Mathematical Inequalities: A Perspective

Mathematical Inequalities......Page 2
Mathematical Inequalities......Page 3
Contents......Page 6
Preface......Page 10
1.1 An Elementary Inequality for Two Numbers......Page 12
1.2 An Elementary Inequality for Three Numbers......Page 14
1.3 A Weighted Inequality for Two Numbers......Page 15
1.4 The Abel Inequality......Page 17
1.5 The Biernacki, Pidek, and Ryll- Nardzewski ( BPR) Inequality......Page 20
1.6 Cebysev\'s Inequality for Synchronous Sequences......Page 22
1.7 The Cauchy-Bunyakovsky- Schwarz (CBS) Inequality for Real Numbers......Page 24
1.8 The Andrica- Badea Inequality......Page 26
1.9 A Weighted Gruss Type Inequality......Page 28
1.10 Andrica- Badea\'s Refinement of the Gruss Inequality......Page 30
1.11 Cebysev Type Inequalities......Page 35
1.12 De Bruijn\'s Inequality......Page 40
1.13 Daykin- Eliezer- Carlitz\'s Inequality......Page 41
1.14 Wagner\'s Inequality......Page 44
1.15 The Polya- Szego Inequality......Page 46
1.16 The Cassels Inequality......Page 48
1.17 Holder\'s Inequality for Sequences of Real Numbers......Page 50
1.18 The Minkowski Inequality for Sequences of Real Numbers......Page 52
1.19 Jensen\'s Discrete Inequality......Page 54
1.20 A Converse of Jensen\'s Inequality for Differentiable Mappings......Page 56
1.21 The Petrovic Inequality for Convex Functions......Page 57
1.22 Bounds for the Jensen Functional in Terms of the Second Derivative......Page 59
1.23 Slater\'s Inequality for Convex Functions......Page 61
1.24 A Jensen Type Inequality for Double Sums......Page 63
2.1 The Hermite-Hadamard Integral Inequality......Page 66
2.2 Hermite-Hadamard Related Inequalities......Page 68
2.3 Hermite- Hadamard Inequality for Log- Convex Mappings......Page 82
2.4 Hermite- Hadamard Inequality for the Godnova- Levin Class of Functions......Page 84
2.5 The Hermite- Hadamard Inequality for Quasi- Convex Functions......Page 87
2.6 The Hermite- Hadamard Inequality of s-Convex Functions in the Orlicz Sense......Page 93
2.7 The Hermite- Hadamard Inequality for s-Convex Functions in the Breckner Sense......Page 105
2.8 Inequalities for Hadamard\'s Inferior and Superior Sums......Page 113
2.9 A Refinement of the Hermite- Hadamard Inequality for the Modulus......Page 120
3.1 Ostrowski\'s Integral Inequality for Absolutely Continuous Mappings......Page 126
3.2 Ostrowski\'s Integral Inequality for Mappings of Bounded Variation......Page 129
3.3 Trapezoid Inequality for Functions of Bounded Variation......Page 131
3.4 Trapezoid Inequality for Monotonic Mappings......Page 133
3.5 Trapezoid Inequality for Absolutely Continuous Mappings......Page 136
3.6 Trapezoid Inequality in Terms of Second Derivatives......Page 138
3.7 Generalised Trapezoid Rule Involving......Page 140
3.8 A Refinement of Ostrowski\'s Inequality for the Cebysev Functional......Page 148
3.9 Ostrowski Type Inequality with End Interval Means......Page 157
3.10 Multidimensional Integration via Ostrowski Dimension Reduction......Page 162
3.11 Multidimensional Integration via Trapezoid and Three Point Generators with Dimension Reduction......Page 168
3.12 Relationships between Ostrowski, Trapezoidal, and Cebysev Functionals......Page 179
3.13 Perturbed Trapezoidal and Midpoint Rules......Page 185
3.14 A Cebysev Functional and Some Ramifications......Page 192
3.15 Weighted Three Point Quadrature Rules......Page 201
4.1 The Gruss Integral Inequality......Page 206
4.2 The Gruss-Cebysev Integral Inequality......Page 209
4.3 Karamata\'s Inequality......Page 211
4.4 Steffensen\'s Inequality......Page 213
4.5 Young\'s Inequality......Page 215
4.6 Gruss Type Inequalities for the Stieltjes Integral of Bounded Integrands......Page 217
4.7 Gruss Type Inequalities for the Stieltjes Integral of Lipschitzian Integrands......Page 224
4.8 Other Gruss Type Inequalities for the Riemann-Stieltjes Integral......Page 234
4.9 Inequalities for Monotonic Integrators......Page 239
4.10 Generalisations of Steffensen\'s Inequality over Subintervals......Page 243
5.1 Schwarz\'s Inequality in Inner Product Spaces......Page 252
5.2 A Conditional Refinement of the Schwarz Inequality......Page 254
5.3 The Duality Schwarz-Triangle Inequalities......Page 257
5.4 A Quadratic Reverse for the Schwarz Inequality......Page 260
5.5 A Reverse of the Simple Schwarz Inequality......Page 266
5.6 A Reverse of Bessel\'s Inequality......Page 270
5.7 Reverses for the Triangle Inequality in Inner Product Spaces......Page 273
5.8 The Boas- Bellman Inequality......Page 279
5.9 The Bombieri Inequality......Page 286
5.10 Kurepa\'s Inequality......Page 299
5.11 Buzano\'s Inequality......Page 303
5.12 A Generalisation of Buzano\'s Inequality......Page 309
5.13 Generalisations of Precupanu\'s Inequality......Page 313
5.14 The Dunkl-Williams Inequality......Page 319
5.15 The Gruss Inequality in Inner Product Spaces......Page 321
5.16 A Refinement of the Gruss Inequality in Inner Product Spaces......Page 324
6.1 A Multiplicative Reverse for the Continuous Triangle Inequality......Page 330
6.2 Additive Reverses for the Continuous Triangle Inequality......Page 335
6.3 Reverses of the Discrete Triangle Inequality in Normed Spaces......Page 341
6.4 Other Multiplicative Reverses for a Finite Sequence of Functionals......Page 350
6.5 The Diaz-Metcalf Inequality for Semi-Inner Products......Page 353
6.6 Multiplicative Reverses of the Continuous Triangle Inequality......Page 356
6.7 Reverses in Terms of a Finite Sequence of Functionals......Page 359
6.8 Generalisations of the Hermite- Hadamard Inequality for Isotonic Linear Functionals......Page 365
6.9 A Symmetric Generalisation......Page 368
6.10 Generalisations of the Hermite- Hadamard Inequality for Isotonic Sublinear Functionals......Page 373
References......Page 382