Exercise 21, § 1

Let $f$ be an increasing convex continuous numerical function in 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 of (0, a), now prove that one has

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