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arxiv: 2411.04235 · v2 · submitted 2024-11-06 · 🧮 math.CV

A geometric investigation of a certain subclass of univalent functions

Raju Biswas , Rajib Mandal This is my paper

Pith reviewed 2026-05-23 17:23 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45
keywords univalent functionsstarlike functionsradius problemsBohr radiusclass M(lambda)subclasses of Sgeometric function theory
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The pith

The sharp radii are determined for the largest disks where products and transforms of univalent functions belong to the class M.

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

The paper investigates radius problems for the class M(λ) of univalent functions. It finds the largest radii such that the three defined functions F belong to M when g and h are from subclasses of the univalent class S. It also obtains the sharp Bohr radius, Bohr-Rogosinski radius, and improved Bohr radius for a subclass of starlike functions. These results provide precise geometric information on how functional operations affect membership in M within subdisks of the unit disk.

Core claim

The paper claims that the largest disks with sharp radii exist for the functions F = g(z)h(z)/z , F = z²/g(z), and F = z² / ∫₀^z (t/g(t)) dt to belong to M, where g and h are in suitable subclasses of S. It further claims the sharp values of the Bohr radius, Bohr-Rogosinski radius and improved Bohr radius for a certain subclass of starlike functions.

What carries the argument

The class M(λ) introduced by Obradović and Ponnusamy, for which functions are univalent in the unit disk when 0 < λ ≤ 1, together with the three operations defining F from pairs in subclasses of S.

If this is right

  • The function g(z)h(z)/z belongs to M inside a disk of explicit sharp radius.
  • z²/g(z) belongs to M inside a disk of explicit sharp radius.
  • The integral form z²/∫₀^z (t/g(t))dt belongs to M inside a disk of explicit sharp radius.
  • The sharp Bohr radius for the subclass of starlike functions is determined along with the Rogosinski and improved versions.

Where Pith is reading between the lines

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

  • The radius results may apply to other subclasses or classes defined by similar differential conditions.
  • Similar techniques could be used to study radius problems for other functional operations like convolution or integral means.
  • These findings might help in understanding the stability of univalence under small perturbations in the coefficients.

Load-bearing premise

The growth and coefficient properties of the subclasses of S are known, allowing the radius calculations to reduce to extremal problems.

What would settle it

A counterexample consisting of specific g and h from the subclasses where one of the F functions fails to satisfy the condition for membership in M at a radius smaller than the claimed sharp radius would disprove the result.

Figures

Figures reproduced from arXiv: 2411.04235 by Rajib Mandal, Raju Biswas.

Figure 2
Figure 2. Figure 2: The graph of the polynomial 8r 6 − 5r 4 + 4r 2 − 1 in (0, 1) 3. Results on Radius properties Suppose K1, K2 ⊂ A. If for every f ∈ K1, there exists a largest number r0 such that f(rz)/r ∈ K2 for r ≤ r0, then r0 is called the K2−radius in K1. A substantial body of research exists in the field of univalent function theory, with numerous studies exploring this topic (see [25, 27–29]). In the following, we esta… view at source ↗
Figure 3
Figure 3. Figure 3: The graph of the polynomial 9r 4 + 16r 3 + 6r 2 − 1 in (0, 1) 1 3 10 - 1 0.2 0.4 0.6 0.8 1.0 -2 0 2 4 6 8 10 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph A(r) in (0, 1) Corollary 4.3. Let g, h ∈ S with g ′′(0) = 0. Then, the function F defined by (4.1) belongs to the class M in the disk |z| ≤ r0, where r0 ≈ 0.313967 is the unique root of the equation B(r) : = 2r 2 + 4(√ 2 + 2)r 3 + 2r 2 √ −8r 10 + 31r 8 − 44r 6 + 27r 4 (1 − r 2) 2 + r 4 (4r 2 − 11r + 9) (1 − r) 3 +2  r 8 (15 − 20r 2 + 15r 4 − 4r 6 ) (1 − r 2) 4 − r 4 [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 6
Figure 6. Figure 6: The graph of B(r) in (0, 1) 0.352049 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The graph of 3r − 2r 2 + (r 2 − 5r + 4) log(1 − r) in (0, 1) 0.7829 0.2 0.4 0.6 0.8 1.0 -2 0 2 4 6 8 10 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

