Mathematical Inequalities: A Perspective

Pietro Cerone and Sever S. Dragomir

Publisher:

Chapman & Hall/CRC

Publication Date:

2011

Number of Pages:

391

Format:

Hardcover

Price:

119.95

ISBN:

9781439848968

Category:

Monograph

MAA Review
Table of Contents

[Reviewed by

Allen Stenger

, on

02/10/2011

]

This is a somewhat specialized monograph on inequalities, focusing on integral inequalities and the use of convexity. The subtitle “A Perspective” means a personal perspective, that is, the book deals with inequalities that the authors have worked on and are interested in, and not with inequalities in general.

The work is deep rather than wide: it usually starts with a familiar inequality and then proves generalizations, analogs, and variants. One of the most interesting aspects is many instances of “reverses” (some would call them “converses”). This is where you have a familiar inequality, A ≤ B, and a reverse is a result B ≤ p A proved under additional hypotheses and with a parameter p that depends on some aspects of the quantities involved.

This book is strictly a reference, with no exercises. One big weakness of the book is that all inequalities are treated in isolation: we don’t know why they were invented or what they might be good for. For example, there is a “Bombieri’s inequality,” which is the key ingredient in the large sieve in number theory and leads to a number of other interesting inequalities, but we are not told any of this.

One peculiarity of the book is that every inequality seems to be named after somebody (as in Bombieri’s inequality). These attributions are probably accurate, but are not standard. Often the inequalities are not referenced back to the original papers, but only to modern treatments by the book’s authors, so it’s not easy to evaluate these namings. There is also some non-standard terminology. For example, on p. 103 we are introduced to Hadamard’s inferior sum and Hadamard’s superior sum, which any calculus teacher will recognize as the midpoint and trapezoidal rules for integration.

Bottom line: a useful book if you are interested in its specific subject matter, but not a good book to start learning about inequalities. The best all-around inequality book is Steele’s The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. It not only has very broad coverage of inequalities, but will teach you how to prove your own inequalities.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

Discrete Inequalities
An Elementary Inequality for Two Numbers
An Elementary Inequality for Three Numbers
A Weighted Inequality for Two Numbers
The Abel Inequality
The Biernacki, Pidek, and Ryll–Nardzewski (BPR) Inequality
Cebysev’s Inequality for Synchronous Sequences
The Cauchy–Bunyakovsky–Schwarz (CBS) Inequality for Real Numbers
The Andrica–Badea Inequality
A Weighted Grüss-Type Inequality
Andrica–Badea’s Refinement of the Grüss Inequality
Cebysev-Type Inequalities
De Bruijn’s Inequality
Daykin–Eliezer–Carlitz’s Inequality
Wagner’s Inequality
The Pólya–Szegö Inequality
The Cassels Inequality
Hölder’s Inequality for Sequences of Real Numbers
The Minkowski Inequality for Sequences of Real Numbers
Jensen’s Discrete Inequality
A Converse of Jensen’s Inequality for Differentiable Mappings
The Petrović Inequality for Convex Functions
Bounds for the Jensen Functional in Terms of the Second Derivative
Slater’s Inequality for Convex Functions
A Jensen-Type Inequality for Double Sums

Integral Inequalities for Convex Functions
The Hermite–Hadamard Integral Inequality
Hermite–Hadamard Related Inequalities
Hermite–Hadamard Inequality for Log-Convex Mappings
Hermite–Hadamard Inequality for the Godnova–Levin Class of Functions
The Hermite–Hadamard Inequality for Quasi-Convex Functions
The Hermite–Hadamard Inequality for s-Convex Functions in the Orlicz Sense
The Hermite–Hadamard Inequality for s-Convex Functions in the Breckner Sense
Inequalities for Hadamard’s Inferior and Superior Sums
A Refinement of the Hermite–Hadamard Inequality for the Modulus

Ostrowski and Trapezoid-Type Inequalities
Ostrowski’s Integral Inequality for Absolutely Continuous Mappings
Ostrowski’s Integral Inequality for Mappings of Bounded Variation
Trapezoid Inequality for Functions of Bounded Variation
Trapezoid Inequality for Monotonic Mappings
Trapezoid Inequality for Absolutely Continuous Mappings
Trapezoid Inequality in Terms of Second Derivatives
Generalised Trapezoid Rule Involving nth Derivative Error Bounds
A Refinement of Ostrowski’s Inequality for the Cebysev Functional
Ostrowski-Type Inequality with End Interval Means
Multidimensional Integration via Ostrowski Dimension Reduction
Multidimensional Integration via Trapezoid and Three Point
Generators with Dimension Reduction
Relationships between Ostrowski, Trapezoidal, and Cebysev Functionals
Perturbed Trapezoidal and Midpoint Rules
A Cebysev Functional and Some Ramifications
Weighted Three Point Quadrature Rules

Grüss-Type Inequalities and Related Results
The Grüss Integral Inequality
The Grüss–Cebysev Integral Inequality
Karamata’s Inequality
Steffensen’s Inequality
Young’s Inequality
Grüss-Type Inequalities for the Stieltjes Integral of Bounded Integrands
Grüss-Type Inequalities for the Stieltjes Integral of Lipschitzian Integrands
Other Grüss-Type Inequalities for the Riemann–Stieltjes Integral
Inequalities for Monotonic Integrators
Generalisations of Steffensen’s Inequality over Subintervals

Inequalities in Inner Product Spaces
Schwarz’s Inequality in Inner Product Spaces
A Conditional Refinement of the Schwarz Inequality
The Duality Schwarz-Triangle Inequalities
A Quadratic Reverse for the Schwarz Inequality
A Reverse of the Simple Schwarz Inequality
A Reverse of Bessel’s Inequality
Reverses for the Triangle Inequality in Inner Product Spaces
The Boas–Bellman Inequality
The Bombieri Inequality
Kurepa’s Inequality
Buzano’s Inequality
A Generalisation of Buzano’s Inequality
Generalisations of Precupanu’s Inequality
The Dunkl–William Inequality
The Grüss Inequality in Inner Product Spaces
A Refinement of the Grüss Inequality in Inner Product Spaces

Inequalities in Normed Linear Spaces and for Functionals
A Multiplicative Reverse for the Continuous Triangle Inequality
Additive Reverses for the Continuous Triangle Inequality
Reverses of the Discrete Triangle Inequality in Normed Spaces
Other Multiplicative Reverses for a Finite Sequence of Functionals
The Diaz–Metcalf Inequality for Semi-Inner Products
Multiplicative Reverses of the Continuous Triangle Inequality
Reverses in Terms of a Finite Sequence of Functionals
Generalisations of the Hermite–Hadamard Inequalities for Isotonic Linear Functionals
A Symmetric Generalisation
Generalisations of the Hermite–Hadamard Inequality for Isotonic Sublinear Functionals

References

Index

Tags:

Analysis