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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.

| | | | created: 2014-02-23 13:49:29 | modified: 2020-06-26 16:35:51 | by: bookofproofs

1.Theorem: Triangle Inequality

2.Proposition: Generalized 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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Bibliography (further reading)