Geometric characterization of functions with positive third derivative

Question

Consider a function $$f:\mathbb R \to \mathbb R$$. Are the following conditions equivalent?

• Q1. The function $$f$$ is continuously differentiable and it second derivative $$f^{(2)}$$ exists at all but countably many points and is increasing on its domain.
• Q2. For any $$a and the quadratic function $$q(x)$$ interpolating the points $$(a,f(a)),(b,f(b)),(c,f(c))$$, the graph of the restriction $$q|_{[b,c]}$$ is contained in the epigraph $$\mathop{epi}(f)$$.
• Q3. For any $$a and a quadratic function $$q(x)$$ if both of the points $$(a,q(a))$$ and $$(c,q(c))$$ are contained in $$\mathop{epi}(f)$$ whilst both of the points $$(b,q(b))$$ and $$(d,q(d))\}$$ are outside of the epigraph then the graph of $$q|_{(-\infty,a]}$$ is contained in $$\mathop{epi}(f)$$.

And is the following condition necessary condition for Q1-Q3?

• Q0. The third derivative $$f^{(3)}$$ exists almost everywhere and is positive on its domain.

Notation:

1. Positivness and monotonicity are considered in the weak sense, i.e. using weak inequalities "$$\geq$$".

2. Almost everywhere refers to a property being satisfied everywhere except for a set of zero Lebesgue measure.

3. Graph of the restriction $$q|_{[b,c]}$$ is the set $$\mathop{graph}(q|_{[b,c]})=\{(x,q(x))\in \mathbb R^2:x\in [b,c]\}.$$

4. Epigraph of $$f$$ is the set $$\mathop{epi}(f) = \{(x,y)\in \mathbb R^2:y \geq f(x)\}$$.

Motivation:

• Condition Q1 implies Q0 due to the fact that a monotone function is differentiable almost everywhere.

• Condition Q2 abstracts from Q0 and Q1 by not referring to any of the function's derivatives explicitly.

• Condition Q3 is an attempt to formulate Q2 by referring solely to the epigraph of $$f$$ and not its boundary – the graph of $$f$$ – similarly as convexity of a set $$S$$ is usually defined by requiring that any segment connecting any two points in $$S$$ is contained in $$S$$ (instead of requiring this property only for any two points on the boundary of $$S$$). Notice that the graph of a quadratic function can be characterized as a parabola with a vertical axis of symmetry.

Observations:

Analogy

Conditions Q0-Q3 were motivated by their analogy to corresponding conditions describing monotonicity and convexity.

Convex Functions:

• C0. The second derivative $$f''$$ exists almost everywhere and is positive on its domain.
• C1. The function $$f$$ is continuous and its derivative $$f'$$ exists at all but countably many points and is increasing on its domain.
• C2. For any $$a and the linear function $$l(x)$$ interpolating the points $$(a,f(a)),(b,f(b))$$, the graph of $$l|_{[a,b]}$$ is contained in $$\mathop{epi}(f)$$.
• C3. For any $$a and a linear function $$l(x)$$ if both of the points $$(a,l(a)),(b,l(b))$$ are in $$\mathop{epi}(f)$$ then the graph of $$l|_{[a,b]}$$ is contained in $$\mathop{epi}(f)$$.

Note that Condition C3 is equivalent to $$\mathop{epi}(f)$$ being a convex set.

Condition C0 is a necessary but not sufficient condition for convexity: https://math.stackexchange.com/a/3425608/1134951

Increasing Functions:

• I0. The derivative $$f'$$ exists almost everywhere and is positive on its domain.
• I1. The function $$f$$ is increasing.
• I2. For any $$a\in \mathbb R$$ and the constant function $$m(x)$$ interpolating the point $$(a,f(a))$$, the graph of $$m|_{(-\infty,a]}$$ is contained in $$\mathop{epi}(f)$$.
• I3. For any $$a\in \mathbb R$$ and a constant function $$m(x)$$ if the point $$(a,m(a))$$ is in $$\mathop{epi}(f)$$ then the graph of $$m|_{(-\infty,a]}$$ is contained in $$\mathop{epi}(f)$$.

Note that Condition I3 is equivalent to $$\mathop{epi}(f)$$ being star-shaped with the vantage point $$(-\infty,0)$$ at the infinity. In other terms $$\mathop{epi}(f) = \mathop{epi}(f) + (-\infty,0]\times\{0\}$$. (The vantage point at infinity represents a direction, since the direction is parallel to the $$x$$-axis, it is possible to informally describe the vantage point as $$(-\infty,0)$$ instead of representing it in the homogenous coordinates as $$(-1,0,0)$$.)

Let me recall that the question was whether Conditions Q1-Q3 are equivalent and if they imply Condition Q0.

• If anyone knows how a function satisfying one of the conditions Q1-Q4 was called, please share, I would like to be able to search more about such functions. I'm surprised that in contrast to how prevalent the concept of convexity is, I've never heard about analytical properties of functions with the third derivative positive. Commented Jan 13, 2023 at 22:13
• Just play with the Lagrange interpolation formula $f(x)=\sum_{k=1}^n f(x_k)\frac{\omega(x)}{(x-x_k)\omega'(x_k)}+\frac 1{n!}f^{(n)}(\xi)\omega(x)$ where $\omega(x)=\prod_k(x-x_k)$ with $n=3$ to derive all the desired results and more for, say, $f\in C^3$. Their generalizations to the case when $f^{(3)}$ is a measure is a bit more delicate and you should not forget the possible singular part in your claims as you did in c0 if you want them to hold. Commented Jan 19, 2023 at 12:34
• I saw in 90s, in a libary book. The author was saying "jerk" for them. Commented Jan 24, 2023 at 19:28