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arxiv: 1907.04636 · v1 · pith:24FKEUZTnew · submitted 2019-07-10 · 🧮 math.CV

Radii of starlikeness and convexity of q-Mittag--Leffler functions

Evrim Toklu This is my paper

Pith reviewed 2026-05-24 23:29 UTC · model grok-4.3

classification 🧮 math.CV
keywords q-Mittag-Leffler functionradii of starlikenessradii of convexityEuler-Rayleigh inequalitiesLaguerre-Pólya classHadamard factorizationstarlike functionsconvex functions
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The pith

Normalized q-Mittag-Leffler functions have tight lower and upper bounds on their radii of starlikeness from Euler-Rayleigh inequalities on the first positive zeros.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines radii of starlikeness and convexity for the q-Mittag-Leffler function under three normalizations chosen so the resulting functions are analytic in the unit disk. It invokes the Hadamard factorization of these normalized functions, which rests on their placement in the Laguerre-Pólya class of real entire functions. Euler-Rayleigh inequalities applied to the first positive zeros then supply explicit lower and upper estimates for the starlikeness radii. The estimates are described as tight and do not require locating every zero. The Laguerre-Pólya class is presented as the key justification for both the factorization and the inequalities.

Core claim

The normalized q-Mittag-Leffler functions belong to the Laguerre-Pólya class, permitting Hadamard factorizations inside the unit disk; Euler-Rayleigh inequalities applied to their first positive zeros then produce tight lower and upper bounds for the radii of starlikeness.

What carries the argument

Hadamard factorization of the three normalized q-Mittag-Leffler functions combined with Euler-Rayleigh inequalities on their first positive zeros.

If this is right

  • Tight lower and upper bounds are obtained for the radii of starlikeness of each of the three normalized functions.
  • The same normalization and factorization framework is used to address radii of convexity.
  • Membership in the Laguerre-Pólya class supplies both the factorization and the applicability of the Euler-Rayleigh inequalities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same zero-based bounds could be tested numerically for concrete values of the parameter q to check sharpness.
  • The technique may transfer to other q-analogues of entire functions whose zeros remain real and negative.
  • If the Laguerre-Pólya membership holds, the distribution of zeros controls geometric properties such as starlikeness in the unit disk.

Load-bearing premise

The normalized q-Mittag-Leffler functions belong to the Laguerre-Pólya class of real entire functions.

What would settle it

A direct computation of the starlikeness radius for one of the normalized functions that falls outside the stated lower and upper bounds would disprove the bounds.

read the original abstract

In this paper we deal with the radii of starlikeness and convexity of the $q-$Mittag--Leffler function for three different kinds of normalization by making use of their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-P\'olya class of real entire functions plays a pivotal role in this investigation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates the radii of starlikeness and convexity for three normalizations of the q-Mittag-Leffler function that are analytic in the unit disk. It invokes the Hadamard factorization of these functions (relying on membership in the Laguerre-Pólya class) and applies Euler-Rayleigh inequalities to their first positive zeros to obtain tight lower and upper bounds on the starlikeness radii.

Significance. If the Laguerre-Pólya membership holds and is properly established, the results would supply explicit, inequality-based bounds for these radii in the setting of q-special functions, extending classical techniques from geometric function theory. The method itself is standard and could be of interest if the foundational assumption is verified.

major comments (2)
  1. [Abstract] Abstract: the statement that the Laguerre-Pólya class 'plays a pivotal role' is not supported by any verification, proof, or citation establishing that the three normalized q-Mittag-Leffler functions have exclusively real negative zeros. This membership is required both for the Hadamard product representation and for the applicability of the Euler-Rayleigh inequalities to the positive real zeros; without it the claimed tight bounds on the starlikeness radii do not follow.
  2. [Main results (factorization and bounds)] The derivation of the radii bounds (presumably in the main results section following the factorization): the Euler-Rayleigh inequalities are applied directly to the first positive zeros after factorization, but the manuscript provides no explicit confirmation that all zeros are real and negative, rendering the bounds conditional on an unproven hypothesis.
minor comments (2)
  1. Clarify the precise definitions of the three normalizations and the corresponding entire functions at the outset, including any parameter restrictions on q.
  2. Ensure that all references to classical inequalities (Euler-Rayleigh) include the exact statements used and any growth-order conditions required for their application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need to explicitly support the Laguerre-Pólya class membership. We address the major comments point by point below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that the Laguerre-Pólya class 'plays a pivotal role' is not supported by any verification, proof, or citation establishing that the three normalized q-Mittag-Leffler functions have exclusively real negative zeros. This membership is required both for the Hadamard product representation and for the applicability of the Euler-Rayleigh inequalities to the positive real zeros; without it the claimed tight bounds on the starlikeness radii do not follow.

    Authors: We agree that the abstract statement requires supporting material. In the revised manuscript we will add either a short proof or an appropriate citation establishing that the three normalized q-Mittag-Leffler functions belong to the Laguerre-Pólya class (i.e., possess only real negative zeros). This will justify both the Hadamard factorization and the subsequent application of the Euler-Rayleigh inequalities. revision: yes

  2. Referee: [Main results (factorization and bounds)] The derivation of the radii bounds (presumably in the main results section following the factorization): the Euler-Rayleigh inequalities are applied directly to the first positive zeros after factorization, but the manuscript provides no explicit confirmation that all zeros are real and negative, rendering the bounds conditional on an unproven hypothesis.

    Authors: The referee is correct that the Euler-Rayleigh bounds rest on the reality and negativity of all zeros. We will insert an explicit confirmation (via proof or reference) of this property for each of the three normalizations immediately before the application of the inequalities, so that the radius bounds are placed on a rigorous footing. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external inequalities to assumed class membership

full rationale

The paper invokes the Laguerre-Pólya class to justify Hadamard factorization and Euler-Rayleigh inequalities on the zeros, then derives radius bounds. This is an external assumption whose verification is not attempted inside the paper; the steps themselves do not reduce by definition, self-citation chain, or fitted-parameter renaming to the target quantities. No self-definitional loops, fitted inputs presented as predictions, or load-bearing self-citations appear. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions from the theory of entire functions; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The q-Mittag-Leffler functions admit a Hadamard factorization as entire functions of order one that can be normalized to be analytic in the unit disk.
    Invoked explicitly to place the functions inside the unit disk before applying zero inequalities.
  • domain assumption The normalized functions belong to the Laguerre-Pólya class.
    Stated to play a pivotal role, enabling the Euler-Rayleigh inequalities on the first positive zeros.

pith-pipeline@v0.9.0 · 5615 in / 1459 out tokens · 27813 ms · 2026-05-24T23:29:54.479078+00:00 · methodology

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Reference graph

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