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.
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