Some of the many faces of the Jensen's inequality

For a real convex function $$\phi$$:

$$\phi(\sum x_i / n) \leq \sum \phi(x_i) / n$$

$$\phi\left( \frac{1}{b-a} \int_a^b g(x) dx \right) \leq$$
$$\leq \frac{1}{b-a} \int_a^b \phi(g(x)) dx$$

$$\phi(\mathrm{E}(X)) \leq \mathrm{E}(\phi(X))$$

$$\phi(\mathrm{E}(X | G)) \leq \mathrm{E}(\phi(X) | G)$$

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