Variants of Jensen's inequality. Let \((X, \mathscr{A}, \mu)\) be a probability space.

(i) Show Jensen's inequality for convex \(V: \mathbb{R} ightarrow \mathbb{R}\), see Example 13.14 (v).

(ii) Show Jensen's inequality for concave \(\Lambda: \mathbb{R} ightarrow \mathbb{R}\), see Example 13.14 (vi).

(iii) Let \(-\infty \leqslant a

For \(u: X ightarrow(a, b)\) such that \(u, V(u) \in \mathcal{L}^{1}(\mu)\) we have \(\int u d \mu \in(a, b)\) and \(V\left(\int u d \muight) \leqslant\) \(\int V(u) d \mu\).

(iv) Let \(-\infty \leqslant a

For \(u: X ightarrow(a, b)\) such that \(u, \Lambda(u) \in \mathcal{L}^{1}(\mu)\) we have \(\int u d \mu \in(a, b)\) and \(\int \Lambda(u) d \mu \leqslant\) \(\Lambda\left(\int u d \muight)\).

Data from example 13.14 (v)

Data from example 13.14 (vi)

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(v) If V: R R is a convex function and u, V(u) L (), then the same proof as for Theorem 13.13 shows that v (Sudu) < [1 < [ v(u)d.