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Let $f$ be a Convex function, $f'' \geq 0$

Let $X$ be a Random variable

Then,

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

where $\mathbb{E}$ denotes Expectation

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(this is a nice case, one can imagine less intuitive cases, but the inequality is still true).

Further, if $f''(X) > 0$ ($f$ is strictly convex), then

$f(\mathbb{E}[X]) = \mathbb{E}[f(X)] \Leftrightarrow X = \mathbb{E}[X]$ w.p. $1$

If $f'' \leq 0$ ($f$ is concave)

$f(\mathbb{E}[X]) \geq \mathbb{E}[f(X)]$

etc.