A Note on the Complete Monotonicity of the Generalized Mittag-Leffler Function

If \(E_{\alpha,\beta}(x)\) is the generalized Mittag-Leffler function, then the complete monotonicity of \(E_{\alpha,\beta}(-x)\) for \(0\le\alpha\le1\), \(\beta\ge\alpha\) is an immediate corollary of a 1948 result due to Pollard. The proof can be accomplished within the framework of real analysis.