Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization: Necessary and Sufficient Conditions
We determine necessary and sufficient conditions for complete monotonicity of the Srivastava--Tomovski extension, a generalized Mittag--Leffler kernel arising in fractional calculus, while separating the defining generalized-Wright series from its positive-axis realization when the series is not entire. For positive parameters, let \(\Delta=1+\alpha-\kappa\). In the entire regime \(\Delta>0\), \(E_{\alpha,\beta}^{\gamma,\kappa}(-x)\) is completely monotone on \((0,\infty)\) if and only if \(\alpha\leq\kappa\) and \(\kappa\beta\geq\alpha\gamma\). For arbitrary positive parameters, the corresponding positive-axis realization satisfies the same criterion; when \(0<\alpha<\kappa\), it is defined by a second-kind Wright kernel and remains meaningful through the finite-radius and zero-radius phases of the defining series. We also identify the Bernstein measure in every admissible case. In the interior region it is, after normalization, the pushforward under \(t\mapsto t^\kappa\) of a power-biased Wright distribution; at \(\alpha=\kappa\) it becomes a powered beta law for \(\beta>\gamma\) and the atom \(\Gamma(\gamma)^{-1}\delta_1\) for \(\beta=\gamma\). Necessity in the excluded regions follows from a Mellin-transform zero, uniqueness of finite signed Laplace transforms, and moment-support asymptotics. The Prabhakar criterion is recovered by setting \(\kappa=1\).