Some results on the complete monotonicity of the Mittag-Leffler functions of Le Roy type

TL;DR

This paper investigates the conditions under which Mittag-Leffler functions of Le Roy type are completely monotonic, extending Pollard's classical results and providing new parameter ranges through analytical and numerical methods.

Contribution

It derives new parameter conditions for complete monotonicity of Le Roy type Mittag-Leffler functions using Laplace transforms and numerical testing, expanding known results.

Findings

Complete monotonicity holds for lpha=1/n and eta  (n+1)/(2n)

Analytical Laplace transform representations are obtained for integer lpha=n

Numerical tests suggest eta  (n+1)/(2n) for all n

Abstract

The paper by R. Garrappa, S. Rogosin, and F. Mainardi, entitled {\em On a generalized three-parameter Wright function of the Le Roy type} and published in [Fract. Calc. Appl. Anal. {\bf 20} (2017) 1196-1215], ends up leaving the open question concerning the range of the parameters $\alpha, \beta$ and $\gamma$ for which Mittag-Leffler functions of Le Roy type $F_{\alpha, \beta}^{(\gamma)}$ 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 $F_{\alpha, \beta}^{(\gamma)}$ for integer $\gamma = n$ and rational $0 < \alpha \leq 1/n$. In this way it is possible to show that Mittag-Leffler functions of Le Roy type are completely monotone for $\alpha = 1/n$ and $\beta \geq (n+1)/(2n)$ as well as for rational $0 < \alpha \leq 1/2$, $\beta = 1$ and $n=2$. For further integer values of $n$ the complete monotonicity is tested numerically for rational $0< \alpha < 1/n$ and various choices of $\beta$. The obtained results suggest that for the complete monotonicity the condition $\beta \geq (n+1)/(2n)$ holds for any value of $n$.