Exercise 19, § 1

On garde les notations du lemme 5 de IV, p. 161. Now prove that if the image of $f$ is contained in $\mathbf{R}$, if $f$ est strictly convex, and if $\varrho_n>0$, then there is equality in the inequality (6) if and only if $a_i=b_i$ for $0\leqslant i\leqslant n$.