Generalized Unimodality and Subordinators, With Applications to Stable Laws and to the Mittag-Leffler Function

Using the differential operator Ωs:=sI-xddx,s>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _s:=s\;I- x \frac{\textrm{d}}{\textrm{d}x}, \; s>0$$\end{document}, we build a new class of infinitely divisible distributions on the half-line. For this class, we give a stochastic interpretation and we provide several monotonicity properties for the associated subordinators. As an application, we solve a problem raised separately by Sendov and Shan in (J Theor Probab 28:1689–1725, 2015) and by Simon in (Math Nachr 285(4): 497–506, 2012) on the distribution of the stable subordinators. Finally, we provide a new complete monotonicity property for the Mittag-Leffler function.