Hilbert-Type Integral Inequalities

In this chapter, we introduce some evolvements for the theory and methods of Hilbert-type inequalities, including Hilbert’s inequality. We must emphasize some excellent works on Hilbert-type inequalities and Hilberttype operators with multi-parameters in recent years, which have made more developments in this Context. This chapter will enhance the understanding of the readers of the content of the following several chapters.

In this chapter, we build some Hilbert-type integral inequalities with two pairs of conjugate exponents and an independent parameter, and consider their operator characterizations. The equivalent forms and the reverses are also given. Some sufficient conditions of bounded operator are established and a number of particular inequalities are deduced. The proofs of some basic theorems are used the limit theorems in the theory of Lebesgue integrals and the technique of real analysis.

In this chapter, by using the way of weight functions and the technique of real analysis, we introduce multi-parameters and give some extended Hilbert -type integral inequalities restricted respectively in the subintervals (a,∞) , (0,b) and (a,b) (0 < a < b) . The strengthened versions, the equivalent forms and some reverses are also considered.

In this chapter, based on some theorems of Chapters 2, by using the technique of real analysis, we discuss how to use some particular parameters to deduce some new Hilbert-type integral inequalities and the reverses with the best constant factors.

In this chapter, we establish some lemmas and obtain two equivalent multiple Hilbert-type integral inequalities with the homogeneous kernels of real number-degree and the reverses, which are the best extensions of the corresponding results in Chapter 2. As applications, two equivalent multiple integral inequalities with the non-homogeneous kernels, two classes of multiple Hardy-type integral inequalities and some particular examples are also considered.

In this chapter, we consider a class of multivariable Hilbert-type integral inequalities and the reverses, which are the best extensions of the corresponding results in Chapter 2. We also consider some equivalent integral inequalities with the non-homogeneous kernels and the Hardy-type integral inequalities.

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