Inequalities: theory of majorization and its applications.
2nd edition.

*(English)*Zbl 1219.26003
Springer Series in Statistics. New York, NY: Springer (ISBN 978-0-387-40087-7/hbk; 978-0-387-68276-1/ebook). xxvii, 909 p. (2011).

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.

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.

Reviewer: Vladimír Janiš (Banská Bystrica)

##### MSC:

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26D15 | Inequalities for sums, series and integrals |

26D20 | Other analytical inequalities |

15B51 | Stochastic matrices |

65F30 | Other matrix algorithms (MSC2010) |