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.
P. R. China