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bibkey: garrappa2025mittagmodulus
authors: Roberto Garrappa, Stefan Gerhold, Marina Popolizio, Thomas Simon
year: 2025
title: On some inequalities for the two-parameter Mittag-Leffler function in the complex plane
doi: null
url: https://arxiv.org/abs/2410.11852v4
claim: "Conjecture 25 characterizes the global reverse modulus inequality by the parameter interval and complete monotonicity of the reciprocal."
strata_touched:
  - D5/S3/Analytic/Asymptotics/MittagLefflerModulusRefutation
license: citation-only
triage: anchor

Verified locator

Source: https://arxiv.org/abs/2410.11852v4

The quoted text is from the fourth arXiv version, pages 1 and 18. The journal version of record has not been read.

Source statement

Page 1, equation (1.1):

In this paper we study several inequalities of the two-parameter Mittag-Leffler function $E_{\alpha ,\beta }$, which is defined on the complex plane for $\alpha >0$ and $\beta \in \mathbb{R}$ by the series representation $E_{\alpha ,\beta }(z)={\sum }_{k=0}^{\infty }\frac{z^k}{\Gamma (\alpha k+\beta )},z\in \mathbb{C}.$

Page 18:

Conjecture 25. For every α ∈ (1, 2), one has The inequality (1.4) holds globally ⇐⇒ β ∈ [α − 1, α] ⇐⇒ 1/Eα,β(x) is CM on (0, ∞).

Here (1.4) is $|E_{\alpha ,\beta }(z)|\ge E_{\alpha ,\beta }(\Re z)$, and globally means every complex $z$. The paper considers $\beta \ge 0$ in this inequality problem. Its CM definition requires a smooth positive-valued function on the positive half-line and nonnegative signed derivatives of every natural order.

Mathematical scope

The Lean series relation uses HasSum for every positive real alpha, arbitrary real beta and complex argument. Reciprocal Gamma is zero at its poles under Mathlib’s totalized division. The claim includes existence of the whole series family, both stated equivalences, and the positive codomain in the CM clause. Real arguments are embedded in the complex plane, with the real part reading the real-valued series there.

The refutation uses alpha = 3/2, beta = 1/2 and z = −115 + 10i. Sixty-four terms and an error at most 4/10^13 separate the modulus at this point from the value at −115. This certifies failure of the interval-to-global-inequality implication. It does not prove the all-alpha family or an interior-beta neighbourhood. The paper reports the other forward implications as proved; those implications are not challenged by this example.

Independent computation and literature boundary

The separate exact rational checker is in the-omega-institute/trureturing-experiments, commit f1be70a8a80297f44ec7a3d651faf1e622920913, at docs/reports/mittag-leffler-modulus-inequality/check.py. Its SHA256 is c6baf5bbab39481e1235752ddd18164291344b7bcfa8a7e11c51fb62a56aa2ff. The preregistration reports python3 check.py, exit 0, ALL_CHECKS_OK, for the larger witnesses r = 1024, 1536 and 2048 on z = r(−1+i/2). These computation readings are distinct from the kernel proof at −115+10i.

The literature screen in issue #22376 is partial. Completeness and absence of an earlier external settlement are ASSUMED-UNVERIFIED. Inaccessible citation APIs, the unread journal version, and unread full texts prevent an exhaustive priority claim. No new general Gamma identity or asymptotic expansion is claimed.