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# HM-Brow

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1 Introductory notions

1.1 The concept of inequality

1.2 The basic properties of an inequality

1.3 Basic inequalities

1.4 Squares are positive

1.5 Some special inequalities and identities for two numbers

1.6 Some special inequalities and identities for more numbers

2 Basic techniques for solving inequalities

2.1 Comparison Method

2.2 Pigeonhole principle

2.3 Majorization and minorization

2.4 The method of undetermined coefficients

2.5 Normalization Method

2.6 Homogenization Method

2.7 The quadratic trinomial

2.8 Using equality to prove inequality

2.9 The mathematical induction

3 Some classical results and their applications

3.1 The AM-GM inequality

3.2 Young’s Inequality

3.3 Cauchy-Bunyakovsky-Schwarz Inequality

3.4 The Rearrangement Inequality

4 Techniques from Algebra

4.1 Breaking the inequality

4.2 Separating the squares

4.3 Working backwards

4.4 The Dual Principle and other standard substitutions

4.5 Mixing variables

4.6 Homogenization and dehomogenization

4.7 The inequalities between the symmetric sums

4.8 The pqr technique

5 Techniques from Analysis

5.1 The principle of extremality and monotonicity

5.2 Limits in inequalities

5.3 Derivatives in Inequalities

5.4 Convexity

5.5 Jensen’s Inequality

5.6 The shrinkage principle and Karamata’s Inequality

5.7 Schur’s Inequality

5.8 The generalized Means

5.9 The tangent line technique and its extensions

6 Glossary

7 Bibliography

1.1 The concept of inequality

1.2 The basic properties of an inequality

1.3 Basic inequalities

1.4 Squares are positive

1.5 Some special inequalities and identities for two numbers

1.6 Some special inequalities and identities for more numbers

2 Basic techniques for solving inequalities

2.1 Comparison Method

2.2 Pigeonhole principle

2.3 Majorization and minorization

2.4 The method of undetermined coefficients

2.5 Normalization Method

2.6 Homogenization Method

2.7 The quadratic trinomial

2.8 Using equality to prove inequality

2.9 The mathematical induction

3 Some classical results and their applications

3.1 The AM-GM inequality

3.2 Young’s Inequality

3.3 Cauchy-Bunyakovsky-Schwarz Inequality

3.4 The Rearrangement Inequality

4 Techniques from Algebra

4.1 Breaking the inequality

4.2 Separating the squares

4.3 Working backwards

4.4 The Dual Principle and other standard substitutions

4.5 Mixing variables

4.6 Homogenization and dehomogenization

4.7 The inequalities between the symmetric sums

4.8 The pqr technique

5 Techniques from Analysis

5.1 The principle of extremality and monotonicity

5.2 Limits in inequalities

5.3 Derivatives in Inequalities

5.4 Convexity

5.5 Jensen’s Inequality

5.6 The shrinkage principle and Karamata’s Inequality

5.7 Schur’s Inequality

5.8 The generalized Means

5.9 The tangent line technique and its extensions

6 Glossary

7 Bibliography

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