#### Foreword

In 1934 G. H. Hardy, J. E. Littlewood and G. Polya published the book entitled "Inequalities", in which a few theorems about Hilbert-type inequalities with the homogeneous kernels of degree -1 and some special cases were considered. In 1991, Lizhi Xu first raised the way of weight coefficients to make a strengthened version of Hilbert’s inequality. In 1998, by applying the way of weight coefficients and introducing an independent parameter and the beta function, a best extension of Hilbert’s integral inequality was published in Journal of Mathematical Analysis and Applications by the author. In 2004 the author added two pairs of conjugate exponents and provided an extended Hilbert’s integral inequality. In 2009, the author published by Science Press (China) his book entitled "The Norm of Operator and Hilbert-Type Inequalities". In that book, two large classes of Hilberttype inequalities with the homogeneous kernels of a negative number degree, including integrals and series were discussed. However the author did not consider it an important case of Hardy-type inequalities. In October 2009, the author published an e-book entitled "Hilbert-Type Integral Inequalities" by Bentham Science Publishers Ltd. In that book, the author studied the broader case of Hilbert-type integral inequalities with the homogeneous kernels of a real number degree, which provide the best recent extensions of corresponding results. A number of equivalent forms as well as their reverses with several extended multiple inequalities in a number of particular cases are considered.

The book is divided into six chapters. In Chapter 1, some preliminary materials on the theory and methods of Hilbert-type inequalities, including the classical Hilbert’s inequality are discussed. Chapter 2 deals with an optimization of the methods of estimating the series and the weight coefficients. Some introductory theorems of improving the methods of Euler-Maclaurin summation formula are analyzed. In Chapter 3, by using the way of weight coefficients some fundamental theorems and corollaries on the discrete Hilbert-type inequalities with the homogeneous kernel of degree -1 are provided. The proofs regarding the best possible property of the constant factors are left to be studied in Chapter 4. In Chapter 4, some discrete Hilbert-type inequalities and their reverses with the general homogeneous kernel and the best constant factors are considered. These provide extensions of certain results of Chapter 3. By applying the improved Euler-Maclaurin summation formula and notions from Real Analysis, some particular examples are given. In Chapter 5 based upon some theorems of Chapter 4 and by applying techniques from Real Analysis, the author has explained how to use particular parameters to formulate some new Hilbert-type inequalities and their reverses with the best constant factors. In addition, a class of Hilberttype inequalities with the general measurable kernels is considered. In Chapter 6, the author has decently formulated some lemmas and obtained two equivalent multiple Hilbert-type inequalities and their reverses with the homogeneous kernel of a real number degree. These inequalities are the best extensions of the corresponding inequalities in Chapter 4. Some special examples are also studied.

The author has succeeded to present in this book an extensive account of several Hilbert-type inequalities in a self contained and rigorous manner. The book will be very useful not only to graduate students who study inequalities in the broader domain of Mathematical Analysis but also to research mathematicians who need the latest information to refer in.

**
***Themistocles M. Rassias*

National Technical University of Athens

Athens

Greece

#### Preface

One Hundred years ago, in 1908, H. Wely published the well known Hilbert’s inequality. In 1925, G. H. Hardy gave a best extension of it by introducing one pair of conjugate exponents (p, q), named as Hardy-Hilbert’s inequality. The Hilbert-type inequalities are a more wide class of analysis inequalities which are with the bilinear kernels, including Hardy-Hilbert’s inequality as the particular case. These inequalities are important in analysis and its applications. By making a great effort of mathematicians in the world at about one hundred years, the theory of Hilbert-type inequalities has now come into being. This book is a monograph about the theory of discrete Hilbert-type inequalities with the homogeneous kernels of real number-degree and its applications. Using the methods of series summation, Real Analysis, Functional Analysis and Operator Theory, and following the way of weight coefficients, the author introduces a few independent parameters to establish a number of discrete Hilbert-type inequalities with the homogeneous kernels of real number-degree and the best constant factors, including some multiple inequalities. The equivalent forms and the reverses with the best constant factors are also considered. As application, the author also considers some discrete Hilbert-type inequalities with the nonhomogeneous kernels and a large number of particular examples.

For reading and understanding this book, readers should hold the basic knowledge of real analysis and functional analysis. This book is suited to the people who are interested in the fields of analysis inequalities and real analysis. The author expects this book to help many readers to make good progress in research for discrete Hilbert-type inequalities and their applications.

**
***Bicheng Yang*

Guangzhou, Guangdong,

P. R. China

#### List of Contributors

##### Author(s):

**Bicheng Yang **

Department of Mathematics

Guangdong Education Institute

Guangdong Guangzhou

P.R, 510303

China

#### Reviews

In this book, several different types of discrete Hilbert – type inequalities with various applications are studied. Special emphasis is given to a number of new results formulated and proved during the last years. In particular, several generalizations, extensions and refinements of discrete Hilbert – type inequalities involving many special functions such as beta, gamma, hyper geometric, trigonometric, hyperbolic, zeta, Bernoulli’s functions and Bernoulli’s numbers, as well as Euler’s constant are studied. The research monograph also studies recent developments of discrete types of operators and inequalities with proofs and discusses a number of examples and applications. For a systematic information of the discrete Hilbert – type inequalities and operators, the reader is referred to this book which provides several new double inequalities with general homogeneous kernels of real numbers as well as two pairs of conjugate exponents and the best constant factors.

**
***Professor Themistocles M. Rassias*

Department of Mathematics

National Technical University of Athens

Personal page: www.math.ntua.gr/~trassias/

One Hundred years ago, H. Wely published the well known Hilbert’s inequality. In 1925, G. H. Hardy gave an extension of it by introducing one pair of conjugate exponents (p, q), named as Hardy-Hilbert’s inequality. The Hilbert-type inequalities are a more wide class of analysis inequalities which are with the bilinear kernels, including Hardy-Hilbert’s inequality as the particular case. By making a great effort of mathematicians, the theory on Hilbert-type inequalities has now come into being in author’s a few publishing books. This book is a monograph about the theory of discrete Hilbert-type inequalities with the homogeneous kernels of real number-degree. Using the methods of Series Summation, Real Analysis, Functional Analysis and Operator Theory, and following the way of weight coefficients, the author introduces a few independent parameters to establish a number of discrete Hilbert-type inequalities with the homogeneous kernels of real number-degree and the best constant factors. The equivalent forms, the reverses as well as some multiple cases are also considered. As application, the author also considers a large number of particular examples. This book is suited to the people who are interested in the fields of Analysis Inequalities and Real Analysis. Reading this book may help readers to make good progress in research for Hilbert –type inequalities and their applications.

**
***Jichang Kuang*

Department of Mathematics, Hunan Normal University

Changsha, Hunan, China