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CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 ii This page intentionally left blank CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 THE CAUCHY–SCHWARZ MASTER CLASS This lively, problem-oriented text is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy–Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered — either by one of history’s famous mathe- maticians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geome- try of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of H¨older, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics. J. Michael Steele is C. F. Koo Professor of Statistics at the Wharton School, University of Pennsylvania. He is the author of more than 100 mathematical publications, including the books Probability Theory and Combinatorial Optimization and Stochastic Calculus and Financial Applications.Heisalso the founding editor of the Annals of Applied Probability. i CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 ii CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 MAA PROBLEM BOOKS SERIES Problem Books is a series of the Mathematical Association of America consisting of collections of problems and solutions from annual mathematical competitions; compilations of problems (including unsolved problems) specific to particular branches of mathematics; books on the art and practice of problem solving, etc. Committee on Publications Gerald Alexanderson, Chair Roger Nelsen Editor Irl Bivens Clayton Dodge Richard Gibbs George Gilbert Gerald Heuer Elgin Johnston Kiran Kedlaya Loren Larson Margaret Robinson Mark Saul AFriendly Mathematics Competition: 35 Years of Teamwork in Indiana, edited by Rick Gillman The Inquisitive Problem Solver, Paul Vaderlind, Richard K. Guy, and Loren C. Larson Mathematical Olympiads 1998–1999: Problems and Solutions from Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 1999–2000: Problems and Solutions from Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 2000–2001: Problems and Solutions from Around the World, edited by Titu Andreescu, Zuming Feng, and George Lee, Jr. The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938–1964, A. M. Gleason, R. E. Greenwood, and L. M. Kelly The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965–1984, Gerald L. Alexanderson, Leonard F. Klosinski, and Loren C. Larson The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions, and Commentary, Kiran S. Kedlaya, Bjorn Poonen, and Ravi Vakil USA and International Mathematical Olympiads 2000, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2001, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2002, edited by Titu Andreescu and Zuming Feng iii CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 iv CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 THE CAUCHY–SCHWARZ MASTER CLASS An Introduction to the Art of Mathematical Inequalities J. MICHAEL STEELE University of Pennsylvania THE MATHEMATICAL ASSOCIATION OF AMERICA v cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK First published in print format isbn-13 978-0-521-83775-0 isbn-13 978-0-521-54677-5 isbn-13 978-0-511-21134-8 © J. Michael Steele 2004 2004 Information on this title: www.cambrid g e.or g /9780521837750 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. isbn-10 0-511-20776-x isbn-10 0-521-83775-8 isbn-10 0-521-54677-x Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback p a p erback p a p erback eBook (Adobe Reader) eBook (Adobe Reader) hardback CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 Contents Preface page ix 1 Starting with Cauchy 1 2 The AM-GM Inequality 19 3 Lagrange’s Identity and Minkowski’s Conjecture 37 4OnGeometry and Sums of Squares 51 5 Consequences of Order 73 6 Convexity — The Third Pillar 87 7Integral Intermezzo 105 8 The Ladder of Power Means 120 9H¨older’s Inequality 135 10 Hilbert’s Inequality and Compensating Difficulties 155 11 Hardy’s Inequality and the Flop 166 12 Symmetric Sums 178 13 Majorization and Schur Convexity 191 14 Cancellation and Aggregation 208 Solutions to the Exercises 226 Chapter Notes 284 References 291 Index 301 vii CY420/Steele-FM CY420/Steele 0 0521837758 January 16, 2004 17:34 Char Count= 0 viii [...]... attention on the nontrivial case where neither of the sequences is identically zero and where both of the sums on the righthand side of the identity (1.8) are finite, then we see that each of the steps we used in the derivation of the bound (1.7) can be reversed Thus one finds that the identity (1.8) implies the identity ∞ k=1 1 ak ˆk = ˆ b 2 ∞ a2 + ˆk k=1 1 2 ∞ ˆ2 = 1 bk (1.9) k=1 By the two-term bound... that one requires of an inner product Moreover, this example only reveals the tip of an iceberg; there are many useful inner products, and they occur in a great variety of mathematical contexts An especially useful example of an inner product can be given by 8 Starting with Cauchy considering the set V = C[a, b] of real-valued continuous functions on the bounded interval [a, b] and by de ning ·, · on. .. has a deeper physical or geometric interpretation that might reveal the reason for its effectiveness? For nonnegative x and y, the direct term-by-term interpretation of the inequality (2.1) simply says that the area of the rectangle with sides x and y is never greater than the average of the areas of the two squares with sides x and y, and although this interpretation is modestly interesting, one can... applied in earnest by anyone was in 1829, when Cauchy used his inequality in an investigation of Newton’s method for the calculation of the roots of algebraic and transcendental equations This eight-year gap provides an interesting gauge of the pace of science; now, each month, there are hundreds — perhaps thousands — of new scientific publications where Cauchy’s inequality is applied in one way or another... is no hard-and-fast rule to govern that association Sometimes the inequality is named after the first finder, but other principles may apply — such as the framer of the final form, or the provider of the best known application If one were to insist on the consistent use of the rule of first finder, then H¨lder’s inequality would become Rogers’s inequality, Jensen’s inequalo ity would become H¨lder’s inequality,... and consider the quadratic polynomial de ned for t ∈ R by p(t) = v + tw, v + tw 16 Starting with Cauchy Observe that this polynomial is nonnegative and use what you know about the solution of the quadratic equation to prove the inner product version (1.16) of Cauchy’s inequality Also, examine the steps of your proof to establish the conditions under which the case of equality can apply Thus, confirm... initial guide, the reader is led through a sequence of interrelated problems whose solutions are presented as they might have been discovered — either by one of history’s famous mathematicians or by the reader The problems emphasize beauty and surprise, but along the way one finds systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential... inequality provides an upper bound for a sum of pairwise products, and a natural sense of confidence is all one needs to guess that there are also upper bounds for the sums of products of three or more terms In this exercise you are invited to justify two prototypical extensions The first of these is de nitely easy, and the second is not much harder, provided that you do not give it more respect than it deserves:... nonzero vectors v and w, one has v, w = v, v 1 2 w, w 1 2 if and only if v = λw for a nonzero constant λ As before, one may be tempted to respond to this challenge by just rattling off a previously mastered textbook proof, but that temptation should still be resisted The challenge offered by Problem 1.3 is important, and it deserves a fresh response — or, at least, a relatively fresh response For example, it... of technique.” A Retraced Passage — Conversion of an Additive Bound Here we are oddly lucky since we have developed only one technique that is even remotely relevant — the normalization method for converting an additive inequality into one that is multiplicative Normalization means different things in different places, but, if we take our earlier analysis as our guide, what we want here is to replace . isbn-13 97 8-0 -5 2 1-8 377 5-0 isbn-13 97 8-0 -5 2 1-5 467 7-5 isbn-13 97 8-0 -5 1 1-2 113 4-8 © J. Michael Steele 2004 2004 Information on this title: www.cambrid g e.or g /9780521837750 This publication is. Press. isbn-10 0-5 1 1-2 0776-x isbn-10 0-5 2 1-8 377 5-8 isbn-10 0-5 2 1-5 4677-x Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites. be given by 8 Starting with Cauchy considering the set V = C[a, b] of real-valued continuous functions on the bounded interval [a, b] and by de ning ·, · on V by setting f,g =  b a f(x)g(x)