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.

  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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