Algebraic Inequalities
This unique collection of new and classical problems provides full coverage of algebraic inequalities. Many of the exercises are presented with detailed author-prepared-solutions, developing creativity and an arsenal of new approaches for solving mathemat
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Hayk Sedrakyan · Nairi Sedrakyan
Algebraic Inequalities
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 i
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