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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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