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...
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.
Contents:
- Preface
- About the Author
- Acknowledgments
- Introduction
- Equivalent Properties of a New Hilbert-Type Integral Inequality with Parameters
- A New Hilbert-Type Integral Inequality Involving One Derivative Function
- A New Hilbert-Type Integral Inequality Involving Two Derivative Functions and Others
- New Hilbert-Type Integral Inequalities Involving Upper-Limit Functions
- Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities
- References
- Index
Readership: Advanced undergraduate students, graduate students, established research mathematicians, researchers working in applied mathematical analysis.
Key Features:
- Guide for further studies and research in the topic
- Valuable overview from known researchers of the inequalities treated in the book
- Up-to-date information in the topics treated
- Particularly useful for seminar use, diploma theses, and as a reference source
