Question 87
Let $g : \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that:
a) if the inequality $g\left(\int_0^1 f(t) dt\right) \le \int_0^1 g(f(t)) dt$ holds for every integrable function $f : [0, 1] \to \mathbb{R}$, then $g$ is convex;
b) if the inequality $g\left(\int_0^1 f(t) dt\right) \le \int_0^1 g(f(t)) dt$ holds for every continuous function $f : [0, 1] \to \mathbb{R}$, then $g$ is convex.
a) if the inequality $g\left(\int_0^1 f(t) dt\right) \le \int_0^1 g(f(t)) dt$ holds for every integrable function $f : [0, 1] \to \mathbb{R}$, then $g$ is convex;
b) if the inequality $g\left(\int_0^1 f(t) dt\right) \le \int_0^1 g(f(t)) dt$ holds for every continuous function $f : [0, 1] \to \mathbb{R}$, then $g$ is convex.