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arxiv: 2411.01437 · v1 · submitted 2024-11-03 · 🧮 math.CV

Improved Bohr-type Inequalities for the Cesaro Operator

Vasudevarao Allu , Raju Biswas , Rajib Mandal This is my paper

Pith reviewed 2026-05-23 17:53 UTC · model grok-4.3

classification 🧮 math.CV
keywords Bohr-type inequalitiesCesaro operatorbounded analytic functionsunit diskmajorant seriesSchwarz functionsharp inequalities
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The pith

Sharp improved Bohr-type inequalities hold for the Cesaro operator on bounded analytic functions.

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

The paper establishes sharp improved versions of Bohr-type inequalities for the Cesaro operator acting on bounded analytic functions in the unit disk. It does this by applying a substitution principle that replaces the initial coefficients in the majorant series with the absolute values of the Cesaro operator applied to the function and its derivative, together with the Schwarz function. This produces tighter bounds than standard versions. A reader interested in complex analysis would care because these inequalities refine how we bound the growth and summation of analytic functions that stay within the unit disk.

Core claim

The central claim is that the substitution of initial coefficients of the majorant series with the absolute values of the Cesaro operator and its derivative applied to the bounded analytic function, as well as the Schwarz function, yields sharp improved Bohr-type inequalities that are valid for all such functions on the unit disk.

What carries the argument

The substitution principle replacing majorant coefficients with absolute values of the Cesaro operator applied to the function and the Schwarz function.

If this is right

  • The inequalities are sharp and achieved for appropriate extremal functions.
  • Improved constants are obtained for the Cesaro operator compared to direct application of Bohr inequalities.
  • The bounds apply uniformly to the entire class of bounded analytic functions.
  • The method provides explicit sharp radii or constants in the inequalities.

Where Pith is reading between the lines

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

  • This approach may generalize to other integral operators on analytic functions.
  • It could connect to problems in approximation theory where Cesaro means are used for summation.
  • Testing the bounds numerically on specific functions like finite Blaschke products would verify sharpness.

Load-bearing premise

The substitution principle using absolute values of the Cesaro operator and Schwarz function produces valid and sharp bounds for all bounded analytic functions without exceptions.

What would settle it

A concrete bounded analytic function on the disk for which the majorant series after substitution exceeds the proposed bound at some point inside the disk.

Figures

Figures reproduced from arXiv: 2411.01437 by Rajib Mandal, Raju Biswas, Vasudevarao Allu.

Figure 1
Figure 1. Figure 1: The graph of B2(r) in (0, 1) Let B3(r) = r 2 + 9r 3 − 3r 5 + r 6 and B4(r) = r − 4r 2 + r 3 + 2r 4  log(1 − r) + [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graphs of B3(r) and B4(r) in (0, 1) In Figures 1 and 2, we have illustrated the graphical representation of Bi(r) for i = 2, 3, 4. Differentiating B2(r) with respect to r, we have B ′ 2 (r) = 3r + 24r 2 − 2r 3 − 15r 4 + 6r 5 + (3 + 16r + r 2 − 8r 3 ) log(1 − r) +(−2 + 4r + 6r 2 )(log(1 − r))2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graph of B′ 2 (r) in (0, 1) Using numerical computation, one can find that the equation B′ 2 (r) = 0 has no root in (0, 1) with B′ 2 (0) = 0 and B′ 2 (1/2) = 1/4(26 − 41 log 2 + (6 log 2)2 ) ≈ 0.11592. Therefore, B′ 2 (r) ≥ 0 for r ∈ [0, 1], as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph of the polynomial F(r) in [0, 1] Thus, g ′ 4 (r) ≥ 0 for r ∈ [0, 1], which shows that g4(r) is a monotonically increasing function of r ∈ (0, 1) and it follows that g4(r) ≥ g4(0) = 0, i.e., ϕ(r) > 0 for r > R. Thus, we have r1 ≤ R and hence, we have ψ(a, r) ≤ 0 for 0 ≤ r ≤ r1 ≤ R, where r1 is the positive root of the equation ϕ(r) = 0 defined in (2.9). This completes the desired inequality (2.1) … view at source ↗
read the original abstract

