A new class of bell-shaped functions

We provide a large class of functions f that are bell-shaped: the nth 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 all extended generalised gamma convolutions, including all density functions of stable distributions. The proof involves representation of f as the convolution of a Pólya 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.