A new class of bell-shaped functions

TL;DR

This paper introduces a broad class of bell-shaped functions characterized by their derivatives' sign changes, unifies known examples, and extends to generalized gamma convolutions, with proofs involving convolution representations.

Contribution

It defines a new extensive class of bell-shaped functions using Stieltjes-type representations, encompassing all known examples and stable distribution densities.

Findings

Includes all previously known bell-shaped functions.

Extends to generalized gamma convolutions.

Provides a convolution-based proof method.

Abstract

We provide a large class of functions $f$ that are bell-shaped: the $n$-th derivative of $f$ changes its sign exactly $n$ times. This class is described by means of Stieltjes-type representation of the logarithm of the Fourier transform of $f$, and it contains all previously known examples of bell-shaped functions, as well as extended generalised gamma convolutions, including all density functions of stable distributions. The proof involves representation of $f$ as the convolution of a P\'olya frequency function and a function which is absolutely monotone on $(-\infty, 0)$ and completely monotone on $(0, \infty)$. In the final part we disprove three plausible generalisations of our result.