On Hilbert Type And Hardy Type Integral Inequalities And Applications
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On Hilbert-Type and Hardy-Type Integral Inequalities and Applications
This book is aimed toward graduate students and researchers in mathematics, physics and engineering interested in the latest developments in analytic inequalities, Hilbert-Type and Hardy-Type integral inequalities, and their applications. Theories, methods, and techniques of real analysis and functional analysis are applied to equivalent formulations of Hilbert-type inequalities, Hardy-type integral inequalities as well as their parameterized reverses. Special cases of these integral inequalities across an entire plane are considered and explained. Operator expressions with the norm and some particular analytic inequalities are detailed through several lemmas and theorems to provide an extensive account of inequalities and operators.
HILBERT-TYPE AND HARDY-TYPE INTEGRAL INEQUALITIES IN THE WHOLE PLANE
Author: Bicheng Yang
language: en
Publisher: Scientific Research Publishing, Inc. USA
Release Date: 2022-07-19
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.
On Extended Hardy-hilbert Integral Inequalities And Applications
Hilbert-type inequalities, including Hilbert's inequalities proved in 1908, Hardy-Hilbert-type inequalities proved in 1934, and Yang-Hilbert-type inequalities first proved around 1998, play an important role in analysis and its applications. These inequalities are mainly divided in three classes: integral, discrete and half-discrete. During the last twenty years, there have been many research advances on Hilbert-type inequalities, and especially on Yang-Hilbert-type inequalities.In the present monograph, applying weight functions, the idea of parametrization as well as techniques of real analysis and functional analysis, we prove some new Hilbert-type integral inequalities as well as their reverses with parameters. These inequalities constitute extensions of the well-known Hardy-Hilbert integral inequality. The equivalent forms and some equivalent statements of the best possible constant factors associated with several parameters are considered. Furthermore, we also obtain the operator expressions with the norm and some particular inequalities involving the Riemann-zeta function and the Hurwitz-zeta function. In the form of applications, by means of the beta function and the gamma function, we use the extended Hardy-Hilbert integral inequalities to consider several Hilbert-type integral inequalities involving derivative functions and upper limit functions. In the last chapter, we consider the case of Hardy-type integral inequalities. The lemmas and theorems within provide an extensive account of these kinds of integral inequalities and operators.Efforts have been made for this monograph hopefully to be useful, especially to graduate students of mathematics, physics and engineering, as well as researchers in these domains.