Hilbert-type inequalities including Hilbert’s inequalities (built-in 1908), Hardy-Hilbert-type inequalities (built-in 1934), and Yang-Hilbert-type inequalities (built-in 1998) played an important role in analysis and their applications, which are mainly divided into three classes of integral, discrete and half-discrete. In recent twenty years, there are many advances in research on Hilbert-type inequalities, especially in Yang-Hilbert-type inequalities.
In this book, applying the weight functions, the parameterized idea, and the techniques of real analysis and functional analysis, we provide three kinds of Hilbert-type and Hardy-type integral inequalities in the whole plane as well as their reverses with parameters, which are extensions of Hilbert-type and Hardy-type integral inequalities in the first quarter. The equivalent forms, the operator expressions, and some equivalent statements of the best possible constant factors related to several parameters are considered. The lemmas and theorems provide an extensive account of these kinds of integral inequalities and operators. There are seven chapters in this book. In Chapter 1, we introduce some recent developments of Hilbert-type integral, discrete, and half-discrete inequalities. In Chapters 2-3, by using the weight function and real analysis, some new Hilbert-type and Hardy-type integral inequalities in the whole plane with the non-homogeneous kernel are given, and the cases of the homogeneous kernel are deduced. The equivalent forms and some equivalent statements of the best possible constant factors related to several parameters are obtained. We also consider the operator expressions as well as the reverses. In Chapters 4-7, the other two kinds of Hilbert-type and Hardy-type integral inequalities in the whole plane are also considered.
We hope that this monograph will prove to be useful especially to graduate students of mathematics, physics, and engineering sciences.
Sample Chapter(s)
Preface (51 KB)
Components of the Book:
- Chapter 1. Introduction
- Chapter 2. Equivalent Properties of a New Hilbert-Type Integral Inequality in the Whole Plane
- Chapter 3. Equivalent Properties of a New Hardy-Type Integral Inequality in the Whole
Plane
- Chapter 4. On Another New Hilbert-Type Integral Inequality in the Whole Plane
- Chapter 5. On Another New Hardy-Type Integral Inequalities in the Whole Plane
- Chapter 6. Equivalent Properties of a Hilbert-Type Integral Inequality in the Whole Plane
with the Exponent Function as the Interval Variables
- Chapter 7. Equivalent Properties of New Hardy-Type Integral Inequalities in the Whole
Plane with the Exponent Function as the Interval Variables
- References
Readership:
Students, academics, teachers, and other people attending or interested in Hilbert-type inequalities.
Bicheng Yang, Guangdong University of Education
Professor Bicheng Yang, he is born in Shanwei, Guangdong China, and his birthday is August 18, 1946. He currently works in the Department of Mathematics at Guangdong University of Education, China. He obtained a BSc in Mathematics from South China Normal University in 1982. His current research interests include analysis inequalities; extensions of Hilbert’s Inequality with best constant factors and applications; extensions of the weight inequalities and applications.