Question
I'm contemplating about what should the third-order analogy to strict quasi-convexity be if it should characterize all the functions with the same ordinal properties as functions with strictly positive third order derivative.
How to characterize all the functions $f\in C^3$, such that $x\mapsto\phi(f(x))$ has strictly positive third-order derivative for some monotonic (increasing) transformation $\phi\in C^3$.
Notation. $C^k$ denotes all the functions $f:\Bbb R \to \Bbb R$ that are $k$ times continuously differentiable.
Analogy
Definition. A function $f\in C^2$ is (second-order) strcitly quasi-convex iff there exists monotonic transformation $\phi\in C^2$ such that $\phi \circ f$ has strictly positive second-order derivative (i.e. $\phi \circ f$ is strictly convex). This property can be characterized as follows:
Characterisation. A function $f$ is strictly quasi-convex iff for any $x<y<z$, $$ f(y) < \max\{f(x),f(z)\}. $$
Context
Quasi-convexity:
Strictly quasi-convex functions $f:\Bbb R \to \Bbb R$ have the same ordinal properties as strictly convex functions do, namely:
- has at most one local minimum,
- every local minimum of $f$ is the global minimum of $f$,
- if $f'(x)=0$ then $x$ is the global,
- has no local / global maximum.
Third-order strict convexity:
We call a function $f\in C^3$ is third-order strictly convex iff $f^{(3)}(x)>0$ for all $x\in \Bbb R$.
Note. This concept can be generalized into functions that are not necessarily three times differentiable, I discuss this in Geometric characterization of functions with positive third derivative. However, for the sake of simplicity, we can restrict our attention only to functions that are three time differentiable.
Let $f$ be a third-order convex function. Notice that since the second order derivative of $f$ is strictly increasing, it is either entirely convex / concave, or there is a point of inflection $x_*$ at which $f$ changes from being concave to being convex. The following properties follow from there:
- a) $f$ has at most one inflection point $x_*$,
- b) $f$ has at most one local maxim and one local minim,
- c) every local maximum of $f$ is its global maximum on $(-\infty,x_*]$ if $x_*$ exists and otherwise it is global maxim on entire $\Bbb R$,
- d) every local minimum of $f$ is its global minimum on $[x_*,\infty)$ if $x_*$ exists and otherwise it is global minum on entire $\Bbb R$,
- e) if $f'(x_0)=0$, then $x_0$ is a local maximum whenever $x_0<x_*$, and it is a local minimum whenever $x_0>x_*$.
Third-order strict quasi-convexity:
Definition. We say that function $f\in C^3$ is third-order strictly quasi-convex iff there exists a strictly increasing function $\phi\in C^3$ such that $\phi\circ f$ has positive third-order derivative.
Apparently, every third-order strictly quasi-convex function has the properties a)-e).
Question. Is there an easy way to characterize the third-order strictly quasi-convex functions as it is in the case of the quasi-convex functions?
Note. I want a characterization that does not refer to the inflection point $x_*$. Of course, the trivial characterization would be that $f$ is strictly quasi-concave on $(-\infty,x_*]$ and strictly quasi-convex on $[x_*,\infty)$.
