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