Reciprocal Mittag-Leffler functions: complete monotonicity below the parameter band, and the exact boundary near order two
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Unrefereed research candidate. In one sentence: the reciprocal 1/Eα,β of the two-parameter Mittag-Leffler function is completely monotone on a strictly larger parameter set than Garrappa, Gerhold, Popolizio and Simon (2025, Conjecture 25) proposed, so their parameter band [α-1, α] is not necessary.
Theorem A. For every 1 < α < 2 and 0 < β ≤ α, log[Γ(β)Eα,β] is a Bernstein function, so every positive power of the normalised reciprocal is the Laplace transform of an infinitely divisible probability law. This covers the whole triangle 0 < β < α-1 below the band. The proof combines a dilation-ratio identity contained in the proof of the authors' Proposition 3 with the recurrence Eα,β = βEα,β+1 + αzE'α,β+1.
Theorem B. For α in a two-sided neighbourhood of 2 the admissible set is exactly (0, b(α)], where b is real analytic, b(2) = 3, and b'(2) = 1 + 2π Si(2π)/(2I(2π)) = 2.8276817311..., with I(a) the integral of (1-cos u)/u over (0,a). The boundary is where two negative zeros near -4π2 collide; the associated laws are self-decomposable. So the band also fails above for α just below 2. A 40-digit numerical continuation of the boundary curve over 1.5 ≤ α ≤ 2.5 supports a replacement conjecture; an exact rational certificate exhibits a failure of the global modulus inequality at (α, β, z) = (3/2, 1/4, -60+i).
Limits: the neighbourhood in Theorem B is not quantified and no specific parameter pair above the band is certified; the equivalence of the band with the global modulus inequality is not addressed; two inputs are cited (the hitting-time construction of Garrappa et al. and the order-two zero theorem). The archive contains the manuscript (PDF and Markdown source), verify.py with three finite checks and a negative control (valid under python -O), their retained outputs, novelty reports, a public review record and SHA-256 sums. Assurance: written proofs; internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text CC0-1.0; code MIT.