Exercise 21, § 1

Let $f$ be a real increasing convex continuous function on an interval $[0, a]$, and such that $f(0) = 0$. If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ is a finite decreasing sequence of points in $[0, a]$, show that one has

(One can restrict oneself to the case where $n = 2m$ is even; remark that for $1 \leq j \leq m$ one has where $\alpha = a_{2j+1} - a_{2j-2} + \cdots + a_{2m-1} - a_{2m}$ and $\beta = \alpha + (a_{2j-1} - a_{2j})$.)