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AIMS Mathematics

2024, Volume 9, Issue 5: 11372-11395. doi: 10.3934/math.2024558
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Classes of completely monotone and Bernstein functions defined by convexity properties of their spectral measures

  • 1.
    Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia; wjedidi@ksu.edu.sa
  • 2.
    Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, ON, N6A 5B7 Canada; hsendov@uwo.ca
  • 3.
    Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, ON, N6A 5B7 Canada; sshan2@uwo.ca
  • Received: 31 December 2023 Revised: 07 March 2024 Accepted: 15 March 2024 Published: 25 March 2024

  • MSC : 26A48, 26D07

  • We were interested in Bernstein and Lévy measures having certain convexity-type properties. The convexity-type properties were an extension of the harmonic convexity property considered in [9]. We characterized the corresponding completely monotone and Bernstein functions. We hope this paper can aid with understanding the analogous properties and open questions presented in [8,9].

    Citation: Wissem Jedidi, Hristo S. Sendov, Shen Shan. Classes of completely monotone and Bernstein functions defined by convexity properties of their spectral measures[J]. AIMS Mathematics, 2024, 9(5): 11372-11395. doi: 10.3934/math.2024558

    Related Papers:

  • We were interested in Bernstein and Lévy measures having certain convexity-type properties. The convexity-type properties were an extension of the harmonic convexity property considered in [9]. We characterized the corresponding completely monotone and Bernstein functions. We hope this paper can aid with understanding the analogous properties and open questions presented in [8,9].


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    [1] W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Basel: Birkhäuser, 2011. http://dx.doi.org/10.1007/978-3-0348-0087-7
    [2] S. Bridaa, W. Jedidi, H. Sendov, Generalized unimodality and subordinators, with applications to stable laws and to the Mittag-Leffler function, J. Theor. Probab., 37 (2024), 1–42. http://dx.doi.org/10.1007/s10959-023-01242-z doi: 10.1007/s10959-023-01242-z
    [3] M. Carter, B. Brunt, The Lebesgue-Stieltjes integral: a practical introduction, New York: Springer-Verlag, 2000. http://dx.doi.org/10.1007/978-1-4612-1174-7
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    [5] M. Merkle, Reciprocally convex functions, J. Math. Anal. Appl., 293 (2004), 210–218. http://dx.doi.org/10.1016/j.jmaa.2003.12.021 doi: 10.1016/j.jmaa.2003.12.021
    [6] R. Rockafellar, Convex analysis, Princeton: Princeton University Press, 1970.
    [7] R. Schilling, R. Song, Z. Vondracek, Bernstein functions theory and applications, 2 Eds., Berlin: De Gruyter, 2012.
    [8] H. Sendov, S. Shan, New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures, J. Theor. Probab., 28 (2015), 1689–1725. http://dx.doi.org/10.1007/s10959-014-0557-9 doi: 10.1007/s10959-014-0557-9
    [9] H. Sendov, S. Shan, Properties of completely monotone and Bernstein functions related to the shape of their measures, J. Convex Anal., 23 (2016), 981–1015.
    [10] T. Simon, On the unimodality of power transformations of positive stable densities, Math. Nachr., 285 (2012), 497–506. http://dx.doi.org/10.1002/mana.201000062 doi: 10.1002/mana.201000062
    [11] D. Widder, The Laplace transform, Princeton: Princeton University Press, 1941.
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  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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