Table Of ContentVasile CCCCîrtoajje
MATHEMMATICAL
INEQQUAALITIES
Volume 1
SYMMMETRIC
PPOOLLYYNNOOMMIIAALL IINNEEQQUUAALLIITTIIEESS
UNIVERSITY OF PLOIESTI, ROMANIA
2015
About the author
“Thesimplerandsharper,themorebeautiful.”
VasileCîrtoaje
VasileCîrtoajeisaProfessorattheDepartmentofAutomaticControlandComputers
fromPetroleum-GasUniversityofPloiesti,Romania,whereheteachesuniversitycourses
suchasControlSystemTheoryandDigitalControlSystems.
Since1970,hepublishedmanymathematicalproblems,solutionsandarticlesinthe
Romanian journals Gazeta Matematica-B, Gazeta Matematica-A and Mathematical Re-
view of Timisoara. In addition, from 2000 to present, Vasile Cîrtoaje has published
many interesting problems and articles in Art of Problem Solving website, Mathemati-
cal Reflections, Crux with Mayhem, Journal of Inequalities and Applications, Journal
of Inequalities in Pure and Applied Mathematics, Mathematical Inequalities and Ap-
plications,BanachJournalofMathematicalAnalysis,JournalofNonlinearScienceand
Applications,JournalofNonlinearAnalysisandApplication,AustralianJournalofMath-
ematicalAnalysisandApplications,BritishJournalofMathematicalandComputerSci-
ence,InternationalJournalofPureandAppliedMathematics,A.M.M.
HecollaboratedwithTituAndreescu,GabrielDospinescuandMirceaLascuinwriting
the book Old and New Inequalities, with Vo Quoc Ba Can and Tran Quoc Anh in writ-
ing the book Inequalities with Beautiful Solution, and he wrote his own books Algebraic
Inequalities-OldandNewMethodsandMathematicalInequalities(Volume1... 4).
NoticethatVasileCîrtoajeistheauthorofsomewell-knownresultsandstrongmeth-
odsforprovingandcreatingdiscreteinequalities,suchas:
-Halfconvexfunctionmethod(HCFmethod)forJensentypediscreteinequalities;
-Partialconvexfunctionmethod(PCFmethod)forJensentypediscreteinequalities;
-Jensentypediscreteinequalitieswithorderedvariables;
-Equalvariablemethod(EVmethod)forrealornonnegativevariables;
-Arithmeticcompensationmethod(ACmethod);
-BestlowerandupperboundsforJensen’sinequality;
-Necessaryandsufficientconditionsforsymmetrichomogeneouspolynomialinequal-
itiesofdegreesixinrealvariables;
i
ABOUTTHEAUTHOR
-Necessaryandsufficientconditionsforsymmetrichomogeneouspolynomialinequal-
itiesofdegreesixinnonnegativevariables;
-Highestcoefficientcancellationmethod(HCCmethod)forsymmetrichomogeneous
polynomialinequalitiesofdegreesixandeightinrealvariables;
-Highestcoefficientcancellationmethodforsymmetrichomogeneouspolynomialin-
equalitiesofdegreesix,sevenandeightinnonnegativevariables;
-Necessaryandsufficientconditionsforcyclichomogeneouspolynomialinequalities
ofdegreefourinrealvariables;
-Necessaryandsufficientconditionsforcyclichomogeneouspolynomialinequalities
ofdegreefourinnonnegativevariables;
- Strong sufficient conditions for cyclic homogeneous polynomial inequalities of de-
greefourinrealornonnegativevariables;
-Inequalitieswithpower-exponentialfunctions.
Foreword
The author, Vasile Cîrtoaje, professor at University of Ploiesti-Romania, has become
well-known for his excellent creations in the mathematical inequality field, ever since
the time when he was student in high school (in Breaza city, Prahova Valley). As a
student (quite some time ago, oh yes!), I was already familiar with the name of Vasile
Cîrtoaje. For me, and many others of my age, it is the name of someone who helped
me to grow in mathematics, even though I never met him face to face. It is a name
synonymoustohardandbeautifulproblemsinvolvinginequalities. WhenyousayVasile
Cîrtoaje (Vasc username on the site Art of Problem Solving), you say inequalities. I
rememberhowhappyIwaswhenIcouldmanagetosolveoneoftheproblemsproposed
byprofessorCîrtoajeinGazetaMatematicaorRevistaMatematicaTimisoara.
