Definition 2

We say that a finite real function $f$ defined on an interval $I \subset \mathbf{R}$ is strictly convex on I if, for any points $x, x'$ of I ($x < x'$), every point $M_z$ of the graph G of $f$ such that $x < z < x'$ lies strictly below the segment $M_x M_{x'}$ (or, what comes to the same, if every point of the segment, apart from the endpoints, lies strictly above G).

In other words, we must have the inequality

for every pair of distinct points $(x, x')$ of $I$ and every $\lambda$ such that $0 < \lambda < 1$.

The remarks that precede def. 2 show that for a convex function $f$ to be strictly convex on $I$ it is necessary and sufficient that there be no interval contained in $I$ (not reducing to a single point) such that the restriction of $f$ to this interval is affine linear.

Of the examples above, the first and third are not strictly convex; on the other hand, $x^2$ is strictly convex on $\mathbf{R}$; a similar calculation shows that $1/x$ is strictly convex on $[0, +\infty[$.