ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1369 on D1719: Expectation
Subresult of R1368: Jensen's inequality
Jensen's inequality for expectation

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $(a, b) \subseteq [-\infty, \infty]$ is an D5146: Open basic interval (ii) $X : \Omega \to (a, b)$ is a D3161: Random real number on $P$ (iii) $$\mathbb{E} |X| < \infty$$
Let $\varphi : (a, b) \to \mathbb{R}$ be a D5606: Subconvex real function.
Then $$\varphi(\mathbb{E} X) \leq \mathbb{E} \varphi(X)$$
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $(a, b) \subseteq [-\infty, \infty]$ is an D5146: Open basic interval (ii) $X : \Omega \to (a, b)$ is a D3161: Random real number on $P$ (iii) $$\mathbb{E} |X| < \infty$$
Let $\varphi : (a, b) \to \mathbb{R}$ be a D5606: Subconvex real function.
This result is a particular case of R1368: Jensen's inequality. $\square$