Mathematical Inequalities: A Perspective

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ISBN: 9781439848968 | 1439848963
Cover: Hardcover
Copyright: 12/1/2010

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This book provides an overview of the expanding field of mathematical inequalities and their applications. Instead of focusing on narrow treatments of various mathematical inequalities, the authors present a number of classical and recent results across the field, covering integral inequalities, discrete inequalities, and inequalities in abstract spaces. They also make new connections and investigate intimate relationships between inequalities. Written in an accessible manner, the text offers simple proofs for young researchers yet incorporates sufficient detail to appeal to experts and graduate students in real and functional analysis.

Preface	p. ix
Discrete Inequalities	p. 1
An Elementary Inequality for Two Numbers	p. 1
An Elementary Inequality for Three Numbers	p. 3
A Weighted Inequality for Two Numbers	p. 4
The Abel Inequality	p. 6
The Biernacki, Pidek, and Ryll-Nardzewski (BPR) Inequality	p. 9
Ceby¿ev's Inequality for Synchronous Sequences	p. 11
The Cauchy-Bunyakovsky-Schwarz (CBS) Inequality for Real Numbers	p. 13
The Andrica-Badea Inequality	p. 15
A Weighted Grüss Type Inequality	p. 17
Andrica-Badea's Refinement of the Grüss Inequality	p. 19
Ceby¿ev Type Inequalities	p. 24
De Bruijn's Inequality	p. 29
Daykin-Eliezer-Carlitz's Inequality	p. 30
Wagner's Inequality	p. 33
The Pólya-Szegö Inequality	p. 35
The Cassels Inequality	p. 37
Hölder's Inequality for Sequences of Real Numbers	p. 39
The Minkowski Inequality for Sequences of Real Numbers	p. 41
Jensen's Discrete Inequality	p. 43
A Converse of Jensen's Inequality for Differentiable Mappings	p. 45
The Petrovic Inequality for Convex Functions	p. 46
Bounds for the Jensen Functional in Terms of the Second Derivative	p. 48
Slater's Inequality for Convex Functions	p. 50
A Jensen Type Inequality for Double Sums	p. 52
Integral Inequalities for Convex Functions	p. 55
The Hermite-Hadamard Integral Inequality	p. 55
Hermite-Hadamard Related Inequalities	p. 57
Hermite-Hadamard Inequality for Log-Convex Mappings	p. 71
Hermite-Hadamard Inequality for the Godnova-Levin Class of Functions	p. 73
The Hermite-Hadamard Inequality for Quasi-Convex Functions	p. 76
The Hermite-Hadamard Inequality of s-Convex Functions in the Orlicz Sense	p. 82
The Hermite-Hadamard Inequality for s-Convex Functions in the Breckner Sense	p. 94
Inequalities for Hadamard's Inferior and Superior Sums	p. 102
A Refinement of the Hermite-Hadamard Inequality for the Modulus	p. 109
Ostrowski and Trapezoid Type Inequalities	p. 115
Ostrowski's Integral Inequality for Absolutely Continuous Mappings	p. 115
Ostrowski's Integral Inequality for Mappings of Bounded Variation	p. 118
Trapezoid Inequality for Functions of Bounded Variation	p. 120
Trapezoid Inequality for Monotonic Mappings	p. 122
Trapezoid Inequality for Absolutely Continuous Mappings	p. 125
Trapezoid Inequality in Terms of Second Derivatives	p. 127
Generalised Trapezoid Rule Involving nth Derivative Error Bounds	p. 129
A Refinement of Ostrowski's Inequality for the Ceby¿ev Functional	p. 137
Ostrowski Type Inequality with End Interval Means	p. 146
Multidimensional Integration via Ostrowski Dimension Reduction	p. 151
Multidimensional Integration via Trapezoid and Three Point Generators with Dimension Reduction	p. 157
Relationships between Ostrowski, Trapezoidal, and Ceby¿ev Functionals	p. 168
Perturbed Trapezoidal and Midpoint Rules	p. 174
A Ceby¿ev Functional and Some Ramifications	p. 181
Weighted Three Point Quadrature Rules	p. 190
Grüss Type Inequalities and Related Results	p. 195
The Grüss Integral Inequality	p. 195
The Grüss-Ceby¿ev Integral Inequality	p. 198
Karamata's Inequality	p. 200
Steffensen's Inequality	p. 202
Young's Inequality	p. 204
Grüss Type Inequalities for the Stieltjes Integral of Bounded Integrands	p. 206
Grüss Type Inequalities for the Stieltjes Integral of Lipschitzian Integrands	p. 213
Other Grüss Type Inequalities for the Riemann-Stieltjes Integral	p. 223
Inequalities for Monotonic Integrators	p. 228
Generalisations of Steffensen's Inequality over Subintervals	p. 232
Inequalities in Inner Product Spaces	p. 241
Schwarz's Inequality in Inner Product Spaces	p. 241
A Conditional Refinement of the Schwarz Inequality	p. 243
The Duality Schwarz-Triangle Inequalities	p. 246
A Quadratic Reverse for the Schwarz Inequality	p. 249
A Reverse of the Simple Schwarz Inequality	p. 255
A Reverse of Bessel's Inequality	p. 259
Reverses for the Triangle Inequality in Inner Product Spaces	p. 262
The Boas-Bellman Inequality	p. 268
The Bombieri Inequality	p. 275
Kurepa's Inequality	p. 288
Buzano's Inequality	p. 292
A Generalisation of Buzano's Inequality	p. 298
Generalisations of Precupanu's Inequality	p. 302
The Dunkl-Williams Inequality	p. 308
The Grüss Inequality in Inner Product Spaces	p. 310
A Refinement of the Grüss Inequality in Inner Product Spaces	p. 313
Inequalities in Normed Linear Spaces and for Functionals	p. 319
A Multiplicative Reverse for the Continuous Triangle Inequality	p. 319
Additive Reverses for the Continuous Triangle Inequality	p. 324
Reverses of the Discrete Triangle Inequality in Normed Spaces	p. 330
Other Multiplicative Reverses for a Finite Sequence of Functionals	p. 339
The Diaz-Metcalf Inequality for Semi-Inner Products	p. 342
Multiplicative Reverses of the Continuous Triangle Inequality	p. 345
Reverses in Terms of a Finite Sequence of Functionals	p. 348
Generalisations of the Hermite-Hadamard Inequality for Isotonic Linear Functionals	p. 354
A Symmetric Generalisation	p. 357
Generalisations of the Hermite-Hadamard Inequality for Isotonic Sublinear Functionals	p. 362
References	p. 371
Index	p. 383
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Mathematical Inequalities: A Perspective

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Mathematical Inequalities: A Perspective

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