Some results on the Complete Monotonicity of Mittag-Leffler Functions of Le Roy Type

Abstract

AbstractThe paper [5] by R. Garrappa, S. Rogosin, and F. Mainardi, entitled “On a generalized three-parameter Wright function of the Le Roy type” and published inFract. Calc. Appl. Anal.20(2017), 1196–1215, ends up leaving the open question concerning the range of the parametersα,βandγfor which Mittag-Leffler functions of Le Roy type$\begin{array}{} F_{\alpha, \beta}^{(\gamma)} \end{array}$are completely monotonic. Inspired by the 1948 seminal H. Pollard’s paper which provides the proof of the complete monotonicity of the one-parameter Mittag-Leffler function, the Pollard approach is used to find the Laplace transform representation of$\begin{array}{} F_{\alpha, \beta}^{(\gamma)} \end{array}$for integerγ=nand rational 0 <α≤ 1/n. In this way it is possible to show that the Mittag-Leffler functions of Le Roy type are completely monotone forα= 1/nandβ≥ (n+ 1)/(2n) as well as for rational 0 <α≤ 1/2,β= 1 andn= 2. For further integer values ofnthe complete monotonicity is tested numerically for rational 0 <α< 1/nand various choices ofβ. The obtained results suggest that for the complete monotonicity the conditionβ≥ (n+ 1)/(2n) holds for any value ofn.