Convex Functions, Partial Orderings, and Statistical Applications (eBook)

Front Cover 1
Convex Functions, Partial Orderings, and Statistical Applications 4
Copyright Page 5
Contents 8
Preface 12
Notation and Numbering System 14
Chapter 1. Convex Functions 16
1.1 One-Variable Convex Functions 16
1.2 Convex Functions on a Normed Linear Space 24
1.3 Convex Functions of Higher Order 29
1.4 Functions Convex with Respect to an ECT System of Functions 38
1.5 Inequalities Involving Derivatives and Differences 45
Chapter 2. Jensen’s and Jensen-Steffensen’s Inequalities 58
2.1 Jensen’s Inequality 58
2.2 Jensen–Steffensen’s Inequality 72
2.3 Companion Inequalities of Jensen’s and Jensen-Steffensen’s Inequalities 78
2.4 Higher–Order Jensen-Type Inequalities 86
Chapter 3. Reversals, Refinements, and Converses of Jensen’s and Jensen–Steffensen’s Inequalities 98
3.1 Reversals of Jensen’s and Jensen–Steffensen’s Inequalities 98
3.2 Some Refinements of Jensen’s and Jensen–Steffensen’s Inequalities 102
3.3 Converses of Jensen’s Inequality 113
Chapter 4. Applications of Jensen's Inequality to Means and Hölder's Inequalities 122
4.1 Inequalities for Means 122
4.2 Hölder's and Minkowski's Inequalities 127
4.3 Dresher's Inequality 134
4.4 Beckenbach's Inequality 137
4.5 Aczél's and Related Inequalities 139
4.6 Further Generalizations of Hölder's and Minkowski's Inequalities 141
4.7 Some Inequalities for Complex Functionals and Norms 143
Chapter 5. Hermite–Hadamard's and Jensen–Petrovic's Inequalities 152
5.1 Hermite–Hadamard's Inequality 152
5.2 Jensen–Petrovic's Inequalities 166
Chapter 6. Popoviciu's, Burkill's, and Steffensen's Inequalities 186
6.1 Inequalities of Popoviciu and Burkill 186
6.2 Steffensen's Inequality 196
Chapter 7. Cebyšev–Grüss', Favard's, Berwald's, Gauss–Winckler's, and Related Inequalities 212
7.1 Cebyšev-Grüss Inequality 212
7.2 Favard's, Berwald's, Gauss–Winckler's, and Related Inequalities 227
Chapter 8. Hardy's, Hilbert's, Opial's, Young's, Nanson's, and Related Inequalities 244
8.1 Hardy's, Hilbert's, Opial's, and Related Inequalities 244
8.2 Young's Inequality 254
8.3 Nanson's Inequality 262
Chapter 9. General Linear Inequalities for Convex Sequences and Functions 268
9.1 Inequalities for m-Convex Sequences and Functions 268
9.2 Some Generalizations and Refinements 277
Chapter 10. Orderings and Convexity–Preserving Transformations 292
10.1 Orderings of Convexity: Generalizations and Related Results 292
10.2 Various Results 303
Chapter 11. Convex Functions and Geometric Inequalities 322
11.1 Old and New Results via Majorization Theory 322
11.2 Concavity via Hyperbolic Forms 329
Chapter 12. Convexity, Majorization, and Schur-Convexity 334
12.1 Majorization and Convex Functions 334
12.2 Schur-Convex Functions 347
12.3 Multivariate Majorization and Convex Functions 351
Chapter 13. Convexity and Log-Concavity Related Moment and Probability Inequalities 354
13.1 Jensen's Inequality 354
13.2 Moment Inequalities for Univariate Random Variables 357
13.3 Dimension-Related Inequalities for Exchangeable Random Variables 358
13.4 Brunn–Minkowski Inequality 362
13.5 A Class of Log-Concave Probability Measures 363
13.6 Some Properties of Log-Concave Density Functions 368
13.7 Some Statistical Applications 370
Chapter 14. Muirhead's Theorem and Related Inequalities 376
14.1 Muirhead's Theorem and Generalizations 376
14.2 Moment Inequalities 379
14.3 Additional Inequalities for Exchangeable Random Variables 381
14.4 Inequalities for a Class of Positively Dependent Random Variables 383
14.5 Applications to Special Families of Random Variables and Distributions 386
Chapter 15. Arrangement Ordering 390
15.1 Definitions and Basic Properties 390
15.2 Preservation Properties of Arrangement Increasing Functions 394
15.3 Arrangement Increasing Property of Overlapping Sums 401
Chapter 16. Applications of Arrangement Ordering 406
16.1 Moment and Geometric Inequalities 406
16.2 Arrangement Increasing Probabilities for AI Families of Densities 410
16.3 Applications to Rank Order Problems 412
16.4 Monotonicity in the Selection of Populations 415
Chapter 17. Multivariate Arrangement Increasing Functions 422
17.1 Definition and Basic Properties of Multivariate Arrangement Increasing Functions 422
17.2 Preservation and Closure Properties of Multivariate Arrangement Increasing Functions 428
17.3 Applications to Measures of Agreement Among s Judges 432
References 434
Author Index 472
Subject Index 478
Mathematics in Science and Engineering 484