# Why are monotonicity and convexity so intensely studied in comparison to their higher-order analogies?

I've been recently contemplating what can be said about functions $$f:\Bbb R \to \Bbb R$$ with positive third-order derivative and their properties:

It puzzles me why monotonicity and convexity, i.e. the properties associate with the first- and second-order derivatives of a function being positive, are frequently used, whilst their higher-order alternatives are so rarely referred to.

I found that the third-order derivative at a point is called jerk in physics and its meaning is discussed here. I found also a discussion on the topic why are third-order concepts so rare. However, what I'm interested in are the uses of the fact that third-order derivative of $$f$$ is positive in analysis.

For example, whilst strict convexity of $$f$$ implies that every critical point of $$f$$ is the unique global minimum of $$f$$ and that $$f$$ has at most two roots, having strictly positive third-order derivative would guarantee that every point with $$f''(x)=0$$ is the unique inflection point of $$f$$, that the function has at most one local minimum and one local maximum, and no more than 3 roots.

One could argue that it would be informative enough to know that there is $$x_2\in \Bbb R$$ such that $$f''(x_2)=0$$ and that $$f$$ is concave on $$(-\infty,x_2]$$ and convex on $$[x_2,\infty)$$. However, analogously one could argue that convexity is not needed for analyzing function's minima because it is enough to know that there is $$x_1\in \Bbb R$$ such that $$f'(x_1)=0$$ and that $$f$$ is decreasing on $$(-\infty,x_1]$$ and increasing on $$[x_1,\infty)$$.

The concept of $$n$$-th order convexity (having positive $$n$$-th order derivative) is essential for the analysis of the number of roots of one-variable functions: A function with positive $n$-th derivative has at most $n$ roots – an inequality version of the Fundamental theorem of Algebra.

Consider the question on what is the maximum number of strict local minima that a degree $$k$$ polynomial $$p(x,y)$$ in two variables can have?

• In case of quadratic polynomial one can readily answer: The polynomial has a strict local minima only if it is strictly convex, and then the minimum must be unique.
• In case of cubic polynomial, there is also at most one strict local minimum because if there were two, say at points $$a$$ and $$b$$, then the third degree polynomial $$q(t)=p\big((1-t)a+tb\big)$$ would need to have two strict local minima – impossible. [1]
• In case of quartic polynomial, none of the above arguments apply and the only ready estimate follows from applying Bézout's theorem to the partial derivatives of $$p$$, and so we can be sure that $$p$$ has is no more than $$3\times 3$$ isolated critical points. [2]

I expected that the above analysis would be trivial for the case of quadratic polynomials thanks to the concept of convexity. However, the analysis is equally trivial in case of cubic polynomial, the only difference is that there is no name for the "third-order convexity" that gives the result. In fact, the analysis becomes difficult as late as in the case of quartic polynomials. This suggests that third-order convexity has practical applications, only the higher-order alternatives would be less practical for the analysis of functions in two or more variables.

Does importance of $$n$$-th derivative drop?

Let me compare the count of search results of terms

• a) convex
• b) monotone or monotonic (sum up the count)

$$\begin{array} {|r|r|r|}\hline & \text{Google} & \text{site:SME} & \text{site:mathoverflow} & \text{G Scholar} \\ \hline \text{a)} & 148M & 79K & 229 & 4.3M \\ \hline \text{b)} & 98M & 77K & 54 & 2.5M \\ \hline \end{array}$$

Why is the second-order property (convexity) more prevalent than the first-oder one (monotonicity) and yet the third-order one is almost never heard of?

Questions:

1. Is there a standardized name for functions with $$f^{(3)}>0$$?
2. Is the main reason that $$f^{(3)}$$ is so rarely analyzed the fact that for most problems it is enough to determine on which regions $$f$$ is convex/concave and the effort needed to analyze $$f^{(3)}$$ would typically not be justified?
3. Is there a known set property of $$\mathop{epi}(f)$$ for $$f$$ with $$f^{(3)}>0$$, akin to strict convexity of $$\mathop{epi}(f)$$ for $$f$$ with $$f^{(2)}>0$$?

Related Posts:

Let me share some observations and conjectures that I came up with when contemplating on this topic:

