Basic Inequalities in Analysis

Theorem 1. (Bernoulli’s inequality, lower bound for exponentiations of \(1 + x\)) \[(1 + x)^r \geq 1 + rx\] for all \(r \in \mathbb{N}\) and \(x \geq -1\), \(x \in \mathbb{R}\).

Theorem 2. (Generalization of Bernoulli’s inequality) \[(1 + x)^r \geq 1 + rx\] for all \(r \in (-\infty, 0] \cup [1, +\infty)\) and \(x \geq -1\), \(x \in \mathbb{R}\). \[(1 + x)^r \leq 1 + rx\] for all \(r \in [0, 1]\) and \(x \geq -1\), \(x \in \mathbb{R}\).

Theorem 3. \[\left(1 + \frac{1}{x}\right)^x < e < \left(1 + \frac{1}{x}\right)^{x+1}\] for all \(x \in \mathbb{R}^+\).

Theorem 4. (Upper bound for exponentiations of \(1 + x\)) \[(1 + x)^r \leq e^{rx}\] for all \(r \in \mathbb{N}\) and \(x \geq 0\), \(x \in \mathbb{R}\).

Theorem 5. For all \(x \in \mathbb{R}\) it holds that \(e^x \geq x + 1\).

Theorem 6. For all \(x \geq 1\) it holds that \((1 - \frac{1}{x})^x \leq e^{-1}\).

Theorem 7. For all \(0 \leq x \leq 1\) it holds that \[e^{-x} \leq 1 - (1 - \frac{1}{e}) \cdot x \leq 1 - \frac{x}{2}\]