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.

- Proposition: Generalized Triangle Inequality
- Theorem: Triangle Inequality
- Theorem: Reverse Triangle Inequalities
- Definition: (Weighted) Arithmetic Mean
- Theorem: Inequality of the Arithmetic Mean
- Theorem: Inequality of Weighted Arithmetic Mean
- Theorem: Bernoulli's Inequality
- Proposition: Generalized Bernoulli's Inequality
- Proposition: Cauchy–Schwarz Inequality
- Definition: Geometric Mean
- Theorem: Inequality Between the Geometric and the Arithmetic Mean
- Lemma: Upper Bound for the Product of General Powers
- Proposition: Hölder's Inequality
- Proposition: Minkowski's Inequality
- Proposition: Hölder's Inequality for Integral p-norms
- Proposition: Cauchy-Schwarz Inequality for Integral p-norms
- Proposition: Minkowski's Inequality for Integral p-norms
- Proposition: Inequality between Square Numbers and Powers of $2$
- Proposition: Inequality between Powers of $2$ and Factorials
- Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials
- Proposition: Bounds for Partial Sums of Exponential Series