• convexity and monotonicity are independent of the differentiability of a function, so your question doesn't make sense. The concept of convexity is related to subsets of vector spaces, where to concept applied to functions comes, and the word monotonicity is related to functions between ordered spaces Commented Jan 17, 2023 at 7:07
• @Masacroso Thanks for sharing your view, however, I must disagree. I restricted attention to three times differentiable functions $f:\Bbb R \to \Bbb R$ for the sake of simplicity. 1) As shown in math.stackexchange.com/questions/4616713/… the property $f^{(n)}\geq 0$ has an analogy that does not require the function to be differentiable. 2) In one of the attached links I also discuss that the property $f^{(3)}>0$ has an ordinal analogy the same way quasi-convexity is an ordinal analogy of convexity. Commented Jan 17, 2023 at 8:37
• 3) The same way convexity of functions $g:\Bbb R^d \to \Bbb R$ can be defined by referring to convexity along lines, the property $d^3 g((1-t)a+tb)/dt^3>0$, where $a\in \Bbb R^d$ and $b\in a+ \Bbb R_{++}^d$ can be studied. Commented Jan 17, 2023 at 8:44
• interesting... I didn't knew this relation of "convexity" related to polynomials of $n$-degree Commented Jan 17, 2023 at 9:13
• @MarianoSuárez-Álvarez But why is that the case? It is understandable that not much is taught about cubic functions in a high-school: There is a practical formula for the roots of a quadratic function, and the concept of convexity is easy to interpret as one only needs to interpolate two points by a line to imagine its meaning. In contrast to interpret "3-convexity" one needs to interpolate 3 points by a quadratic function to imagine its meaning. Such a concept is perhaps too challenging for most high-school students, but how about graduates? Commented Jan 19, 2023 at 2:53

The thing with considering how higher-order derivatives behave and what that implies is that we lose information in the process. Our definition of the function becomes more broad or arbitrary. It's similar to how higher-order differential equations have more arbitrary solutions, with one more linearly independent family if we increase the order by one for linear ones for example.

To picture this, we will consider the intervals at which the function is increasing and at which it's decreasing given that an $$n^\text{th}$$ derivative, $$D^n$$ is positive everywhere for example and consider what we need to do to be able to get a sketch for it, it's like trying to expand the function in some sense.

For simplicity, let $$C_k\subseteq\mathbb{R}$$ be a continuous interval in the domain, where $$C_1\cup C_2\cup\cdots\cup C_n=\mathbb{R}$$ given $$D^n>0.$$

\begin{align} &D^1>0 \begin{cases} D^1>0&\mathbb R \end{cases}\\ &D^2>0 \begin{cases} D^1>0&C_1\\ D^1<0&C_2\\ \end{cases}\\ &D^3>0 \begin{cases} D^2<0&C_1\\ D^2>0&C_2 \end{cases}\\ \implies&D^3>0 \begin{cases} D^1<0&C_1\cup C_2\\ D^1>0&C_3\cup C_4 \end{cases}\\ &\qquad\qquad\vdots \end{align} It's like a tree diagram where going up an order adds a new branch, or a better analogy is a piecewise function that gets choppier the higher the order gets.

For the first derivative, it's increasing everywhere; to graph it, we need to evaluate $$D^0$$ at a point. For the second we need to solve $$D^1=0$$ and evaluate $$D^0$$ at $$2$$ points. For the third derivative, we need to solve $$D^2=0$$ and $$D^1=0$$ and evaluate $$D^0$$ at $$4$$ points and so on.

It can also get increasingly more complex as the order increases to solve for the turning points at which the derivatives are $$0,$$ but that's not what the main reason we don't consider higher-order derivatives.

• What you describe could be seen as an advantage as well: Suppose that I have a third degree polynomial with a positive leading coefficient. It is neither convex nor concave. However, its third-order derivative is positive, so I can use that property when analyzing it. Imagine that you asked about why are only quadratic polynomials and not higher-degree ones used to interpolate points of a graph, and I answered that it is because higher-degree polynomials have more complex behavior. Well, more degrees of freedom come with more complexity, that is natural. Commented Jan 25, 2023 at 4:03
• Well, I don't think I agree that it's an advantage. What I am trying to point out is that, for example, if I only give you that a cubic polynomial has positive second derivative and that another one has positive third derivative, which one has less possibilities? Which one is more useful for analysing a particular function, which I think is what most of the problems in analysis are? For your arguement about quadratics, I think that saying that higher-degree polynomials are more complex is a fair and sensible answer. Commented Jan 25, 2023 at 4:26
• When I Google a) convex I receive 148M results; b) for monotone and monotonic the total is only 97.7M. In Google Scholar these counts are a) 4.27M and b) 2.48M. So the second order property (convexity) is studied / used more than the first-order property (monotonicity). How could this be consistent with your argument? Commented Jan 25, 2023 at 6:22
• Because convexity is more advanced. You wouldn't search for solutions of seperable differential equations, but you would search for that of homogeneous and second-order ones, right? You are still not getting my point. A monotonic function is a very strict definition that's very exact, at least more than a concave/convex one which encompasses more possibilities. Monotonicity is stronger than convexity, but for example, some theorems might only require the weaker condition which makes the theorems themselves stronger etc. Commented Jan 25, 2023 at 7:52
• I think looking for Google results is misleading since convexity is a geometric property that has meaning in many situations outside of derivatives (see functional analysis and convex hulls, elementary geometry and many others etc), while monotonicity depends on order so generally speaking, is just a real-valued property; Commented Jan 25, 2023 at 16:37