Inequalities: Theory of Majorization and Its Applications PDF
Preview Inequalities: Theory of Majorization and Its Applications
Springer Series in Statistics Advisors: P.Bickel,P.Diggle,S.Feinberg,U.Gather, I.Olkin,S.Zeger Forothertitlespublishedinthisseries,goto http://www.springer.com/series/692 Albert W. Marshall Ingram Olkin • Barry C. Arnold Inequalities: Theory of Majorization and Its Applications Second Edition ABC AlbertW.Marshall BarryC.Arnold DepartmentofStatistics DepartmentofStatistics UniversityofBritishColumbia UniversityofCalifornia Vancouver,BCV6T1Z2 Riverside,CA92521 Canada USA and mailing address [email protected] 2781W.ShoreDrive LummiIsland,WA98262 [email protected] IngramOlkin DepartmentofStatistics StanfordUniversity Stanford,CA94305 USA [email protected] ISSN0172-7397 ISBN978-0-387-40087-7 e-ISBN978-0-387-68276-1 DOI10.1007/978-0-387-68276-1 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2010931704 (cid:2)c SpringerScience+BusinessMedia,LLC2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To our long-suffering wives for their patience with this project: Sheila (AWM), Anita (IO), Carole (BCA) To the memory of Z.W. (Bill) Birnbaum and Edwin Hewitt who initiated my interest in inequalities (AWM) To my students and colleagues who have energized, enriched and enlivened my life (IO) To the memory of Peggy Franklin (BCA) Preface and Acknowledgments from the First Edition Preface Although they play a fundamental role in nearly all branches of math- ematics, inequalities are usually obtained by ad hoc methods rather than as consequences of some underlying “theory of inequalities.” For certain kinds of inequalities, the notion of majorization leads to such a theory that is sometimes extremely useful and powerful for deriv- ing inequalities. Moreover, the derivation of an inequality by methods of majorization is often very helpful both for providing a deeper understanding and for suggesting natural generalizations. As the 1960s progressed, we became more and more aware of these facts. Our awareness was reinforced by a series of seminars we gave while visiting the University of Cambridge in 1967–1968. Because the ideas associated with majorization deserve to be better known, we decided by 1970 to write a little monograph on the subject—one that might have as many as 100 pages—and that was the genesis of this book. Theideaof majorization is aspecial caseof several moregeneral no- tions,butthesegeneralizations arementioned inthisbookonlyforthe perspective they provide. We have limited ourselves to various aspects of majorization partly because we want to emphasize its importance and partly because its simplicity appeals to us. However, to make the vii viii Preface and Acknowledgments from the First Edition book reasonably self-contained, five chapters at the end of the book are included which contain complementary material. Becausethebasicideasofmajorization areelementary,weoriginally intended to write a book accessible at least to advanced undergradu- ate or beginning graduate students. Perhaps to some degree we have succeeded in this aim with the first 10 chapters of the book. Most of thesecond 10 chapters involve moresophistication, and there the level andrequiredbackgroundarequiteuneven.However,anyonewishingto employmajorization asatoolinapplicationscanmakeuseofthetheo- remswithoutstudyingtheirproofs;forthemostpart,theirstatements are easily understood. The book is organized so that it can be used in a variety of ways for a variety of purposes. Sequential reading is not necessary. Extensive crossreferencinghasbeenattemptedsothatrelatedmaterialcaneasily be found; we hope this will enhance the book’s value as a reference. For the same purpose, a detailed table of contents and an extensive index are also provided. Basic background of interest to all readers is found in Chapters 1 and 4, with Chapter 5 as a reference. See also the Basic Notation and Terminology immediately following the Acknowledgments. Technical details concerning majorization are given in Chapters 2 and 3 (especially important are Sections 2.A, 2.B, and 3.A). Added perspective is given in Chapters 14 and 15. Analytic inequalities are discussed in Chapter 3 and in Sections 16.A–16.D, with Chapter 6 also of some relevance. Elementary geometric inequalities are found in Chapter 8. Combinatorics arediscussedprimarilyinChapter7,butChapters2, 6, and Section 5.D are also pertinent. Matrix theory is found especially in Chapters 9 and 10, but also in Chapters 2, 19, 20, and Sections 16.E and 16.F. Numerical analysis is found in Chapter 10; Chapters 2 and 9 and Sections 16.E and 16.F may also be of interest. Probability and statistics are discussed primarily in Chapters 11–13, and also in Chapters 15, 17, and 18. Partly for historical interest, we have tried to give credit to original authors and to cite their original writings. This policy resulted in a bibliography of approximately 450 items. Nevertheless, it is surely far from being complete. As Hardy, Littlewood, and Po´lya (1934, 1952) say in the preface to the first edition of their book on inequalities: Historical and bibliographical questions are particularly troublesome in a subject like this, which has application in Preface and Acknowledgments from the First Edition ix every part of mathematics but has never been developed systematically. It is often really difficult to trace the origin of a familiar inequality. It is quite likely to occur first as an auxiliary proposition, often without explicit statement, in a memoir on geometry or astronomy; it may have been rediscovered, many years later, by half a dozen different authors,... We apologize for the inevitable errors of omission or commission that have been made in giving credits for various results. Occasionally the proofs provided by original authors have been re- produced. More often, new proofs are given that follow the central theme of majorization and build upon earlier results in the book. Acknowledgments The photographs in this book were collected only through the gen- erosity of a number of people. G. Po´lya provided the photos of himself and of I. Schur. A. Gillespie was instrumental in tracing members of the family of R. F. Muirhead; photos of him were loaned to us by W. A. Henderson, and they were expertly restored by John Coury. Trinity College provided a photo of J. E. Littlewood and a photo of G.H.HardyandJ.E.Littlewood together.ThephotosofG.H.Hardy and H. Dalton were obtained from the Radio Times Hulton Picture Library, London. We have been heavily influenced by the books of Hardy, Littlewood, and Po´lya (1934, 1952), Beckenbach and Bellman (1961), and Mitri- novi´c (1970); to these authors we owe a debt of gratitude. We are also indebted to numerous colleagues for comments on various versions of the manuscript. In addition to many errors that were called to our at- tention, very significant substantive comments were made, enabling us toconsiderablyimprovethemanuscript.Inparticular,weacknowledge such help from KumarJogdeo, Frank Proschan, RobertC.Thompson, YungLiangTong, andRobertA.Wijsman.Koon-Wing Chengwas es- pecially helpful with Chapters 2 and 3, and Michael D. Perlman gave us insightful comments aboutChapters 11 and 12. Moshe Shaked read a number of drafts and contributed both critical comments and bibli- ographic material over a period of several years. Perceptive comments about several chapters were made by Tom Snijders; in particular, Chapter 17 would not have been written in its present form without his comments. Friedrich Pukelsheim read nearly all of the manuscript; his meticulously detailed comments were invaluable to us. x Preface and Acknowledgments from the First Edition As visitors to Western Washington State University and Imperial College, we were generously granted the same use of facilities and services as were regular members of the faculty. Much work on the manuscript was accomplished at these institutions. The National Science Foundation has contributed essential financial support throughout the duration of this project, and support has also beenprovided forthepasttwo years bytheNational Research Council of Canada. Typing of the manuscript has been especially difficult because of our many revisions and corrections. The dependability, enduring pa- tience, and accurate and efficient services of Carolyn Knutsen and Nancy Steege through the duration of this project are most gratefully acknowledged. Albert W. Marshall Ingram Olkin
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