For more than a century, the study of various types of inequalities has been the focus of great attention by many researchers, interested both in the theory and its applications. In particular, there exists a very rich literature related to the well known Cebysev, Gruss, Trapezoid, Ostrowski, Hadamard and Jensen type inequalities. The present monograph is an attempt to organize recent progress related to the above inequalities, which we hope will widen the scope of their applications. The field to be covered is extremely wide and it is impossible to treat all of these here. The material included in the monograph is recent and hard to find in other books. It is accessible to any reader with a reasonable background in real analysis and an acquaintance with its related areas. All results are presented in an elementary way and the book could also serve as a textbook for an advanced graduate course. The book deserves a warm welcome to those who wish to learn the subject and it will also be most valuable as a source of reference in the field. It will be invaluable reading for mathematicians and engineers and also for graduate students, scientists and scholars wishing to keep abreast of this important area of research.
Author(s): B. G. Pachpatte
Edition: 2012
Publisher: Atlantis Press
Year: 2012
Language: English
Pages: 314
Tags: Математика;Прочие разделы математики;
Cover......Page 1
Title Page......Page 2
Analytic Inequalities......Page 4
Preface......Page 8
Contents......Page 10
Introduction......Page 12
1.2 Gr¨ uss-type inequalities......Page 18
1.3 C ebysev-type inequalities......Page 25
1.4 Inequalities of the Gru¨ ss- and C ebysev-type......Page 32
1.5 More inequalities of the Gru¨ ss-and C ebysev-type......Page 42
1.6 Discrete Inequalities of the Gru¨ ss-and C ebysev-type......Page 56
1.7.1 Estimation of the remainder in the Trapezoid formula......Page 62
1.7.2 Bounds for a perturbed generalized Taylor’s formula......Page 64
1.7.3 Some inequalities for expectation and the cumulative distribution functions......Page 69
1.8.3 Pachpatte [113]......Page 72
1.8.6 Pachpatte [137]......Page 73
1.8.8 Pachpatte [137]......Page 74
1.8.11 Pachpatte [124]......Page 75
1.8.13 Pachpatte [124]......Page 76
1.8.15 Pachpatte [132]......Page 77
1.8.17 Pachpatte [119]......Page 78
1.8.20 Dragomir and Khan [51]......Page 79
1.8.22 Pachpatte [105]......Page 80
1.9 Notes......Page 81
2.2 Some Gr¨ uss-type inequalities in inner product spaces......Page 82
2.3 Gru¨ ss-and C ebysev-type inequalities in two and three variables......Page 91
2.4 Trapezoid-type inequalities in two variables......Page 104
2.5 Some multivariate Gr¨ uss-type integral inequalities......Page 111
2.6 Multivariate Gru¨ ss-and C ebysev-type discrete inequalities......Page 118
2.7.1 Some integral inequalities via Gr¨uss inequality in inner product spaces......Page 125
2.7.2 Application to numerical integration......Page 128
2.7.3 Approximation for the finite Fourier transform of two independentvariables......Page 131
2.8.2 Ujevi´c [153]......Page 136
2.8.4 Dragomir [55]......Page 137
2.8.6 Hanna, Dragomir and Roumeliotis [67]......Page 138
2.8.8 Hanna, Dragomir and Cerone [62]......Page 139
2.8.10 Pachpatte [94]......Page 140
2.9 Notes......Page 141
3.2 Inequalities of the Ostrowski-type......Page 144
3.3 Ostrowski-type inequalities involving the product of two functions......Page 154
3.4 Inequalities of the Ostrowski-and Gr¨ uss-type......Page 162
3.5 Further inequalities of the Ostrowski-type......Page 174
3.6 Discrete Ostrowski-type inequalities......Page 180
3.7.1 Applications for some special means......Page 185
3.7.2 Applications in numerical integration......Page 187
3.7.3 More applications in numerical integration......Page 191
3.8.2 Cheng [19]......Page 192
3.8.5 Pachpatte [123]......Page 193
3.8.8 Dragomir and Barnett [28]......Page 194
3.8.10 Pachpatte [100]......Page 195
3.8.12 Pachpatte [124]......Page 196
3.8.14 Pachpatte [102]......Page 197
3.8.17 Pachpatte [120]......Page 198
3.8.18 Pachpatte [120]......Page 199
3.9 Notes......Page 200
4.2 Ostrowski-type inequalities in two variables......Page 202
4.3 Ostrowski-type inequalities in three variables......Page 216
4.4 Ostrowski-type inequalities in several variables......Page 228
4.5 More Ostrowski-type inequalities in several variables......Page 233
4.6 Discrete inequalities of the Ostrowski-type in several variables......Page 238
4.7.1 Applications for cubature formulae......Page 248
4.7.2 More applications for cubature formulae......Page 249
4.7.3 Application to numerical integration......Page 251
4.8.2 Hanna and Roumeliotis [63]......Page 253
4.8.3 Hanna and Dragomir [65]......Page 254
4.8.5 Milovanovi´c [76]......Page 255
4.8.8 Anastassiou [2]......Page 256
4.8.11 Pachpatte[83]......Page 257
4.8.14 Pachpatte [95]......Page 258
4.9 Notes......Page 259
5.2 Integral inequalities involving convex functions......Page 260
5.3 Further integral inequalities involving convex functions......Page 268
5.4 Integral inequalities involving log-convex functions......Page 272
5.5 Discrete inequalities involving log-convex functions......Page 282
5.6 Discrete inequalities for differentiable convex functions......Page 287
5.7.1 Applications for special means......Page 292
5.7.2 Applications for some inequalities related to Ky Fan’s inequality......Page 294
5.7.3 Applications for Shannon’s entropy......Page 296
5.8.1 Klarici´c Bakula and Pecari´c [70]......Page 298
5.8.4 Pachpatte [98]......Page 299
5.8.7 Pachpatte [82]......Page 300
5.8.11 Pachpatte [107]......Page 301
5.8.12 Pachpatte [107]......Page 302
5.9 Notes......Page 303
Bibliography......Page 304
Subject Index......Page 312
