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

The 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 in Fract. Calc. Appl. Anal. 20 (2017), 1196-1215, ends up leaving the open question concerning the range of the parameters alpha, beta and. 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 <= 1/n. In this way it is possible to show that the Mittag-Leffler functions of Le Roy type are completely monotone for a = 1/n and beta >= (n + 1)/(2n) as well as for rational 0 < alpha <= 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 >= (n + 1)/(2n) holds for any value of n.