Table Of ContentProblem Books in Mathematics
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
(cid:129)
Algebraic Inequalities
123
Hayk Sedrakyan Nairi Sedrakyan
University Pierre andMarie Curie Yerevan, Armenia
Paris, France
ISSN 0941-3502 ISSN 2197-8506 (electronic)
Problem Booksin Mathematics
ISBN978-3-319-77835-8 ISBN978-3-319-77836-5 (eBook)
https://doi.org/10.1007/978-3-319-77836-5
LibraryofCongressControlNumber:2018934928
MathematicsSubjectClassification(2010): 97U40,00A07,26D05
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Preface
Inmathematicsoneoftendealswithinequalities.Thisbookisdesignedtoteachthe
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,andquadraticmeans,theprincipleofmathematicalinduction,
the change of variable(s) method, techniques using the Cauchy–Bunyakovsky–
Schwarz inequality, Jensen’s inequality, and Chebyshev’s properties offunctions,
amongothers.Themainideabehindoftheprooftechniquesdiscussedinthisbook
is making the complicated simple, so that even a beginner can understand com-
plicatedinequalities,theirproofsandapplications.Thisapproachmakesitpossible
notonlytoprovealargevarietyofinequalities,butalsotosolveproblemsrelatedto
inequalities. To explain each technique of proof, we provide examples and prob-
lems 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.Insomecases,theproofsofChapter14useseveralprooftechniquesfrom
thepreceding chapters simultaneously. One hundred selectedinequalities 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
refertotheprovidedproofsonlywhendifficultiesarise.Werecommend tousethe
proofs provided in the book, paying more attention to the choice of the mathe-
matical 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
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viii Preface
nameoftheauthorofindividualinequalitiesisunknowntous,wecitethesebooks
as the main references. However, for well-known inequalities we have tried to
providethe name of theauthors.This book was published inSeoul inKorean [13,
14]andisbasedon[16],whichwaslaterpublishedinMoscowinRussian[15].The
historical origins provided at the beginning of some chapters are mostly based on
[10] or our personal knowledge.
Itwasconsideredappropriatetogivetheproofsofeachchapterattheendofthe
same chapter.
Paris, France Hayk Sedrakyan
Yerevan, Armenia 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
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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.HereceivedhisdoctorateinmathematicsattheUniversitéPierre
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,havingservedasanInternationalMathematicalOlympiad(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
andproblemselectioncommitteememberoftheZhautykovInternationalMathematical
Olympiad (ZIMO), jury member and problem selection committee member of the
International Olympiad of Metropolises, and the president of the International
MathematicalOlympiadTournamentoftheTownsinArmenia.Heisalsotheauthorof
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.
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Overview
This book is designed to teach the reader new and classical mathematical proof
techniques for proving inequalities, in particular, to prove algebraic inequalities.
Theseprooftechniquesandmethodsareappliedtoproveinequalitiesofvarious
types.Themainideabehindthisbookandtheprooftechniquesdiscussedismaking
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.
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