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

Using the differential operator Omega(s) := s I - xd/dx, s > 0, 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.