## A new class of bell-shaped functions

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- by Mateusz Kwaśnicki PDF
- Trans. Amer. Math. Soc.
**373**(2020), 2255-2280 Request permission

## Abstract:

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 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.## References

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## Additional Information

**Mateusz Kwaśnicki**- Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
- Email: mateusz.kwasnicki@pwr.edu.pl
- Received by editor(s): February 23, 2018
- Received by editor(s) in revised form: November 26, 2018, and January 9, 2019
- Published electronically: January 28, 2020
- Additional Notes: This work was supported by the Polish National Science Centre (NCN) grant no. 2015/19/B/ST1/01457
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 2255-2280 - MSC (2010): Primary 26A51, 60E07; Secondary 60E10, 60G51
- DOI: https://doi.org/10.1090/tran/7825
- MathSciNet review: 4069218

Dedicated: In memory of Augustyn Kałuża