# Half-discrete Hilbert-type inequalities [electronic resource] / Bicheng Yang, Lokenath Debnath

- Author:
- Yang, Bicheng
- Published:
- Singapore ; Hackensack, N.J. : World Scientific Pub. Co., [2014]
- Copyright Date:
- ©2014
- Physical Description:
- xiv, 333 pages
- Additional Creators:
- Debnath, Lokenath and World Scientific (Firm)

##### Access Online

- Contents:
- 1. Recent developments of Hilbert-Type inequalities with applications. 1.1. Introduction. 1.2. Hilbert's inequality and Hilbert's operator. 1.3. Modern research for Hilbert-type inequalities. 1.4. Some new applications for Hilbert-type inequalities. 1.5. Concluding remarks -- 2. Improvements of the Euler-Maclaurin summation formula and applications. 2.1. Introduction. 2.2. Some special functions relating Euler-Maclaurin's summation formula. 2.3. Estimations of the residue term about a class series. 2.4. Two classes of series estimations -- 3. A half-discrete Hilbert-type inequality with a general homogeneous kernel. 3.1. Introduction. 3.2. Some preliminary lemmas. 3.3. Some theorems and corollaries -- 4. A half-discrete Hilbert-type inequality with a non-homogeneous kernel. 4.1. Introduction. 4.2. Some preliminary lemmas. 4.3. Some theorems and corollaries. 4.4. Some particular examples -- 5. Multi-dimensional half-discrete Hilbert-type inequalities. 5.1. Introduction. 5.2. Some preliminary results and lemmas. 5.3. Some inequalities related to a general homogeneous kernel. 5.4. Some inequalities relating a general non-homogeneous kernel -- 6. Multiple half-discrete Hilbert-type inequalities. 6.1. Introduction. 6.2. First kind of multiple Hilbert-type inequalities. 6.3. Second kind of multiple Hilbert-type inequalities. 6.4. Some examples with the particular kernels.
- Summary:
- In 1934, G. H. Hardy et al. published a book entitled "Inequalities", in which a few theorems about Hilbert-type inequalities with homogeneous kernels of degree-one were considered. Since then, the theory of Hilbert-type discrete and integral inequalities is almost built by Prof. Bicheng Yang in their four published books. This monograph deals with half-discrete Hilbert-type inequalities. By means of building the theory of discrete and integral Hilbert-type inequalities, and applying the technique of Real Analysis and Summation Theory, some kinds of half-discrete Hilbert-type inequalities with the general homogeneous kernels and non-homogeneous kernels are built. The relating best possible constant factors are all obtained and proved. The equivalent forms, operator expressions and some kinds of reverses with the best constant factors are given. We also consider some multi-dimensional extensions and two kinds of multiple inequalities with parameters and variables, which are some extensions of the two-dimensional cases. As applications, a large number of examples with particular kernels are also discussed. The authors have been successful in applying Hilbert-type discrete and integral inequalities to the topic of half-discrete inequalities. The lemmas and theorems in this book provide an extensive account of these kinds of inequalities and operators. This book can help many readers make good progress in research on Hilbert-type inequalities and their applications.
- Subject(s):
- Genre(s):
- ISBN:
- 9789814504980 electronic bk.

9789814504973 hbk - Note:
- AVAILABLE ONLINE TO AUTHORIZED PSU USERS.
- Bibliography Note:
- Includes bibliographical references (pages 321-330) and index.
- Reproduction Note:
- Electronic reproduction. Singapore : World Scientific Publishing Co., 2014. System requirements: Adobe Acrobat Reader. Mode of access: World Wide Web. Available to subscribing institutions.

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