Convex Functions, Partial Orderings, and Statistical Applications (Mathematics in Science & Engineering)

This research-level book presents up-to-date information concerning recent developments in convex functions and partial orderings and some applications in mathematics, statistics, and reliability theory. The book will serve researchers in mathematical and statistical theory and theoretical and applied reliabilists. Key Features * Presents classical and newly published results on convex functions and related inequalities * Explains partial ordering based on arrangement and their applications in mathematics, probability, statsitics, and reliability * Demonstrates the connection of partial ordering with other well-known orderings such as majorization and Schur functions * Will generate further research and applications Convex Functions, Partial Orderings, and Statistical Applications Copyright Page Contents Preface Notation and Numbering System Chapter 1. Convex Functions 1.1 One-Variable Convex Functions 1.2 Convex Functions on a Normed Linear Space 1.3 Convex Functions of Higher Order 1.4 Functions Convex with Respect to an ECT System of Functions 1.5 Inequalities Involving Derivatives and Differences Chapter 2. Jensen’s and Jensen-Steffensen’s Inequalities 2.1 Jensen’s Inequality 2.2 Jensen–Steffensen’s Inequality 2.3 Companion Inequalities of Jensen’s and Jensen-Steffensen’s Inequalities 2.4 Higher–Order Jensen-Type Inequalities Chapter 3. Reversals, Refinements, and Converses of Jensen’s and Jensen–Steffensen’s Inequalities 3.1 Reversals of Jensen’s and Jensen–Steffensen’s Inequalities 3.2 Some Refinements of Jensen’s and Jensen–Steffensen’s Inequalities 3.3 Converses of Jensen’s Inequality Chapter 4. Applications of Jensen's Inequality to Means and Hölder's Inequalities 4.1 Inequalities for Means 4.2 Hölder's and Minkowski's Inequalities 4.3 Dresher's Inequality 4.4 Beckenbach's Inequality 4.5 Aczél's and Related Inequalities 4.6 Further Generalizations of Hölder's and Minkowski's Inequalities 4.7 Some Inequalities for Complex Functionals and Norms Chapter 5. Hermite-Hadamard's and Jensen-Petrović's Inequalities 5.1 Hermite-Hadamard's Inequality 5.2 Jensen-Petrović's Inequalities Chapter 6. Popoviciu's, Burkill's, and Steffensen's Inequalities 6.1 Inequalities of Popoviciu and Burkill 6.2 Steffensen's Inequality Chapter 7. Cebyšev–Grüss', Favard's, Berwald's, Gauss–Winckler's, and Related Inequalities 7.1 Cebyšev-Grüss Inequality 7.2 Favard's, Berwald's, Gauss–Winckler's, and Related Inequalities Chapter 8. Hardy's, Hilbert's, Opial's, Young's, Nanson's, and Related Inequalities 8.1 Hardy's, Hilbert's, Opial's, and Related Inequalities 8.2 Young's Inequality 8.3 Nanson's Inequality Chapter 9. General Linear Inequalities for Convex Sequences and Functions 9.1 Inequalities for m-Convex Sequences and Functions 9.2 Some Generalizations and Refinements Chapter 10. Orderings and Convexity–Preserving Transformations 10.1 Orderings of Convexity: Generalizations and Related Results 10.2 Various Results Chapter 11. Convex Functions and Geometric Inequalities 11.1 Old and New Results via Majorization Theory 11.2 Concavity via Hyperbolic Forms Chapter 12. Convexity, Majorization, and Schur-Convexity 12.1 Majorization and Convex Functions 12.2 Schur-Convex Functions 12.3 Multivariate Majorization and Convex Functions Chapter 13. Convexity and Log-Concavity Related Moment and Probability Inequalities 13.1 Jensen's Inequality 13.2 Moment Inequalities for Univariate Random Variables 13.3 Dimension-Related Inequalities for Exchangeable Random Variables 13.4 Brunn–Minkowski Inequality 13.5 A Class of Log-Concave Probability Measures 13.6 Some Properties of Log-Concave Density Functions 13.7 Some Statistical Applications Chapter 14. Muirhead's Theorem and Related Inequalities 14.1 Muirhead's Theorem and Generalizations 14.2 Moment Inequalities 14.3 Additional Inequalities for Exchangeable Random Variables 14.4 Inequalities for a Class of Positively Dependent Random Variables 14.5 Applications to Special Families of Random Variables and Distributions Chapter 15. Arrangement Ordering 15.1 Definitions and Basic Properties 15.2 Preservation Properties of Arrangement Increasing Functions 15.3 Arrangement Increasing Property of Overlapping Sums Chapter 16. Applications of Arrangement Ordering 16.1 Moment and Geometric Inequalities 16.2 Arrangement Increasing Probabilities for AI Families of Densities 16.3 Applications to Rank Order Problems 16.4 Monotonicity in the Selection of Populations Chapter 17. Multivariate Arrangement Increasing Functions 17.1 Definition and Basic Properties of Multivariate Arrangement Increasing Functions 17.2 Preservation and Closure Properties of Multivariate Arrangement Increasing Functions 17.3 Applications to Measures of Agreement Among s Judges References Author Index Subject Index Mathematics in Science and Engineering Presents information concerning developments in convex functions and partial orderings and some applications in mathematics, statistics, and reliability theory. This book explains partial ordering based on arrangement and their applications in mathematics, probability, statsitics, and reliability.