Convex Lipschitz Integrands

In the setting of The Real Numbers: Standing Notation and Background, write $[0,\infty)$ for the set of nonnegative real numbers; for $a,b\in[0,\infty)$ and real $t$ with $0\le t\le1$ the number $ta+(1-t)b$ lies in $[0,\infty)$, as a sum of products of nonnegative reals. Let $L\in\mathbb{R}$ be nonnegative.

(Convex Lipschitz integrand)¶ A convex Lipschitz integrand with constant $L$ is a function $\Phi:[0,\infty)\to\mathbb{R}$ with $\Phi(0)=0$ such that, for all $a,b\in[0,\infty)$ and every real $t$ with $0\le t\le1$,

and, whenever $a\le b$,