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arxiv: 2604.24776 · v1 · submitted 2026-04-18 · 🧮 math.CV

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Sharp Coefficient and Inverse Problems for Holomorphic Semigroup Generators

Molla Basir Ahamed, Sanju Mandal

Pith reviewed 2026-05-10 07:15 UTC · model grok-4.3

classification 🧮 math.CV
keywords holomorphic semigroup generatorscoefficient problemsclass A_βfunctions of bounded turninginverse coefficientsFekete-Szegö functionalhypergeometric functionsgeometric function theory
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The pith

The class A_β of holomorphic semigroup generators admits sharp bounds on its logarithmic coefficients, inverse coefficients, and a generalized Fekete-Szegö functional.

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

This paper examines extremal coefficient problems within the class A_β of holomorphic semigroup generators on the unit disk, a natural filtration of the broader class G_0 of infinitesimal generators where the class R of bounded turning functions is the minimal element. It establishes sharp estimates for the initial logarithmic coefficients γ_n, the inverse coefficients A_n, and the logarithmic inverse coefficients Γ_n for n up to 3, as well as sharp upper and lower bounds on the successive differences |A_{n+1}| - |A_n| for n=1,2. Additionally, sharp bounds are given for a generalized Fekete-Szegö functional in the class R. These findings are supported by explicit extremal functions involving Gauss hypergeometric functions and they unify several prior results in geometric function theory by linking coefficient problems for bounded turning functions to the dynamics of holomorphic semigroups.

Core claim

Within the class A_β, sharp bounds are obtained for the logarithmic coefficients γ_n (n=1,2,3), the inverse coefficients A_n (n=1,2,3), and the logarithmic inverse coefficients Γ_n (n=1,2,3). Sharp upper and lower estimates are derived for |A_{n+1}| - |A_n| when n=1,2. Sharp bounds are also established for a generalized Fekete-Szegö functional in the subclass R of functions of bounded turning, with extremality shown by constructions related to Gauss hypergeometric functions.

What carries the argument

The class A_β itself, serving as a parameterized filtration between the generators G_0 and the bounded turning class R, with extremal problems solved via Gauss hypergeometric functions.

If this is right

  • The coefficient bounds provide precise control over the growth and behavior of functions in A_β.
  • The successive coefficient differences offer new insights into how coefficients evolve in this class.
  • The Fekete-Szegö bound in R extends the theory of bounded turning functions.
  • These results connect the dynamics of semigroups to classical problems in univalent function theory.

Where Pith is reading between the lines

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

  • The approach using hypergeometric functions may generalize to other subclasses of generators or higher-order problems.
  • Connections between semigroup generators and bounded turning could inspire new differential equation models in geometric function theory.
  • Numerical verification of the bounds for specific β values beyond the given constructions would test their sharpness.

Load-bearing premise

The constructed Gauss hypergeometric functions belong to the class A_β and attain the extremal values for the coefficient functionals considered.

What would settle it

Finding a function in A_β for which the absolute value of the first logarithmic coefficient γ_1 exceeds the sharp bound provided, or proving that the hypergeometric extremal is not in A_β, would falsify the claims.

read the original abstract

In this paper, we study extremal problems for coefficient functionals associated with a distinguished subclass of holomorphic semigroup generators, denoted by $\mathcal{A}_{\beta}$ ($0 \le \beta \le 1$), defined on the unit disk $\mathbb{D}$. This class forms a natural filtration of the class $\mathcal{G}_0$ of infinitesimal generators, with the class $\mathcal{R}$ of functions of bounded turning arising as its minimal element. We obtain sharp bounds for the initial logarithmic coefficients $\gamma_n$, the inverse coefficients $A_n$, and the logarithmic inverse coefficients $\Gamma_n$ for $n = 1,2,3$ within the class $\mathcal{A}_{\beta}$. In addition, we address the successive coefficient problem by deriving sharp upper and lower estimates for the differences $|A_{n+1}| - |A_n|$ for $n = 1,2$. Furthermore, we establish sharp bounds for a generalized Fekete--Szeg\"o functional in the class $\mathcal{R}$. The extremality of the obtained results is demonstrated by explicit constructions, including functions related to Gauss hypergeometric functions. Our results unify and extend several earlier contributions in geometric function theory and reveal a structural connection between coefficient problems for functions of bounded turning and the dynamics of holomorphic semigroup generators.

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

3 major / 2 minor

Summary. The paper studies extremal coefficient problems in the class A_β (0 ≤ β ≤ 1) of holomorphic semigroup generators on the unit disk, a filtration of G_0 with R (bounded turning functions) as the β=0 case. It derives sharp bounds on the logarithmic coefficients γ_n, inverse coefficients A_n and logarithmic inverse coefficients Γ_n for n=1,2,3; sharp upper and lower estimates for the successive differences |A_{n+1}|−|A_n| (n=1,2); and sharp bounds on a generalized Fekete–Szegő functional in R. Sharpness is asserted via explicit extremal functions constructed from Gauss hypergeometric series.