The first three volumes of this book are a great opportunity to see and know many
old and new elementary methods for solving mathematical inequalities: Volume 1 -
Symmetric polynomial inequalities (in real variables and nonnegative real variables),
Volume2-Symmetricrationalandnonrationalinequalities,Volume3-Cyclicandnon-
cyclicinequalities. Asarule,theinequalitiesfromeachsectionofthesevolumesarein-
creasinglyorderedbythenumberofvariables: two,three,four,five,sixandn-variables.
Thelastvolume(Volume4)containsnewbeautifulandefficientoriginalmethodsfor
creatingandsolvinginequalities: halforpartiallyconvexfunctionmethod-forJensen’s
type inequalities, equal variable method - for nonnegative or real variables, arithmetic
compensation method - for symmetric inequalities, the highest coefficient cancellation
method - for symmetric homogeneous polynomial inequalities of degree six, seven and
eightinnonnegativeorrealvariables,methodsinvolvingeitherstrongornecessaryand
sufficientconditions-forcyclichomogeneousinequalitiesofdegreefourinrealornon-
negativevariablesandsoon.
Many problems, the majority I would say, are made up by the author himself. The
chaptersandvolumesarerelativelyindependent,andyoucanopenthebooksomewhere
to solve an inequality or only read its solution. If you carefully make a thorough study
of the book, then you will find that your skills in solving inequalities are considerably
improved.
Thebookisarichandmeaningfulcollectionofmorethan1000beautifulinequalities,
hints, solutions and methods, some of them being posted in the last ten years by the
authorandotherinventivemathematiciansonArtofProblemSolvingwebsite(VoQuoc
iii
FOREWORD
Ba Can, Pham Kim Hung, Michael Rozenberg, Nguyen Van Quy, Gabriel Dospinescu,
Darij Grinberg, Pham Huu Duc, Tran Quoc Anh, Le Huu Dien Khue, Marius Stanean,
CezarLupu,NguyenAnhTuan,PhamVanThuan,BinZhao,JiChenetc.)
Mostinequalitiesandmethodsareoldandrecentowncreationsoftheauthor. Among
these,Iwouldliketopointoutthefollowinginequalities:
(a2+b2+c2)2≥3(a3b+b3c+c3a), a,b,c∈(cid:82);
(cid:88)
(a−kb)(a−kc)(a−b)(a−c)≥0, a,b,c,k∈(cid:82);
(cid:16) a (cid:17)2 (cid:129) b (cid:139)2 (cid:16) c (cid:17)2 (cid:129) d (cid:139)2
+ + + ≥1, a,b,c,d ≥0;
a+b b+c c+d d+a
4
(cid:88) 1
≥1, a ,a ,a ,a >0, a a a a =1;
1+a +a2+a3 1 2 3 4 1 2 3 4
i=1 i i i
a a a
1 + 2 +···+ n ≥1, a ,a ,···,a ≥0;
a +(n−1)a a +(n−1)a a +(n−1)a 1 2 n
1 2 2 3 n 1
aea+beb ≥aeb+bea, a,b>0, e≈2.7182818.
a3b+b3a ≤2, a,b≥0, a+b=2.
The book represents a rich source of beautiful, serious and profound mathematics,
dealingwithclassicalandnewapproachesandtechniqueswhichhelpthereadertode-
velop his inequality-solving skills, intuition and creativity. As a result, it is suitable for
a wide variety of audiences; high school students and teachers, college and university
students, mathematics educators and mathematicians will find something of interest
here. Each problem has a hint, and many problems have multiple solutions, almost all
of which are, not surprisingly, quite ingenious. Almost all inequalities require careful
thought and analysis, making the book a rich and rewarding source for anyone inter-
estedinOlympiad-typeproblemsandinthedevelopmentoftheinequalityfield. Many
problemsandmethodscanbeusedinaclassroomforadvancedhighschoolstudentsas
groupprojects.
