Inequalities

1. Jensen's Inequality

If $\varphi(\cdot)$ is convex, then

• For any $x_1, x_2$ and $t \in [0,1]$

$$\varphi(t x_1 + (1-t)x_2) \leq t \varphi(x_1) + (1-t)\varphi(x_2)$$

• If $X$ is a random variable

$$\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)]$$

2. H\ddot{\mathrm{o}}lder's Inequality

$p, q \in [1, \infty]$ with $\frac{1}{p} + \frac{1}{q} = 1$, then

$$\|f g\| \leq \|f\|_p \|g\|_q$$

Also, for random variables $X ,Y$

$$\mathbb{E}\left[ |XY | \right] \leq \left(\mathbb{E}[|X|^p]\right)^{\frac{1}{p}} \left(\mathbb{E}[|Y|^q]\right)^{\frac{1}{q}}$$

Note that you can choose $p = 1$ and $q = \infty$.

3. Young's Inequality

$p, q > 0$, $a, b \geq 0$ with $\frac{1}{p} + \frac{1}{q} = 1$, then

$$a b \leq \frac{a^p}{p} + \frac{b^q}{q}$$

4. data processing inequality