In Probability

The weighted Hardy-Hilbert type inequalities (including discrete form and integral form) are extended by introducing proper weight functions and parameters, and by means of the theory of analytic function and analytical technique. The constant factor is proved to be the best possible, and the reverse inequalities of them and some special results are given. And then Hardy-Hilbert inequalities are improved by applying the positive definiteness of Gram matrix. As applications, the classical Hardy-Littlewood theorem and probability inequalities are extended.The following questions are studied in this dissertation:Some new Hardy-Hilbert inequalities are established by introducing gamma function and parameters; Some sharp results of Hardy-Hilbert type inequalities are given by applying refined Holder inequality; Hardy-Littlewood theorem are extended by using analytical technique. The layout of the dissertation is as follows:Chapter Ⅰ:The aim, background, methods and results of the dissertation are introduced.Chapter Ⅱ:Some generalized Hardy-Hilbert type inequalities (including discrete form and integral form) are established by means of the weight function method and by introducing parameters and proper weight functions. Some important and special results are enumerated. In particular, when p=2, the generalizations of the classical Hilbert inequalities (including discrete form and integral form) are obtained.Chapter Ⅲ:The proof of newly built inequality is given by applying the Hardy technique and Holder inequality. The constant factor is proved to be the best possible by using analytic method. The expression of the weight function is given with the help of psi function. And the generalized reverse Hardy-Hilbert type inequalities (including discrete form and integral form) are proved by applying the reverse Holder inequality.Chapter IV:Some significant refinements on the generalized Hardy-Hilbert type inequalities (including discrete form and integral form) are built by means of the positive definiteness of Gram matrix and the refined Holder inequality and by selecting properly variable unit-vector.Chapter V:As applications, the classical Hardy-Littlewood inequalities (including discrete form and integral form) and probability inequalities are extended.