Let $(X, \mathcal{A}, \mu)$ be a measure space. Suppose $\mu(X) = 1$, and let $f: X \to \mathbb{R}$ be Lebesgue integrable. Are the following inequalities true or not?
$$(\int f)^2 \le \int f^2$$ $$e^{\int f} \le \int e^f$$
2026-03-27 04:18:50.1774585130
Is it true that $(\int f)^2 \le \int f^2$, $e^{\int f} \le \int e^f$?
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These follow from Jensen's Inequality and the fact that $x^2$ and $e^x$ are convex functions.