Inequalities : theory of majorization and its applications PDF
Preview Inequalities : theory of majorization and its applications
This is Volume 143 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request. Inequalities: Theory of Majorization and Its Applications Albert W. Marshall Department of Mathematics University of British Columbia Vancouver, British Columbia Canada Ingram Olkin Department of Statistics Stanford University Stanford, California 1979 ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers New York London Toronto Sydney San Francisco COPYRIGHT © 1979, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX Library of Congress Cataloging in Publication Data Marshall, Albert W Inequalities. (Mathematics in science and engineering) Bibliography: p. Includes index. 1. Inequalities (Mathematics) I. Olkin, Ingram, joint author. II. Title. III. Title: Majorization and its applications. IV. Series. QA295.M42 1979 515'.26 79-50218 ISBN 0-12-473750-1 PRINTED IN THE UNITED STATES OF AMERICA 79 80 81 82 9 8 7 6 5 4 3 2 1 To Sheila Rhoda Preface Although they play a fundamental role in nearly all branches of mathematics, 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 deriving in equalities. Moreover, the derivation of an inequality by methods of majorization is often very helpful both for providing a deeper understand ing 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 as sociated 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. The idea of majorization is a special case of several more general no tions, but these generalizations are mentioned in this book only for the 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 book reasonably self-contained, five chapters at the end of the book are in cluded which contain complementary material. Because the basic ideas of majorization are elementary, we originally intended to write a book accessible at least to advanced undergraduate or xiii XIV PREFACE beginning graduate students. Perhaps to some degree we have succeeded in this aim with the first ten chapters of the book. Most of the second ten chapters involve more sophistication, and there the level and required background is quite uneven. However, anyone wishing to employ majori- zation as a tool in applications can make use of the theorems without studying their proofs; for the most part, their statements are easily under stood. 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 cross referencing has been attempted so that related material can easily 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 Ter minology immediately following the Acknowledgments. Theoretical details concerning majorization are given in Chapters 2 and 3 (especially important are Sections 2.A, 2.B, and 3.A). Added perspec tive 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 are discussed primarily in Chapter 7, but Chapters 2, 6, and Section 5.D are also pertinent. Matrix theory is found especially in Chapters 9 and 10, but also in Chap ters 2, 19, 20, and Sections 16.E and 16.F. Numerical analysis is found in Chapter 10; Chapters 2 and 9 and Sec tions 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 bibliog raphy of approximately 450 items. Nevertheless, it is surely far from being complete. As Hardy, Littlewood, and Poly a (1934, 1952) say in the pre face to the first edition of their book on inequalities: Historical and bibliographical questions are particularly trouble some in a subject like this, which has applications in 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 with out explicit statement, in a memoir on geometry or astronomy; it may PREFACE XV 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 credit for various results. Occasionally the proofs provided by original authors have been repro duced. 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 genero sity of a number of people. G. Polya provided the photos of himself and of I. Schur. A. Gillespie was instrumental in tracing menjbers 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. Little wood and a photo of G. H. Hardy and J. E. Littlewood together. The photos of G. H. Hardy and H. Dalton were ob tained from the Radio Times Hulton Picture Library, London. We have been heavily influenced by the books of Hardy, Littlewood, and Polya (1934, 1952), Beckenbach and Bellman (1961), and Mitnnovic (1970); to these authors we owe a debt of gratitude. We are also indebted to numerous colleagues for comments on various versions of the manu script. In addition to the many errors that were called to our attention, very significant substantive comments were made, enabling us to considerably improve the manuscript. In particular, we acknowledge such help from Kumar Jogdeo, Frank Proschan, Robert C. Thompson, Yung Liang Tong, and Robert A. Wijsman. Koon-Wing Cheng was especially helpful with Chapters 2 and 3, and Michael D. Perlman gave us insightful comments about Chapters 11 and 12. Moshe Shaked read a number of drafts and contributed both critical comments and bibliographic 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 xvii XV111 ACKNOWLEDGMENTS Pukelsheim read nearly all of the manuscript; his meticulously detailed comments were invaluable to us. As visitors to Western Washington State University and Imperial Col lege, 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 been provided for the past two years by the National Research Council of Canada. Typing of the manuscript has been especially difficult because of our many revisions and corrections. The dependability, enduring patience, and accurate and efficient services of Carolyn Knutsen and Nancy Steege through the duration of this project are most gratefully acknowledged. Basic Notation and Terminology The following notation is used throughout this book. It is given here in tabular form for easy reference. # = (-00,00), # =[0,oo), + # =(0,oo), ++ #" = {(xi, x ):Xi e 01 for all /}, n <T = {(x x ):x· > 0 for all /}, + l9 n t 0\ ={(x , χ ):χι > 0 for all /}, + 1 η ®={(X!, x„):xi > -·>χ„}, ® = {(Xi, x„):xi > *· * > x > 0}, + n ^ = {(x x„):xi > · · · > x„ > 0}. ++ l9 Throughout this book, increasing means nondecreasing and decreasing means nonincreasing. Thus if f\0t -> #, / is increasing ifx<y^f(x)<f(y\ strictly increasing if x<y=>f(x)<f(y), decreasing ifx<y=>f(x)>f(y\ strictly decreasing if x<y=>f(x)>f{y).
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