Exercise 23, § 1

. b) Prove that there exists a convex function h in (0, 1) such that $h \geq f$ and that

(For every $a$ such that $0 < a \leq 1$, let $f_a$ be the function equal to $f$ for $0 \leq t \leq a$ and to $f(a)$ for $a \leq t \leq 1$; let $A$ be the set of the $a \in ]0, 1[$ for which there exists a convex function $h_a$ in $(0, 1)$ such that $h_a \geq f_a$ and $\int_0^1 h_a(t) \, dt \leq 2 \int_0^1 f_a(t) \, dt$. Prove that the upper bound $b$ of $A$ still belongs to $A$, using exerc. 1 of I, p. 52. Proving subsequently that $f_b = f$ by reasoning by contradiction. For this, one reduces to proving the following result: if $\varphi$ is continuous and increasing in $(0, 1]$, nonconstant and such that $\varphi(0) = 0$, there is a point $c \in ]0, 1[$ such that $\varphi(c) > 0$ and that the linear function $\psi$ such that $\psi(c) = \varphi(c)$ and

verifies the relation $\psi(t) \geq \varphi(t)$ for $\max(0, 2c - 1) \leq t \leq c.$

23) Let $\mathbf{f}$ be a continuously differentiable vector function in an interval $(a, b) \subset \mathbf{R}$, with values in a complete normed space. a) Prove that for $a \leq t \leq b$, one has

b) Deduce the inequalities

and