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# HM-Brow

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Preface

Acknowledgements

Abbreviations and Notations

I The basic Inequalities

1 AM-GM nequality

1.1 AM-GM Inequality and Applications

1.2 The Cauchy Reverse Technique

2 Cauchy-Schwarz and Holder inequalities

2.1 Cauchy-Schwarz inequality and Applications

2.2 Holder Inequality

3 Chebyshev Inequality

3.1 Chebyshev Inequality and Applications

3.2 The Chebyshev Associate Technique

4 Inequalities with Convex Functions

4.1 Convex functions and Jensen inequality

4.2 Convex Functions and Inequalities with Variables Restricted to an Interval

5 Abel Formula and Rearrangement Inequality

5.1 Abel formula

5.2 Rearrangement Inequality

6 The Method of Balanced Coefficients

6.1 Balancing coefficients by AM-GM inequality

6.2 Balancing coefficients by Cauchy-Schwarz and Holder inequalities

7 Derivative and Applications

7.1 Derivative of one-variable functions

7.2 Derivative of n-Variable Functions

8 A note on symmetric inequalities

8.1 Getting started

8.2 Primary symmetric polynomials

8.3 Normalization skill

8.4 Symmetric separation

9 Problems and Solutions

Glossary

Acknowledgements

Abbreviations and Notations

I The basic Inequalities

1 AM-GM nequality

1.1 AM-GM Inequality and Applications

1.2 The Cauchy Reverse Technique

2 Cauchy-Schwarz and Holder inequalities

2.1 Cauchy-Schwarz inequality and Applications

2.2 Holder Inequality

3 Chebyshev Inequality

3.1 Chebyshev Inequality and Applications

3.2 The Chebyshev Associate Technique

4 Inequalities with Convex Functions

4.1 Convex functions and Jensen inequality

4.2 Convex Functions and Inequalities with Variables Restricted to an Interval

5 Abel Formula and Rearrangement Inequality

5.1 Abel formula

5.2 Rearrangement Inequality

6 The Method of Balanced Coefficients

6.1 Balancing coefficients by AM-GM inequality

6.2 Balancing coefficients by Cauchy-Schwarz and Holder inequalities

7 Derivative and Applications

7.1 Derivative of one-variable functions

7.2 Derivative of n-Variable Functions

8 A note on symmetric inequalities

8.1 Getting started

8.2 Primary symmetric polynomials

8.3 Normalization skill

8.4 Symmetric separation

9 Problems and Solutions

Glossary

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