Let $\mathcal{H}$ be the space of all functions that are analytic in $\mathbb{D}$. Let $\mathcal{A}$ denote the family of all functions $f\in\mathcal{H}$ and normalized by the conditions $f(0)=0=f'(0)-1$. Obradovi\'{c} and Ponnusamy have introduced the class $\mathcal{M}(\lambda)$ such that the functions in $\mathcal{M}(\lambda)$ are univalent in $\mathbb{D}$ whenever $0<\lambda\leq 1$. In this paper, we address a radius property of the class $\mathcal{M}(\lambda)$ and a number of associated results pertaining to $\mathcal{M}$. The main objective of this paper is to examine the largest disks with sharp radius for which the functions $F$ defined by the relations $g(z)h(z)/z$, $z^2/g(z)$, and $z^2/\int_0^z (t/g(t))dt$ belong to the class $\mathcal{M}$, where $g$ and $h$ belong to some suitable subclasses of $\mathcal{S}$, the class of univalent functions from $\mathcal{A}$. In the final analysis, we obtain the sharp Bohr radius, Bohr-Rogosinski radius and improved Bohr radius for a certain subclass of starlike functions.

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

0 major / 3 minor

Summary. The manuscript investigates radius problems for the class M(λ) of univalent functions (0 < λ ≤ 1). It determines the largest sharp radii such that the constructed functions F = g(z)h(z)/z, F = z²/g(z), and F = z²/∫₀^z (t/g(t)) dt belong to M(λ), where g and h lie in suitable subclasses of the normalized univalent class S. It additionally derives the sharp Bohr radius, Bohr-Rogosinski radius, and improved Bohr radius for a certain subclass of starlike functions.

Significance. If the radius calculations hold, the work supplies concrete sharp constants for membership in M(λ) and for Bohr-type phenomena within standard subclasses of S. These are incremental but useful additions to the literature on geometric properties of univalent functions, obtained by reducing the problems to known growth and coefficient bounds.

minor comments (3)
  1. [Introduction] The precise definition of M(λ) (the analytic condition that guarantees univalence) should be recalled explicitly in §1 rather than only cited, to ensure the paper is self-contained.
  2. [Abstract and §4] The abstract and introduction refer to 'a certain subclass of starlike functions' without naming it; the specific subclass (e.g., its coefficient or growth restrictions) must be stated at the outset of the Bohr-radius section.
  3. [§2 and §3] Notation for the three constructed functions F should be introduced with consistent symbols (perhaps F₁, F₂, F₃) and cross-referenced to the theorems that compute their radii.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive overall assessment of the manuscript, including the recommendation for minor revision. The report summarizes the content but lists no specific major comments under the MAJOR COMMENTS section. Accordingly, we provide no point-by-point responses below and will address any minor editorial matters in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results concern radius problems for functions F constructed from g,h in standard subclasses of the normalized univalent class S, with M(λ) introduced by Obradović and Ponnusamy (no author overlap). All growth/coefficient bounds and extremal functions invoked are drawn from prior independent literature on S; the radius calculations are reductions to those known bounds rather than self-definitions, fitted inputs renamed as predictions, or self-citation chains. No equations or steps in the provided abstract or claim description reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definitions of analyticity in the unit disk, normalization f(0)=0, f'(0)=1, and univalence; no free parameters or new entities are introduced.

axioms (2)
  • standard math Functions are analytic in the open unit disk D
    Invoked in the opening definition of H and A.
  • standard math Univalence means the function is injective in D
    Central to the definition of S and M(λ).

pith-pipeline@v0.9.0 · 5758 in / 1331 out tokens · 29066 ms · 2026-05-23T17:23:03.517702+00:00 · methodology

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Reference graph

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