I tried to characterize mathematically what makes a function bell-shaped. I found the following:
Definition. A $C^\infty$-function $f:\Bbb R\to\Bbb R$ is called bell-shaped if its $n$-th derivative has exactly $n$ zeros (counted with multiplicity) for all $n\in\Bbb N_0$.
I do not claim that any intuitively bell-shaped curve is included (e.g. non-smooth functions), but I felt pretty confident that there are no intuitively non-bell-shaped curves in this class. However, my imagination is limited, so I want to ask whether anyone can find a pathological example of some curve in this class for which it might be debatable to call it bell-shaped in an intuitive sense. Upside down or asymmetric bells are okay for me.
Also, is it true that any such function must be bounded? I would consider an unbounded function as a counterexample. Then the next question would be whether it suffices to extend the definition by the predicate "bounded".
Update
There were unbounded counterexamples provided by this follow-up question on MO. For example
- $f(x)=(1+x^2)^s$ for $s\in(0,1/2)$ and
- $f(x)=\log(1+x^2)$.
So I will include boundedness into the definition.
Further, I will provide some reasoning why I believe above definition (+ boundedness) characterizes bell-shaped functions. Let $f$ be an intuitively bell-shaped $C^\infty$-function. This includes (at least for me) that
- $f$ has a single maximum (or minimum),
- $f$ has no saddle points, and
- $f^{(n)}$ vanishes for $|x|\to\infty$.
Because of 1. and 2., the derivative $f'$ has a single zero.
Note that a function $g$ which vanishes at infinity and has exactly $n>0$ zeros will yield a derivative $g'$ with at least $n+1$ zeros. This is because of rolle's theorem there is a stationary point between any two zeros, as well as between a zero and infinity (because $g\to 0$ for $|x|\to 0$ can be considered as zeros of $g$ in some sense).
These considerations showed that an intuitively bell-shaped function has an increasing number of zeros for its derivatives. Turning this around made me hope that the bell-shaped functions are exactly those functions for which this zero count increases as slowly as possible. As mentioned, I learned that I will need boundedness to make $f$ vanish at infinity. But I still hope that these restrictions on the zeros make the functions as well-behaved as possible.