Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval

Let $I\subseteq \mathbb{R}$ be an interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f:I\to \mathbb{R}$. For $x,y\in I$ and $\lambda \in [0,1]$, the point $\lambda x+(1-\lambda )y$ belongs to $I$. The function $f$ is convex when the convex-combination inequality holds for every weight in $[0,1]$:

$f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)\mathrm{.}$

It is strictly convex when this inequality is strict whenever $x\mathrm{\ne }y$ and $0<\lambda <1$. It is concave (respectively strictly concave) when $-f$ is convex (respectively strictly convex). It is midpoint convex when the displayed inequality is required only at $\lambda =1\mathrm{/}2$.

The endpoint weights $0$ and $1$ impose equalities, so strict convexity does not ask for strictness there. A singleton interval satisfies the convexity and strict-convexity conditions vacuously.