Problem Books in Mathematics Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA More information about this series at http://www.springer.com/series/714
Hayk Sedrakyan Nairi Sedrakyan Algebraic Inequalities 123
Hayk Sedrakyan University Pierre and Marie Curie Paris, France Nairi Sedrakyan Yerevan, Armenia ISSN 0941-3502 ISSN 2197-8506 (electronic) Problem Books in Mathematics ISBN 978-3-319-77835-8 ISBN 978-3-319-77836-5 (ebook) https://doi.org/10.1007/978-3-319-77836-5 Library of Congress Control Number: 2018934928 Mathematics Subject Classification (2010): 97U40, 00A07, 26D05 Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To Margarita, a wonderful wife and a loving mother To Ani, a wonderful daughter and a loving sister
Preface In mathematics one often deals with inequalities. This book is designed to teach the reader new and classical techniques for proving algebraic inequalities. Moreover, each chapter of the book provides a technique for proving a certain type of inequality. The book includes techniques of using the relationship between the arithmetic, geometric, harmonic, and quadratic means, the principle of mathematical induction, the change of variable(s) method, techniques using the Cauchy Bunyakovsky Schwarz inequality, Jensen s inequality, and Chebyshev s properties of functions, among others. The main idea behind of the proof techniques discussed in this book is making the complicated simple, so that even a beginner can understand complicated inequalities, their proofs and applications. This approach makes it possible not only to prove a large variety of inequalities, but also to solve problems related to inequalities. To explain each technique of proof, we provide examples and problems with complete proofs or hints. At the end of each chapter there are problems for independent study. In Chapter 14 (Miscellaneous Inequalities) are included inequalities whose proofs employ various techniques not covered in the preceding chapters. In some cases, the proofs of Chapter 14 use several proof techniques from the preceding chapters simultaneously. One hundred selected inequalities and their hints are also provided in the end of Chapter 14, and interested readers are encouraged to choose and provide any methods of proofs they prefer. In each chapter we have tried to include inequalities belonging to the same topic and to present them in order of increasing difficulty, using principles similar to those in [11]. This allows the reader to try to prove these inequalities step by step and to refer to the provided proofs only when difficulties arise. We recommend to use the proofs provided in the book, paying more attention to the choice of the mathematical proof technique. Most of the inequalities in this book were created by the authors. Nevertheless, some of the inequalities were proposed in different mathematical olympiads in different countries or have been published elsewhere (including author-created inequalities). However, the provided solutions are different from the original ones. Most such inequalities are included in the books [2, 5, 6, 7, 8, 9], and since the vii
viii Preface name of the author of individual inequalities is unknown to us, we cite these books as the main references. However, for well-known inequalities we have tried to provide the name of the authors. This book was published in Seoul in Korean [13, 14] and is based on [16], which was later published in Moscow in Russian [15]. The historical origins provided at the beginning of some chapters are mostly based on [10] or our personal knowledge. It was considered appropriate to give the proofs of each chapter at the end of the same chapter. Paris, France Yerevan, Armenia Hayk Sedrakyan Nairi Sedrakyan
Contents 1 Basic Inequalities and Their Applications... 1 2 Sturm s Method... 9 3 The HM-GM-AM-QM Inequalities... 21 4 The Cauchy Bunyakovsky Schwarz Inequality... 45 5 Change of Variables Method... 59 6 Using Symmetry and Homogeneity... 71 7 The Principle of Mathematical Induction... 81 8 A Useful Inequality... 107 9 Using Derivatives and Integrals... 127 10 Using Functions... 143 11 Jensen s Inequality... 155 12 Inequalities of Sequences... 177 13 Algebraic Inequalities in Number Theory... 187 14 Miscellaneous Inequalities... 197 Appendix Power Sums Triangle... 239 Bibliography... 243 ix
About the Authors Hayk Sedrakyan is an IMO medal winner, a professor of mathematics in Paris, France, and a professional math olympiad coach in the greater Boston area, Massachusetts, USA. He received his doctorate in mathematics at the Université Pierre et Marie Curie, Paris, France. Hayk is a Doctor of Mathematical Sciences in USA, France, Armenia. He has been awarded master s degrees in mathematics from Germany, Austria, and Armenia and completed part of his doctoral studies in Italy. Hayk has authored several books on the topic of problem-solving and olympiad-style mathematics published in USA and South Korea. Nairi Sedrakyan has long been involved in national and international mathematical olympiads, having served as an International Mathematical Olympiad (IMO) problem selection committee member and the president of Armenian Mathematics Olympiads. He is the author of one of the hardest problems ever proposed in the history of International Mathematical Olympiads (the fifth problem of the 37th IMO). He has been the leader of the Armenian IMO Team, jury member of the IMO, jury member and problem selection committee member of the Zhautykov International Mathematical Olympiad (ZIMO), jury member and problem selection committee member of the International Olympiad of Metropolises, and the president of the International Mathematical Olympiad Tournament of the Towns in Armenia. He is also the author of a large number of problems proposed in these olympiads and has authored several books on the topic of problem-solving and olympiad-style mathematics published in United States, Russia, Armenia, and South Korea. The students of Nairi Sedrakyan have obtained 20 medals (1 gold medal, 4 silver medals, 15 bronze medals) in IMO. For his outstanding teaching, Nairi Sedrakyan received the title of best teacher of the Republic of Armenia and was awarded special recognition by the prime minister. xi
Overview This book is designed to teach the reader new and classical mathematical proof techniques for proving inequalities, in particular, to prove algebraic inequalities. These proof techniques and methods are applied to prove inequalities of various types. The main idea behind this book and the proof techniques discussed is making the complicated simple, so that even a beginner can understand complicated inequalities, their proofs and applications. The book Algebraic Inequalities is also devoted to the topic of inequalities and can be considered a continuation of the book Geometric Inequalities: Methods of proving [12]. It can serve teachers, high-school students, and mathematical competitors. xiii