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

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 $n$.