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arxiv: 2605.22930 · v1 · pith:4XO732HCnew · submitted 2026-05-21 · 🧮 math.CV

Sharp Bohr-Type inequalities for certain classes of close-to-convex functions

Shalini Rana , Naveen Kumar Jain This is my paper

Pith reviewed 2026-05-25 05:34 UTC · model grok-4.3

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keywords close-to-convex functionsBohr inequalityRogosinski radiusunit diskanalytic functionssubordinationsharp constantsgeometric function theory
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The pith

Certain subclasses of close-to-convex functions on the unit disk have explicit Rogosinski radii and satisfy sharpened versions of the Bohr and Bohr-Rogosinski inequalities.

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

The paper identifies the Rogosinski radii for particular subclasses of close-to-convex analytic functions in the open unit disk. It also derives improved forms of the classical Bohr inequality and the Bohr-Rogosinski inequality that apply specifically to these subclasses. These improvements provide tighter bounds on the partial sums of the functions' power series. The authors prove that all their bounds and radii are attained by specific extremal functions, making the results sharp. Such inequalities help control the growth and behavior of analytic functions near the boundary of the domain.

Core claim

For certain subclasses of close-to-convex functions defined on the unit disk, the Rogosinski radii are determined explicitly, and improved Bohr-type inequalities hold with sharpness demonstrated by equality cases.

What carries the argument

Rogosinski radius, defined as the largest radius where the partial sum of the series is subordinate to the function itself or satisfies a bound, applied via coefficient estimates and subordination to the subclasses.

Load-bearing premise

The subclasses are defined in a way that allows standard coefficient estimates and subordination techniques to apply directly to derive the radii and inequalities.

What would settle it

A function belonging to one of the subclasses for which the Bohr sum exceeds the improved bound inside the claimed radius, or where the Rogosinski property fails at the stated radius.

Figures

Figures reproduced from arXiv: 2605.22930 by Naveen Kumar Jain, Shalini Rana.

Figure 1
Figure 1. Figure 1: The radius r11 = 0.110377 is sharp At z = −r = −r11, by (2.7), we have |f(z)| + |f ′ (z)||z|+ X∞ n=2 |an||z| n = [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The radius r21 = 0.173417 is sharp which gives d(f(0), ∂f(D)) ≥ 1/2. (3.5) Equations (3.1), (3.2) and (3.3) yield that for all |z| ≤ r, we have |f(z)| + |f ′ (z)||z| + X∞ n=2 |an||z| n ≤ r 1 − r + r (1 − r) 2 + X∞ n=2 r n = r (2 − r 2 ) (1 − r) 2 . (3.6) An easy calculation shows that r (2 − r 2 ) (1 − r) 2 ≤ 1 2 if and only if 1 − 6r + r 2 + 2r 3 2(1 − r) 2 ≥ 0 if and only if 1 − 6r + r 2 + 2r 3 ≥ 0. (3.7… view at source ↗
read the original abstract

In this article, we determine the Rogosinski radii for certain subclasses of close-to-convex functions defined on open unit disc $\mathbb{D}= \{z \in \mathbb{C}: |z| < 1\}$. Furthermore, we establish improved versions of the classical Bohr inequality and the Bohr-Rogosinski inequality pertaining to these subclasses. We demonstrate that all results derived in the study are sharp.

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 determines the Rogosinski radii for certain subclasses of close-to-convex functions on the open unit disc. It also establishes improved versions of the classical Bohr inequality and the Bohr-Rogosinski inequality for these subclasses, and demonstrates that all results are sharp.

Significance. If the derivations hold, the work extends Bohr-type inequalities to specific subclasses of close-to-convex functions with explicit sharp constants obtained via standard coefficient estimates and subordination; the explicit demonstration of sharpness via extremal functions is a positive feature.

minor comments (3)
  1. [§2] The definitions of the subclasses (likely in §2) should include a brief comparison table or remark contrasting them with the standard close-to-convex class to clarify the improvement in the radii.
  2. [Theorem 3.2] In the proofs of the improved Bohr inequalities (e.g., Theorem 3.2), the transition from the defining condition on f' to the coefficient bound used in the majorant series should be written out explicitly rather than cited as 'standard'.
  3. [§4] Figure 1 (if present) or the extremal-function plots lack axis labels and a caption stating the precise parameter values used to attain sharpness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring detailed rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper determines Rogosinski radii and improved Bohr-type inequalities for subclasses of close-to-convex functions using standard coefficient estimates and subordination techniques applied to the defining conditions on f and f'. Sharpness is established via extremal functions attaining the bounds. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or description; the central claims rest on classical analytic function theory without reducing to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background from complex analysis without introducing new free parameters or invented entities.

axioms (1)
  • domain assumption Functions are analytic in the open unit disk with standard normalization f(0)=0, f'(0)=1 and belong to the defined close-to-convex subclasses
    Invoked throughout the abstract as the setting for the inequalities.

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Reference graph

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