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  "slug": "reciprocal-mittag-leffler-complete-monotonicity",
  "title": "Reciprocal Mittag–Leffler functions: complete monotonicity below the parameter band, and the exact boundary near order two",
  "shortTitle": "The band is too narrow: when the reciprocal Mittag–Leffler function is completely monotone",
  "url": "https://evidencepress.org/releases/reciprocal-mittag-leffler-complete-monotonicity/",
  "oneLine": "The reciprocal of the Mittag–Leffler function E is completely monotone for every 0 < β ≤ α when 1 < α < 2, and, for α near 2, exactly for 0 < β ≤ b(α) with b(2) = 3, so the conjectured band α − 1 ≤ β ≤ α is too narrow on both sides.",
  "abstract": "For which parameters is the reciprocal 1/E of the two-parameter Mittag–Leffler function completely monotone, and so (after normalising) the Laplace transform of a probability law? Garrappa, Gerhold, Popolizio and Simon (2025, Conjecture 25) proposed that, for 1 < α < 2, this happens exactly when α − 1 ≤ β ≤ α. This candidate proves two results showing the band is too narrow. Theorem A: for every 1 < α < 2 and 0 < β ≤ α, log[Γ(β)E] is a Bernstein function, so every positive power of the normalised reciprocal is the Laplace transform of an infinitely divisible law; this covers the whole triangle below the band. The proof combines a dilation-ratio identity contained in the proof of the authors' Proposition 3 with a classical recurrence that lowers β by one. Theorem B: for α in some neighbourhood of 2, the admissible set is exactly the interval (0, b(α)], where b is real analytic, b(2) = 3 and b′(2) = 1 + 2π Si(2π)/(2I(2π)) ≈ 2.8277, with I(a) the integral of (1 − cos u)/u over (0, a); the boundary is where two negative zeros near −4π² collide, and the associated laws are self-decomposable. Since b(α) > α just below 2, the band fails above as well. A numerical continuation of the boundary curve supports a replacement conjecture, and an exact rational certificate exhibits a failure of the global modulus inequality where Theorem A holds. The neighbourhood in Theorem B is not quantified. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-04",
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  "authors": [
    "Anonymous"
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  "keywords": [
    "Mittag-Leffler function",
    "complete monotonicity",
    "Bernstein function",
    "infinite divisibility",
    "self-decomposability",
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    "zeros of special functions",
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    "Theorem A (proof): for every 1 < α < 2 and 0 < β ≤ α, log[Γ(β)E] is a Bernstein function, so 1/E is completely monotone and every positive power of the normalised reciprocal is an infinitely divisible probability Laplace transform; this covers the triangle 0 < β < α − 1 below the conjectured band.",
    "Theorem B (proof): for α in a neighbourhood of 2 the admissible set is exactly (0, b(α)], with b real analytic, b(2) = 3, and self-decomposable laws throughout; the band fails above as well as below.",
    "Boundary slope (proof and 90-digit check): b′(2) = 1 + 2π Si(2π)/(2I(2π)) = 2.82768173110428…, where the two negative zeros near −4π² collide.",
    "Separation of properties (proof and exact certificate): below the band, complete monotonicity holds while the global inequality |E(z)| ≥ E(Re z) fails, certified at (α, β, z) = (3/2, 1/4, −60 + i).",
    "Numerics and conjecture (evidence only): the double-zero curve continues smoothly over 1.5 ≤ α ≤ 2.5 with b(α) > α below 2, suggesting C_α = (0, b(α)] for all 1 < α < 2."
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  "relatedWorks": [
    {
      "citation": "Bartholmé and Patie (2021). Turán Inequalities and Complete Monotonicity for a Class of Entire Functions. Analysis Mathematica. Bernstein correspondence and the normalised diagonal identity.",
      "url": "https://doi.org/10.1007/s10476-021-0088-4",
      "doi": "10.1007/s10476-021-0088-4"
    },
    {
      "citation": "Flajolet and Sedgewick (2009). Analytic Combinatorics. Cambridge University Press. Pringsheim's theorem (Theorem IV.6), used for necessity.",
      "url": "https://doi.org/10.1017/cbo9780511801655",
      "doi": "10.1017/cbo9780511801655"
    },
    {
      "citation": "Garrappa, Gerhold, Popolizio and Simon (2025). On some inequalities for the two-parameter Mittag-Leffler function in the complex plane. Journal of Mathematical Analysis and Applications. The band result, the hitting-time construction and Conjecture 25 addressed here.",
      "url": "https://doi.org/10.1016/j.jmaa.2025.129588",
      "doi": "10.1016/j.jmaa.2025.129588"
    },
    {
      "citation": "Konstantopoulos, Patie and Sarkar (2024). A new class of solutions to the van Dantzig problem, the Lee–Yang property, and the Riemann hypothesis. Annales de l'Institut Fourier. Earlier Mittag–Leffler constructions and their probabilistic interpretations.",
      "url": "https://doi.org/10.5802/aif.3600",
      "doi": "10.5802/aif.3600"
    },
    {
      "citation": "Pierre Patie (2009). Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Annales de l'Institut Henri Poincaré. The diagonal case β = α.",
      "url": "https://doi.org/10.1214/08-aihp182",
      "doi": "10.1214/08-aihp182"
    },
    {
      "citation": "Polson (2026). Thorin Quantisation. SSRN preprint on related hitting-time laws; read at abstract level only, a recorded priority uncertainty.",
      "url": "https://doi.org/10.2139/ssrn.7277540",
      "doi": "10.2139/ssrn.7277540"
    },
    {
      "citation": "Schilling, Song and Vondraček (2012). Bernstein Functions. De Gruyter monograph. Closure properties of Bernstein functions.",
      "url": "https://doi.org/10.1515/9783110269338",
      "doi": "10.1515/9783110269338"
    }
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