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Publisher:

Chapman & Hall/CRC

Publication Date:

2011

Number of Pages:

391

Format:

Hardcover

Price:

119.95

ISBN:

9781439848968

Category:

Monograph

[Reviewed by , on ]

Allen Stenger

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**

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