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arxiv: 2605.21978 · v1 · pith:L3JYUESInew · submitted 2026-05-21 · 🧮 math.CV

Certain subclass of Meromorphic function associated with Wright function

Anish Kumar This is my paper

Pith reviewed 2026-05-22 02:54 UTC · model grok-4.3

classification 🧮 math.CV
keywords meromorphic functionsWright functioncoefficient estimatesstarlikenessconvexityCarathéodory functionsgeometric function theory
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The pith

A Wright-function operator defines a new meromorphic class with exact coefficient bounds and starlikeness radii

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

The paper defines a subclass Σ(θ, λ, γ) of meromorphic functions in the punctured unit disk by means of a generalized operator built from the Wright function. It derives an integral representation for the class together with necessary and sufficient convolution conditions and supplies sufficient conditions for membership. Coefficient bounds are obtained via Carathéodory-function properties and induction; these bounds are applied to compute the precise radii of meromorphic starlikeness and convexity of order ρ. The constructions generalize earlier results on meromorphic univalent functions.

Core claim

The operator W_{α,β} associated with the Wright function is used to construct the class Σ(θ, λ, γ) of meromorphic functions; the class admits an exact integral representation, satisfies convolution conditions, obeys coefficient estimates derived by Carathéodory functions and induction, and possesses explicit radii of meromorphic starlikeness and convexity of order ρ.

What carries the argument

The generalized operator W_{α,β} built from the Wright function, which defines the class Σ(θ, λ, γ) and supplies the integral and convolution relations used to obtain coefficient bounds and radii.

If this is right

  • Coefficient estimates hold for every function in the class Σ(θ, λ, γ).
  • The radius of meromorphic starlikeness of order ρ is determined exactly from the coefficient bounds.
  • The radius of meromorphic convexity of order ρ is determined exactly from the coefficient bounds.
  • The results recover and extend several known theorems on meromorphic starlike and convex functions.

Where Pith is reading between the lines

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

  • The same operator technique could be applied to other special functions to generate analogous meromorphic classes.
  • The radii obtained may serve as explicit constants in distortion theorems or growth estimates for meromorphic mappings.
  • Numerical verification of the radii for sample functions would provide direct checks of the coefficient estimates.

Load-bearing premise

The operator W_{α,β} maps meromorphic functions into itself so that the integral representation and convolution conditions remain valid for the chosen parameter ranges.

What would settle it

An explicit function belonging to Σ(θ, λ, γ) for concrete parameter values whose second coefficient exceeds the derived bound or whose starlikeness radius is smaller than the claimed value.

Figures

Figures reproduced from arXiv: 2605.21978 by Anish Kumar.

Figure 1
Figure 1. Figure 1: The graphical representation of the maximal radius of meromorphic starlikeness r1 as a function of the order ρ for the dominant leading term (n = 1) [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graphical representation of the maximal radius of meromorphic convexity r2 as a function of the order ρ for a higher-order dominant term (n = 2). 4. Conclusion In the present investigation, we have successfully defined and systematically studied a new subclass Σ(θ, λ, γ) of meromorphic functions in the punctured unit disk D∗ , utilizing the gen￾eralized linear operator Wα,β. The primary achievements of… view at source ↗
read the original abstract

In this paper, we introduce and investigate a novel subclass $\Sigma(\theta, \lambda, \gamma)$ of meromorphic functions defined in the punctured unit disk ${D}^*$. This class is constructed utilizing a specialized generalized operator $W_{\alpha, \beta}$ associated with Wright function. We derive the exact integral representation and establish necessary and sufficient convolution (Hadamard product) conditions. Furthermore, sufficient conditions involving strict inequalities are provided for functions to be members of this class $\Sigma(\theta, \lambda, \gamma)$. Additionaly, by employing the properties of Carath\'eodory functions and the principle of mathematical induction, we establish coefficient estimates for functions belonging to this new class. Finally, as an applications, we the established coefficient bounds, we determine the precise radii of meromorphic starlikeness and meromorphic convexity of order $\rho$. The results presented in this study generalize several existing outcomes in geometric Function Theory.

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 introduces a new subclass Σ(θ, λ, γ) of meromorphic functions in the punctured unit disk D* defined via a generalized operator W_{α,β} built from the Wright function. It claims an exact integral representation for functions in the class, derives necessary and sufficient convolution (Hadamard product) conditions, provides sufficient conditions for membership, obtains coefficient estimates by combining Carathéodory-function properties with mathematical induction, and applies the bounds to compute precise radii of meromorphic starlikeness and meromorphic convexity of order ρ. The results are presented as generalizations of earlier work in geometric function theory for meromorphic functions.

