Inequalities: theory of majorization and its applications
type: Item
instance of: scholarly article
title: Inequalities: theory of majorization and its applications
zbMATH Open document ID: 1219.26003
DOI: 10.1007/978-0-387-68276-1
author: Albert W. Marshall
author: Barry C. Arnold
author: Ingram Olkin
published in: Springer Series in Statistics
publication date: 2010-12-28T00:00:00Z
review text: The book present a comprehensive (909 pages) overview of majorization theory, including definitions and (in some places) proofs. The book is divided into five parts: Theory of majorization (Chapters 1--6), Mathematical applications (7--10), Stochastic applications (11--13), Generalizations (14--15), and Complementary topics (16--20). Chapter 1 (Introduction) offers an overview of basic results. Chapter 2 (Doubly stochastic matrices) discusses these matrices and offers an insight into the geometry of majorization. Chapter 3 (Schur-convex functions) studies functions preserving the order of majorization. Chapters 4 (Equivalent conditions for majorization), 5 (Preservation and generation of majorization) and 6 (Rearrangements and majorization) show equivalent conditions and some ways in which the ordering can arise. The following chapters are devoted to mathematical applications: Chapter 7 (Combinatorial analysis), 8 (Geometric inequalities), 9 (Matrix theory), and to applications in statistics: Chapter 11 (Stochastic majorizations), 12 (Probabilistic, statistical and other applications), and 13 (Additional statistical applications). Generalizations are included in Chapter 14 (Orderings extending majorization) and 15 (Multivariate majorization). The aim of the final part is to provide necessary and complementary topics on convex functions and classical inequalities. It includes Chapter 16 (Convex functions and some classical inequalities), 17 (Stochastic ordering), 18 (Total positivity), 19 (Matrix factorizations, compounds, direct products, and \(M\)-matrices), and 20 (Extremal representations of matrix functions). For a review of the first edition (1979) see Zbl 0437.26007.
reviewed by: Vladimír Janiš
Mathematics Subject Classification ID: 26-01
Mathematics Subject Classification ID: 26-02
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D20
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 5829875
zbMATH Keywords: majorization
zbMATH Keywords: rearrangements
zbMATH Keywords: doubly stochastic matrices
zbMATH Keywords: convex functions
zbMATH Keywords: inequalities
zbMATH Keywords: combinatorial analysis
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-387-68276-1
OpenAlex ID: W4292058084
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description: scientific article; zbMATH DE number 5829875