Take two weakly convex, weakly increasing non-negative functions, $g(x)$ and $h(x)$, domain of $x$ is $[a,d]$, $g(x)=0$ for $x \in [a,b]$, $h(x)=0$ for $x \in [a,c]$, and $g(d)=h(d)$. So $g$ and $h$ start at zero, stay at zero over some (possibly different) domain(s), and intersect again at $d$. Can we make a general statement on the relative convexities and slopes of $g$ and $h$ that achieves a weak ranking of $g$ and $h$ over all $x$ in $[a,d]$? i.e. $g(x)\geq h(x)$?
Clearly if $b<c$ and $g'(x)\leq h'(x)$ for $x \in [c,d]$ that will do it, but i'm more interested in situations where the first derivatives can't necessarily be ranked in this way (and, preferably, where we don't have to invoke a condition on $b$ or $c$). Intuitively it seems like there must be some condition on the scaled second derivative? like on $g''(x)/g'(x)$ vs $h''(x)/h'(x)$?