Significance. If the hypergeometric constructions are shown to lie in A_β for all β and to attain equality in each functional, the results would unify and extend several earlier coefficient estimates in geometric function theory while clarifying the link between bounded-turning functions and semigroup-generator dynamics. The explicit constructions constitute a strength when accompanied by complete membership and equality verifications.

major comments (3)
  1. [Sections 3–4 (main results on coefficients and inverse coefficients)] The central sharpness claims rest on the assertion that the hypergeometric constructions belong to A_β and saturate the stated bounds. The manuscript must supply a complete, self-contained verification that each candidate satisfies the defining analytic condition of A_β (the appropriate real-part or subordination condition on the generator) for the full interval 0 ≤ β ≤ 1; without this, the extremality statements for γ_n, A_n and Γ_n (n=1,2,3) remain unestablished.
  2. [Section 4 (successive coefficient problem)] For the successive-coefficient differences |A_{n+1}|−|A_n| (n=1,2), direct substitution of the Taylor coefficients of the proposed extremal function into the functional must be carried out explicitly to confirm that the claimed upper and lower values are attained; the current presentation leaves this equality case unverified.
  3. [Section 5 (Fekete–Szegő functional)] The generalized Fekete–Szegő bound in the class R (β=0 case) requires an explicit check that the hypergeometric construction reduces to a function of bounded turning and that the functional attains the stated value; any gap here undermines the unification claim with earlier results on R.
minor comments (2)
  1. [Abstract and Section 1] The abstract refers to “functions related to Gauss hypergeometric functions” without giving the precise series or parameter choices; these should be stated explicitly already in the introduction or preliminaries.
  2. [Section 2 (preliminaries)] Notation for the inverse coefficients A_n and logarithmic inverse coefficients Γ_n should be introduced with a brief reminder of the standard inversion formulas to aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that additional explicit verifications are needed to fully establish the sharpness claims and will revise the paper accordingly. Below we address each major comment point by point.

read point-by-point responses

  1. Referee: [Sections 3–4 (main results on coefficients and inverse coefficients)] The central sharpness claims rest on the assertion that the hypergeometric constructions belong to A_β and saturate the stated bounds. The manuscript must supply a complete, self-contained verification that each candidate satisfies the defining analytic condition of A_β (the appropriate real-part or subordination condition on the generator) for the full interval 0 ≤ β ≤ 1; without this, the extremality statements for γ_n, A_n and Γ_n (n=1,2,3) remain unestablished.

    Authors: We agree that a complete, self-contained verification is required. In the revised manuscript, we will add a detailed proof establishing that each hypergeometric candidate satisfies the defining real-part condition of A_β for every β ∈ [0,1], using the series representation and known subordination properties of the Gauss hypergeometric function. We will also explicitly verify that equality holds in the stated bounds for γ_n, A_n, and Γ_n (n=1,2,3). revision: yes

  2. Referee: [Section 4 (successive coefficient problem)] For the successive-coefficient differences |A_{n+1}|−|A_n| (n=1,2), direct substitution of the Taylor coefficients of the proposed extremal function into the functional must be carried out explicitly to confirm that the claimed upper and lower values are attained; the current presentation leaves this equality case unverified.

    Authors: We will incorporate explicit calculations in the revision. By substituting the Taylor coefficients of the extremal hypergeometric function, we will directly compute |A_{n+1}| − |A_n| for n=1,2 and confirm that the proposed upper and lower bounds are attained, thereby verifying the equality cases. revision: yes

  3. Referee: [Section 5 (Fekete–Szegő functional)] The generalized Fekete–Szegő bound in the class R (β=0 case) requires an explicit check that the hypergeometric construction reduces to a function of bounded turning and that the functional attains the stated value; any gap here undermines the unification claim with earlier results on R.

    Authors: We accept that an explicit verification is necessary for the unification claim. In the revised version, we will demonstrate that the hypergeometric construction for β=0 belongs to the class R of bounded-turning functions and compute the value of the generalized Fekete–Szegő functional to show that the bound is attained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from class definitions via standard analytic inequalities, with sharpness verified by independent explicit examples.

full rationale

The paper defines the class A_β via an analytic condition on holomorphic semigroup generators (a natural filtration of G_0 with R as the β=0 case), then applies standard techniques such as subordination or coefficient inequalities to obtain sharp bounds on γ_n, A_n, Γ_n (n=1,2,3), successive differences |A_{n+1}|−|A_n| (n=1,2), and a generalized Fekete–Szegő functional. Extremality is shown by separate explicit constructions (Gauss hypergeometric functions) that are verified to lie in A_β and attain equality; these verifications are independent checks, not inputs to the bound derivations. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain. The results are self-contained against the class definition and external analytic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms of complex analysis and the definition of the class A_β without introducing new entities or fitted parameters; all bounds are derived from these.

axioms (2)
  • standard math Holomorphic functions on the unit disk satisfy Cauchy's integral formula, maximum modulus principle, and standard growth estimates.
    Invoked for extracting coefficients and establishing bounds.
  • domain assumption The class A_β forms a natural filtration of G_0 with R as its minimal element for 0 ≤ β ≤ 1.
    This is the foundational definition of the class under study.

pith-pipeline@v0.9.0 · 5526 in / 1539 out tokens · 89667 ms · 2026-05-10T07:15:14.167471+00:00 · methodology

discussion (0)

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Reference graph

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