Some Improvements And Extensions Of Hilbert-type Inequalities

Hilbert inequalities are highly important in analysis.By introducing pa-rameters and the method of weight function,improving some Hilbert inequali-ties,obtaining new inequalities and equivalent conditions of Hilbert inequalities.Chapter 1:The significance,background and main results of the paper are introduced.Chapter 2:An extended Hilbert's integral inequality in the whole plane with parameters is established by introducing interval variables and independent parameters,applying the weight functions and the technique of real analysis The constant factor is the best.The related homogeneous forms with particular parameters,the equivalent forms and the reverses are considered.Moreover,an extended Hilbert's integral operator in the whole plane is defined,and the operator expressions are givenChapter 3:By means of the methods of real analysis and the weight func-tions,a few equivalent conditions of a reverse Hilbert-type integral inequality with the general non-homogeneous kernel in the whole plane are obtained.As applications,a.few equivalent conditions of a reverse Hilbert-type integral in-equality with the general homogeneous kernel in the whole plane are deduced Some particular cases and examples are consideredChapter 4:By introducing the weight function,using the way of real anal-ysis,Holder inequality and Hermite-Hadamard's inequality,a more accurate Mulholland-type inequality is improved.New inequalities are givenChapter 5:We summarize the main contents of this paper,and outlook the future research.