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 $α, β$ and $γ$ for which Mittag-Leffler functions of Le Roy type $F_{α, β}^{(γ)}$ 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 $F_{α, β}^{(γ)}$ for integer $γ= n$ and rational $0 < α\leq 1/n$. In this way it is possible to show that Mittag-Leffler functions of Le Roy type are completely monotone for $α= 1/n$ and $β\geq (n+1)/(2n)$ as well as for rational $0 < α\leq 1/2$, $β= 1$ and $n=2$. For further integer values of $n$ the complete monotonicity is tested numerically for rational $0< α< 1/n$ and various choices of $β$. The obtained results suggest that for the complete monotonicity the condition $β\geq (n+1)/(2n)$ holds for any value of $n$.