Significance. If the central derivations are valid, the manuscript adds a concrete example of how Wright-function operators can be used to generate new subclasses of meromorphic functions and to obtain explicit coefficient bounds and radii. The combination of an integral representation, convolution characterization, induction-based estimates, and radius applications follows a recognizable pattern in the field and supplies a modest but usable extension of existing results on meromorphic starlikeness and convexity.

major comments (2)
  1. [Section introducing the operator W_{α,β} and the class Σ(θ, λ, γ)] The exact integral representation asserted for the operator W_{α,β} (used to define Σ(θ, λ, γ)) is obtained by term-by-term integration of the Wright series. No explicit verification of absolute convergence or dominated convergence in D* is supplied for the full range of parameters α, β, θ, λ, γ; if the Wright function decays too slowly, the representation may hold only in a smaller punctured disk or under additional restrictions not stated in the manuscript. This assumption is load-bearing for all subsequent convolution conditions and coefficient estimates.
  2. [Application section on radii of starlikeness and convexity] The radii of meromorphic starlikeness and convexity of order ρ are derived directly from the coefficient bounds obtained by induction. Because those bounds rest on the integral representation and the convolution characterization, any gap in the justification of the representation propagates to the radius formulas; the manuscript does not discuss the dependence of the radii on possible restrictions needed for convergence.
minor comments (2)
  1. [Abstract] The abstract contains typographical errors and an incomplete sentence: “Additionaly” should read “Additionally” and the clause “we the established coefficient bounds, we determine” is missing words.
  2. [Introduction and definitions] Notation for the free parameters θ, λ, γ and the operator parameters α, β should be introduced with a single consistent list at the beginning of the paper rather than scattered across definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We have addressed each major concern point by point below. Revisions have been made to strengthen the justification of the integral representation and to clarify the domain of validity for the radius results.

read point-by-point responses

  1. Referee: [Section introducing the operator W_{α,β} and the class Σ(θ, λ, γ)] The exact integral representation asserted for the operator W_{α,β} (used to define Σ(θ, λ, γ)) is obtained by term-by-term integration of the Wright series. No explicit verification of absolute convergence or dominated convergence in D* is supplied for the full range of parameters α, β, θ, λ, γ; if the Wright function decays too slowly, the representation may hold only in a smaller punctured disk or under additional restrictions not stated in the manuscript. This assumption is load-bearing for all subsequent convolution conditions and coefficient estimates.

    Authors: We appreciate this observation. The Wright function parameters in the manuscript are chosen such that the series coefficients satisfy |a_n| ≤ C / (n!)^ρ for some ρ > 0, ensuring absolute convergence of the integrated series uniformly on compact subsets of D* by the Weierstrass M-test. To make this rigorous, we have inserted a new Lemma 2.1 in the revised manuscript that proves absolute convergence of the term-by-term integral for the stated parameter ranges using the known asymptotic estimates for the Wright function. This lemma directly supports the integral representation, convolution conditions, and all subsequent estimates without requiring further restrictions on the domain. revision: yes

  2. Referee: [Application section on radii of starlikeness and convexity] The radii of meromorphic starlikeness and convexity of order ρ are derived directly from the coefficient bounds obtained by induction. Because those bounds rest on the integral representation and the convolution characterization, any gap in the justification of the representation propagates to the radius formulas; the manuscript does not discuss the dependence of the radii on possible restrictions needed for convergence.

    Authors: We agree that the radius formulas inherit their validity from the coefficient bounds. In the revised manuscript we have added a remark immediately after Theorems 4.1 and 4.2 stating that the computed radii hold throughout the punctured unit disk under the parameter conditions for which the integral representation is justified (now established in the new Lemma 2.1). No additional domain restrictions are needed beyond those already implicit in the definition of the class Σ(θ, λ, γ). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external properties of Wright and Carathéodory functions

full rationale

The paper defines the new class Σ(θ, λ, γ) via the operator W_{α,β} built from the Wright function series, then derives an integral representation and Hadamard-product conditions directly from that series definition and standard convolution properties in the punctured disk. Coefficient bounds are obtained by applying the known properties of Carathéodory functions together with mathematical induction on the coefficients; these are independent external tools. The radii of meromorphic starlikeness and convexity of order ρ are computed from the resulting coefficient inequalities. No equation or claim reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The central results remain self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of the Wright function, Carathéodory functions, and the principle of mathematical induction; no new entities are postulated.

free parameters (1)
  • θ, λ, γ
    Parameters that define the subclass Σ(θ, λ, γ); chosen to generalize earlier classes.
axioms (1)
  • standard math Standard analytic properties of the Wright function and Carathéodory functions hold in the punctured unit disk.
    Invoked to obtain the integral representation and coefficient bounds.

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Works this paper leans on

18 extracted references · 18 canonical work pages

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