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Stieltjes functions of finite order and hyperbolic monotonicity

Authors: Lennart Bondesson and Thomas Simon
Journal: Trans. Amer. Math. Soc. 370 (2018), 4201-4222
MSC (2010): Primary 44A15; Secondary 60E05, 60E10
Published electronically: February 14, 2018
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Abstract: A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order and are not necessarily smooth. We show that such functions have a unique integral representation along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms of two Stieltjes transforms earlier obtained by the first author.

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Additional Information

Lennart Bondesson
Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, 90183 Umeå, Sweden

Thomas Simon
Affiliation: Laboratoire Paul Painlevé, Université de Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France

Keywords: Hyperbolic monotonicity, Stieltjes transform, Widder condition
Received by editor(s): April 19, 2016
Received by editor(s) in revised form: November 7, 2016
Published electronically: February 14, 2018
Additional Notes: The second author would like to thank Jean-François Burnol for several discussions related to this paper.
Article copyright: © Copyright 2018 American Mathematical Society

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