College of education mathematics department

PREFACE 1
THANK YOU 2
INTRODUCTION TO INEQUALITY 5
CHAPTER 1. ARITHMETIC MEAN - GEOMETRIC MEAN (AM-GM) INEQUALITY 6
1.1. Theorem: 6
1.2. Lemma: 7
1.3. Theorem: 8
1.4. Exercises: 9
CHAPTER 2. BERNOULLI’S INEQUALITY & THE CAUCHY–SCHWARZ INEQUALITY 11
2.1. The Inequality and Bernoulli's Inequality 11
2.1.1. Theorem: 11
2.1.2. Theorem (Bernoulli's inequality): 13
2.1.3. Corollary (Bernoulli's inequality) 13
2.1.4. Exercises: 14
2.2. The Cauchy–Schwarz inequality: 17
2.2.1. Theorem (Cauchy-Schwarz inequality) 17
2.2.2. Corollary: 19
2.2.3. Corollary: 20
2.2.4. Corollary: 20
2.2.5. Theorem (Chebishev's inequality) 21
2.2.6. Exercises: 22
CHAPTER 3: HOLDER’S INEQUALITY, MINKOWSKI’S INEQUALITY 25
3.1. Holder’s inequality 25
3.1.1. Young's Inequality 25
3.1.2. (Holder's inequality) 26
3.1.3. Exercises 28
3.2. Minkowski's inequality 30
3.2.1. Theorem (First Minkowski's inequality) : 30
3.2.2. Theorem (Second Minkowski's inequality) : 31
3.2.3. Theorem (Third Minkowski's inequality) : 32
3.2.4. Exercises: 32
CHAPTER 4: GEOMETRIC ( TRIANGLE ) INEQUALITY 34
4.1. Proposition: 36
4.2. Exercises: 37
CHAPTER 5: APPLICATIONS OF INEQUALITIES 40
5.1. Applications of Cauchy inequality 40
5.2. Application of holder inequality 44
5.2.1. Applications in calculus 44
5.2.2. Applications in geometry 46
5.2.3. Applications in trigonometry 47
5.2.4. Applications in arithmetic. 48
5.2.5. Applications in Algebra 49
5.2.6. Applications in Analytic Geometry 51
5.2.7. Applications in combinatorial analysis 52
5.3. Application of Minkowski’s inequality 54
5.3.1. Applications in trigonometry: 54
5.3.2. Applications in analytics: 55
5.3.3. Applications in algebra: 58
5.4. Application of Bernoulli Inequality 60
5.4.1. Common application: 60
5.4.2. Applications of Bernoulli distribution 64
REFERENCES 65