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Chapter: Useful Inequalities

Inequalities belong to the most important tools when practicing analysis, especially for the purpose of estimating boundaries of mathematical expressions (e.g. functions, sums, or products) from above or from below. In this section, we will present some theorems regarding inequalities that can be applied in such cases.

Table of Contents

1. Proposition: Generalized Triangle Inequality
2. Theorem: Triangle Inequality
3. Theorem: Reverse Triangle Inequalities
4. Definition: (Weighted) Arithmetic Mean
5. Theorem: Inequality of the Arithmetic Mean
6. Theorem: Inequality of Weighted Arithmetic Mean
7. Theorem: Bernoulli's Inequality
8. Proposition: Generalized Bernoulli's Inequality
9. Proposition: Cauchy–Schwarz Inequality
10. Definition: Geometric Mean
11. Theorem: Inequality Between the Geometric and the Arithmetic Mean
12. Lemma: Upper Bound for the Product of General Powers
13. Proposition: Hölder's Inequality
14. Proposition: Minkowski's Inequality
15. Proposition: Hölder's Inequality for Integral p-norms
16. Proposition: Cauchy-Schwarz Inequality for Integral p-norms
17. Proposition: Minkowski's Inequality for Integral p-norms
18. Proposition: Inequality between Square Numbers and Powers of $2$
19. Proposition: Inequality between Powers of $2$ and Factorials
20. Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials
21. Proposition: Bounds for Partial Sums of Exponential Series

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