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

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