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 0-511-21311-5 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 (EBL) eBook (EBL) 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 [...]... informative With the Cauchy–Schwarz inequality as the 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... is defined by the sum 2 pθ (k; θ) p(k; θ) I(θ) = p(k; θ), (1.30) k∈D where pθ (k; θ) = ∂p(k; θ)/∂θ The quantity defined by the left side of the bound (1.29) is called the variance of the unbiased estimator g, and the quantity I(θ) is known as the Fisher information at θ of the model e Mθ The inequality (1.29) is known as the Cram´r–Rao lower bound, and it has extensive applications in mathematical statistics... b 2 , n 1 2 and there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics A central aim of this course — or master class — is to suggest a path to mastery of this inequality, its many extensions, and its many applications — from the most basic to the most sublime The Typical Plan The typical chapter in this course is built around the solution of... 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 only riotous confusion would o result The most... Count= 0 Preface In the fine arts, a master class is a small class where students and coaches work together to support a high level of technical and creative excellence This book tries to capture the spirit of a master class while providing coaching for readers who want to refine their skills as solvers of problems, especially those problems dealing with mathematical inequalities The most important prerequisite... Cauchy’s proof via the imaginative leap-forward fall-back induction is a priceless part of the world’s mathematical inheritance, some of the alternative proofs are just as well loved One The AM-GM Inequality 23 Fig 2.1 The line y = 1 + x is tangent to the curve y = ex at the point x = 0, and the line is below the curve for all x ∈ R Thus, we have 1 + x ≤ ex for all x ∈ R, and, moreover, the inequality... to one 24 The AM-GM Inequality In the AM-GM inequality (2.9) the left-hand side contains a product of terms, and the analytic inequality 1 + x ≤ ex stands ready to bound such a product by the exponential of a sum Moreover, there are two ways to exploit this possibility; we could write the multiplicands ak in the form 1 + xk and then apply the analytic inequality (2.8), or we could modify the inequality... takes the sequence with kj copies of aj for each 1 ≤ j ≤ n and then applies the plain vanilla AM-GM inequality (2.3); there is nothing more to it, or, at least there is nothing more if we attend strictly to the stated problem Nevertheless, there is a further observation one can make Once the result (2.7) is established for rational values, the same inequality follows for general values of pj “just by... 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 xy ≤ (x2 + y 2 )/2 , we also know that 1... a o nonnegative function, except P´lya calls on the function x → ex rather o than the function x → x2 The graph of y = ex in Figure 2.1 illustrates o the property of y = ex that is the key to P´lya’s proof; specifically, it shows that the tangent line y = 1 + x runs below the curve y = ex , so one has the bound 1 + x ≤ ex for all x ∈ R (2.8) Naturally, there are analytic proofs of this inequality; for . 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. 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. attention on the nontrivial case where neither of the sequences is identically zero and where both of the sums on the right- hand side of the identity (1.8) are finite, then we see that each of the steps