Discrete Hilbert-type inequalities including Hilbert's inequality are important in mathematical analysis and its applications. In 1998, the author presented an extension of Hilbert's integral inequality with an independent parameter. In 2004, some new extensions of Hilbert's inequality were presented by introducing two pairs of conjugate exponents and additional independent parameters. Since then, a number of new discrete Hilbert-type inequalities have arisen. In this book, the author explains how to use the way of weight coefficients and introduce specific parameters to build new discrete Hilbert-type inequalities and consider some applications. The book is intended to augment the reader's understanding of Hilbert-type inequalities.

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