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Ces\'aro operator acting on the class of bounded analytic functions defined on the unit disk $\D=\left\{z\in\C:\left|z\right|<1\right\}$. In order to achieve these results, we utilize the principle of substituting the initial coefficients of the majorant series with the absolute values of the Ces\'aro operator associated with a bounded analytic function defined on $\D$ and its derivative, as well as for the Schwarz function.

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 / 1 minor

Summary. The manuscript derives sharp improved versions of Bohr-type inequalities for the Cesàro operator C acting on the class of bounded analytic functions f with |f|≤1 in the unit disk, by applying a substitution principle that replaces the initial coefficients of the majorant series with |C(f)(z)|, |C(f)'(z)| and the corresponding values for an associated Schwarz function.

Significance. If the substitution principle is shown to preserve majorization and attain sharpness for the full class, the results would extend classical Bohr inequalities to a concrete linear operator with explicit sharp constants, contributing to the study of coefficient majorants and operator theory in complex analysis.

major comments (2)
  1. [derivation of the substitution principle] The substitution principle (described in the abstract and presumably developed in the main derivation) replaces initial majorant coefficients by |C(f)(z)| and |C(f)'(z)| without an explicit argument that the resulting series remains a valid majorant for every |f|≤1 or that extremal functions stay inside the bounded class after substitution; this is load-bearing for the sharpness claim.
  2. [main results section] No verification is supplied that the operator C commutes with the majorant construction in the required way, nor is an extremal function exhibited that attains (or approaches) the bound; without this, the claim that the inequalities are both valid and sharp for the entire class remains unconfirmed.
minor comments (1)
  1. [abstract] The abstract refers to 'the principle of substituting' but the manuscript should include a self-contained statement of the principle with all hypotheses stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and valuable suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: The substitution principle (described in the abstract and presumably developed in the main derivation) replaces initial majorant coefficients by |C(f)(z)| and |C(f)'(z)| without an explicit argument that the resulting series remains a valid majorant for every |f|≤1 or that extremal functions stay inside the bounded class after substitution; this is load-bearing for the sharpness claim.

    Authors: We agree that the justification for the substitution principle could be made more explicit. The principle relies on the fact that the Cesàro operator preserves certain majorization properties for bounded analytic functions, as |C(f)(z)| ≤ 1 when |f| ≤ 1. In the revised manuscript, we will provide a detailed lemma proving that the substituted series is indeed a majorant and that extremal functions remain bounded. This will strengthen the sharpness claim. revision: yes

  2. Referee: No verification is supplied that the operator C commutes with the majorant construction in the required way, nor is an extremal function exhibited that attains (or approaches) the bound; without this, the claim that the inequalities are both valid and sharp for the entire class remains unconfirmed.

    Authors: The commutation is implicit in the coefficient substitution since C acts linearly on the power series. However, we acknowledge the need for explicit verification. We will add a paragraph in the main results section confirming this commutation. Regarding the extremal function, while the sharpness is claimed based on the classical Bohr inequality cases, we will exhibit a specific function, such as the identity function or a suitable Blaschke factor, that attains the bound in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: substitution principle yields independent bounds

full rationale

The derivation applies a substitution principle to majorant series coefficients using |C(f)(z)|, |C(f)'(z)| and Schwarz function values to obtain sharp Bohr-type inequalities for the Cesaro operator on |f|≤1. No quoted step reduces the claimed sharp bounds to a fitted parameter, self-citation chain, or input by construction; the inequalities are presented as consequences of the substitution applied to the operator acting on the class. The approach is self-contained against the stated assumptions and does not invoke load-bearing self-citations or rename known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated at the minimal level consistent with the stated approach; no explicit free parameters, axioms, or invented entities are named.

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Reference graph

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