What makes this book so attractive? The answer is simple: the great number of in-
equalities, their quality and freshness, as well as the new approaches and methods for
solving mathematical inequalities. Nevertheless, you will find this book to be delight-
ful, inspired, original and enjoyable. Certainly, any interested reader will remark the
tenacity, enthusiasm and ability of the author in creating and solving nice and difficult
inequalities. Andthisbookisneithermore,norlessthanaWorkofaMaster.
The author must be congratulated for publishing such interesting and original in-
equalitybook. Ihighlyrecommendit.
MarianTetiva
Contents
About the author i
Foreword iii
1 Some Classic and New Inequalities and Methods 1
2 Symmetric Polynomial Inequalities in Real Variables 19
2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Symmetric Polynomial Inequalities in Nonnegative Variables 149
3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149
3.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
4 Bibliography 309
v
vi VasileCîrtoaje
Chapter 1
Some Classic and New Inequalities
and Methods
1. AM-GM (ARITHMETIC MEAN-GEOMETRIC MEAN) INEQUALITY
If a ,a ,···,a arenonnegativerealnumbers,then
1 2 n
a +a +···+a ≥n(cid:112)n a a ···a ,
1 2 n 1 2 n
withequalityifandonlyif a =a =···=a .
1 2 n
2. WEIGHTED AM-GM INEQUALITY
Let p ,p ,···,p bepositiverealnumberssatisfying
1 2 n
p +p +···+p =1.
1 2 n
If a ,a ,···,a arenonnegativerealnumbers,then
1 2 n
p a +p a +···+p a ≥ap1ap2···apn,
1 1 2 2 n n 1 2 n
withequalityifandonlyif a =a =···=a .
1 2 n
3. AM-HM (ARITHMETIC MEAN-HARMONIC MEAN) INEQUALITY
If a ,a ,···,a arepositiverealnumbers,then
1 2 n
(cid:129) 1 1 1 (cid:139)
(a +a +···+a ) + +···+ ≥n2,
1 2 n
a a a
1 2 n
withequalityifandonlyif a =a =···=a .
1 2 n
1
2 VasileCîrtoaje
4. POWER MEAN INEQUALITY
Thepowermeanoforder k ofpositiverealnumbers a ,a ,···,a ,thatis
1 2 n
(cid:16)a1k+a2k+···+ank(cid:17)1k , k(cid:54)=0
n
Mk = (cid:112) ,
n a a ···a , k=0
1 2 n
is an increasing function with respect to k ∈ (cid:82). For instant, M2 ≥ M1 ≥ M0 ≥ M−1 is
equivalentto
(cid:118)
(cid:117)(cid:116)a12+a22+···+an2 ≥ a1+a2+···+an ≥(cid:112)n a a ···a ≥ n .
n n 1 2 n 1 1 1
+ +···+
a a a
1 2 n
5. BERNOULLI’S INEQUALITY
Foranyrealnumber x ≥−1,wehave
a) (1+x)r ≥1+rx for r ≥1and r ≤0;
b) (1+x)r ≤1+rx for0≤r ≤1.
Inaddition,ifa ,a ,···,a arerealnumberssuchthateithera ,a ,···,a ≥0or−1≤
1 2 n 1 2 n
a ,a ,···,a ≤0,then
1 2 n
(1+a )(1+a )···(1+a )≥1+a +a +···+a .
1 2 n 1 2 n
6. SCHUR’S INEQUALITY
For any nonnegative real numbers a, b, c and any positive number k, the inequality
holds
ak(a−b)(a−c)+bk(b−c)(b−a)+ck(c−a)(c−b)≥0,
withequalityfor a= b=c,andfor a=0and b=c (oranycyclicpermutation).
For k=1,wegetthethirddegreeSchur’sinequality,whichcanberewrittenasfollows
a3+b3+c3+3abc≥ab(a+b)+bc(b+c)+ca(c+a),
(a+b+c)3+9abc≥4(a+b+c)(ab+bc+ca),
9abc
a2+b2+c2+ ≥2(ab+bc+ca),
a+b+c
(b−c)2(b+c−a)+(c−a)2(c+a−b)+(a−b)2(a+b−c)≥0.
