A new class of bellshaped functions
Abstract
We provide a large class of functions $f$ that are bellshaped: the $n$th derivative of $f$ changes its sign exactly $n$ times. This class is described by means of Stieltjestype representation of the logarithm of the Fourier transform of $f$, and it contains all previously known examples of bellshaped 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 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.11023
 arXiv:
 arXiv:1710.11023
 Bibcode:
 2017arXiv171011023K
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Complex Variables;
 Mathematics  Probability
 EPrint:
 24 pages, 2 figures