A new class of bell-shaped functions

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 $(-\mathrm{\infty },0)$ and completely monotone on $(0,\mathrm{\infty })$. In the final part we disprove three plausible generalisations of our result.