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On Hilbert-Type and Hardy-Type Integral Inequalities and Applications

Bicheng Yang and Themistocles M. Rassias
Publication Date: 
Number of Pages: 
SpringerBriefs in Mathematics
[Reviewed by
Allen Stenger
, on
This is a specialist monograph on generalizations of two classical inequalities: the integral Hilbert inequality,
\( \int_{0}^{\infty} \int_{0}^{\infty} \frac{f(x)g(y)}{x+y}\, dx \, dy \leq \pi \left(  \int_{0}^{\infty} | f(x) | ^{2} \, dx \right)^{1/2} \left( \int_{0}^{\infty} | g(x) |^{2} \, dx \right)^{1/2} \,\, f, g \in L^{2}, \)
and the integral Hardy inequality,
\( \int_{0}^{\infty} \left( \frac{1}{x} \int_{0}^{x} f(u) \, du \right)^{2} \leq 4 \int_{0}^{T} f^{2}(x) \, dx, \, \, f \in L^{2}.  \)
The constants \( \pi \) and \( 4 \) are the best possible. (These inequalities also have discrete analogs for sequences in \( \ell^{2} \); the constants are the same.) Generalizations of these inequalities have been an active research area in the past twenty years; this book gives special attention to results that were proved in the years 2009–2012. The methods of proof are generally those of classical real analysis (Fubini theorem, Hölder inequality), although the book also uses some language from functional analysis (operator, kernel, norm).
The first half of the book deals with generalizations of the Hilbert inequality, in which we replace the factor \( 1/(x + y) \) with a general kernel, and raise the functions to other powers (similarly to the way that Hölder’s inequality generalizes the Cauchy–Schwarz inequality). Most of the material is devoted to determining the best-possible constant on the right, which is expressed in terms of a Mellin transform of the kernel. In the examples given, most of the constants are expressed in terms of the gamma function and the Riemann or Hurwitz zeta functions. There is also some work on “reverse” inequalities, that is, conditions under which the inequality holds with the sense of the inequality sign reversed and a different constant on the right. 
The second half of the book follows the same plan, but for generalizations of the Hardy inequality. These are divided into a first and second kind, depending in whether the inner integral runs from \( 0 \) to \( x \) or from \( x \) to \( \infty \).  
The book is well-done, although it starts in the middle of things and doesn’t give much background or motivation. A couple of weak areas: (1) No index. (2) The title promises Applications, by which it seems to mean examples; that is, it considers a variety of specific kernels and then calculates the best possible constant. The book does not give applications of these inequalities to other problems in mathematics.
Less-general treatments of the Hilbert and Hardy inequalities are in many books on inequalities, for examples the following two (which do include applications): Hardy, Littlewood, and Pólya, Inequalities 2nd edition, Chapter IX; and J. Michael Steele, The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, Chapters 10 and 11.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web site is His mathematical interests are number theory and classical analysis.