  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.
