Mathematical Inequalities: A Perspective, 1st Edition (Hardback) book cover

Mathematical Inequalities

A Perspective, 1st Edition

By Pietro Cerone, Silvestru Sever Dragomir

Chapman and Hall/CRC

391 pages

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pub: 2010-12-01
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Description

Drawing on the authors’ research work from the last ten years, Mathematical Inequalities: A Perspective gives readers a different viewpoint of the field. It discusses the importance of various mathematical inequalities in contemporary mathematics and how these inequalities are used in different applications, such as scientific modeling.

The authors include numerous classical and recent results that are comprehensible to both experts and general scientists. They describe key inequalities for real or complex numbers and sequences in analysis, including the Abel; the Biernacki, Pidek, and Ryll–Nardzewski; Cebysev’s; the Cauchy–Bunyakovsky–Schwarz; and De Bruijn’s inequalities. They also focus on the role of integral inequalities, such as Hermite–Hadamard inequalities, in modern analysis. In addition, the book covers Schwarz, Bessel, Boas–Bellman, Bombieri, Kurepa, Buzano, Precupanu, Dunkl–William, and Grüss inequalities as well as generalizations of Hermite–Hadamard inequalities for isotonic linear and sublinear functionals.

For each inequality presented, results are complemented with many unique remarks that reveal rich interconnections between the inequalities. These discussions create a natural platform for further research in applications and related fields.

Reviews

"Written by two leading experts in this subject, this book is both a text on a topic of current interest in inequalities and a great overview of the techniques and striking applications of the inequalities theory."

—Dumitru Acu, Zentralblatt MATH 1298

"Cerone and Dragomir have successfully produced an extensive list of inequalities used in analysis. … The writing is straightforward and the text often references results given elsewhere. … Recommended."

— J.R. Burke, CHOICE, December 2011

"… a well-written and welcome addition to the literature. … to have the results in one place is a service to all interested parties."

— P.S. Bullen, Mathematical Reviews, Issue 2011m

"One of the most interesting aspects is many instances of ‘reverses’ … a useful book if you are interested in its specific subject matter …"

MAA Reviews, February 2011

Table of Contents

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

About the Authors

Pietro Cerone is a professor of mathematics at Victoria University, where he served as head of the School of Computer Science and Mathematics from 2003 to 2008. Dr. Cerone is on the editorial board of a dozen international journals and has published roughly 200 refereed works in the field. His research interests include mathematical modeling, population dynamics, and applications of mathematical inequalities.

Sever S. Dragomir is a professor of mathematics and chair of the international Research Group in Mathematical Inequalities and Applications at Victoria University. Dr. Dragomir is an editorial board member of more than 30 international journals and has published over 600 research articles. His research in pure and applied mathematics encompasses classical mathematical analysis, operator theory, Banach spaces, coding, adaptive quadrature and cubature rules, differential equations, and game theory.

Subject Categories

BISAC Subject Codes/Headings:
MAT037000
MATHEMATICS / Functional Analysis