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arxiv: 2504.19133 · v1 · submitted 2025-04-27 · 🧮 math.CV

The Bohr's Phenomenon involving multiple Schwarz functions

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

Pith reviewed 2026-05-22 19:04 UTC · model grok-4.3

classification 🧮 math.CV
keywords Bohr inequalitySchwarz functionsanalytic functionsunit diskRogosinski inequalitysubordinationcoefficient majorization
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The pith

Bounded analytic functions in the unit disk satisfy several sharp Bohr inequalities involving multiple Schwarz functions.

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

This paper proves sharp versions of Bohr inequalities that bound the sum of absolute Taylor coefficients for analytic functions f with |f| ≤ 1 inside the unit disk, where the bounds are expressed using several Schwarz functions. A sympathetic reader would care because the classical Bohr phenomenon gives an explicit radius inside which the coefficient sum is controlled by the function bound alone, and the multiple-function version allows more flexible majorization while keeping the inequalities sharp. The work also derives an improved Rogosinski inequality for the partial sums of the power series. If these results hold, they supply tighter tools for estimating growth and remainders in classes of bounded analytic functions.

Core claim

For f analytic in the unit disk with |f(z)| ≤ 1 and for Schwarz functions ω_j that fix the origin and map the disk to itself, the paper establishes sharp inequalities of the form sum |a_n| r^n ≤ |f(0)| plus a term controlled by the ω_j, valid up to explicit positive radii, with equality attained for suitable extremal choices; it likewise sharpens the classical Rogosinski inequality on the remainder after partial sums.

What carries the argument

Multiple Schwarz functions (analytic self-maps of the disk fixing the origin) used to majorize coefficients of the target function via subordination.

If this is right

  • The classical Bohr radius 1/3 is recovered or adjusted as a special case when one or more Schwarz functions are introduced.
  • The inequalities remain sharp, so equality cases exist and are attained by explicit extremal functions.
  • The sharpened Rogosinski inequality supplies a strictly better estimate on the size of partial sums than the classical version.

Where Pith is reading between the lines

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

  • The technique could extend to other majorants besides Schwarz functions, such as those arising from fixed-point conditions in different domains.
  • Numerical checks with finite Blaschke products as test cases would give concrete values for the radii involved.
  • The results may connect to coefficient problems in univalent function theory where multiple subordinations appear naturally.

Load-bearing premise

The Schwarz functions must be analytic, fix the origin, and map the unit disk into itself while the main function stays bounded by one.

What would settle it

A concrete bounded analytic function together with Schwarz functions such that the left-hand side of one of the claimed Bohr sums strictly exceeds the right-hand side for some radius inside the stated interval would disprove the inequality.

Figures

Figures reproduced from arXiv: 2504.19133 by Rajib Mandal, Raju Biswas, Vasudevarao Allu.

Figure 1
Figure 1. Figure 1: Location of Rm,p,k in (0, 1) 0.414 0.717 0.745 0.949 0.961 0.2 0.4 0.6 0.8 1.0 -2 -1 0 1 2 r1,1 r3,3 r2,3 r5,30 r30,20 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Location of rm,p in (0, 1) In the following result, we establish the sharp refined version of the Bohr inequality in the context of several Schwarz functions. Theorem 3.2. Suppose that f(z) = P∞ n=0 anz n is analytic in D with |f(z)| ≤ 1 in D and ωk ∈ Bk for k ≥ 1. Then |f(ωm(z))| 2 + |ωp(z)||f ′ (ωm(z))| + X∞ n=2 |an||ωk(z)| n +  1 1 + |a0| + |ωk(z)| 1 − |ωk(z)| X∞ n=1 |an| 2 |ωk(z)| 2n ≤ 1 for r ≤ R2,m… view at source ↗
Figure 3
Figure 3. Figure 3: Location of R2,m,p,k in (0, 1) 0.618 0.819 0.852 0.962 0.981 0.2 0.4 0.6 0.8 1.0 -2 -1 0 1 2 r2,1,1 r2,3,3 r2,2,3 r2,5,30 r2,30,20 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Location of r2,m,p in (0, 1) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Location of R3,m,k in (0, 1) In the following result, we establish the sharp improved version of the classical Ro￾gosinski inequality containing two Schwarz functions. Theorem 3.5. Suppose that f(z) = P∞ n=0 anz n is analytic in D with |f(z)| ≤ 1 in D and ωk ∈ Bk for k ≥ 1. Then for N ≥ 1, we have |f(ωm(z))| + X N n=1 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Location of R5,m,k in (0, 1) [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
read the original abstract

The primary objective of this paper is to establish several sharp versions of Bohr inequalities for bounded analytic functions in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$ involving multiple Schwarz functions. Moreover, we obtain an improved version of the classical Rogosinski inequality for analytic functions in $\mathbb{D}$.

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 / 3 minor

Summary. The manuscript establishes several sharp versions of Bohr inequalities for analytic functions f bounded by 1 in the unit disk, expressed in terms of multiple Schwarz functions (analytic maps fixing the origin and sending the disk to itself). It also derives an improved form of the classical Rogosinski inequality for analytic functions in the disk, with explicit constants and extremal examples.

Significance. If the claimed sharpness holds, the work meaningfully extends the Bohr phenomenon to settings with several subordinate Schwarz functions, supplying a natural generalization that may aid coefficient estimates and subordination arguments in geometric function theory. The explicit construction of extremal functions that simultaneously attain all bounds and satisfy the Schwarz-lemma hypotheses is a clear strength, as is the use of standard tools (Schwarz lemma, coefficient majorization, subordination) without introducing ad-hoc parameters. The stress-test concern about missing derivation details does not land on the full manuscript, which supplies the required proofs and verifications.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 (two-Schwarz case): the proof applies the Schwarz lemma separately to each function and then invokes majorization on the coefficients of f; while the extremal construction is given, the simultaneous attainment of the bound for all three functions at the same radius should be verified explicitly to confirm the constant is not merely an upper estimate.
  2. [§4] §4, improved Rogosinski inequality: the refinement over the classical constant is stated, but the derivation relies on the same majorization technique used earlier; a short remark confirming that the improvement is strict for non-constant functions would strengthen the central claim.
minor comments (3)
  1. [Introduction] Introduction: a one-sentence recall of the classical Bohr inequality (with its radius 1/3) would help readers situate the new multi-Schwarz versions.
  2. [Notation] Notation section: the precise hypotheses on the Schwarz functions (analyticity, φ(0)=0, |φ(z)|<1) are used repeatedly but are not restated in a single displayed list before the first theorem.
  3. [References] References: several recent papers on generalized Bohr inequalities with subordinate functions are omitted; adding two or three would improve context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (two-Schwarz case): the proof applies the Schwarz lemma separately to each function and then invokes majorization on the coefficients of f; while the extremal construction is given, the simultaneous attainment of the bound for all three functions at the same radius should be verified explicitly to confirm the constant is not merely an upper estimate.

    Authors: We agree that an explicit check of simultaneous attainment is useful for clarity. In the revised manuscript we have inserted a short paragraph immediately after the extremal construction in the proof of Theorem 3.1. There we substitute the explicit forms f(z)=z, φ₁(z)=e^{iθ}z, φ₂(z)=e^{iψ}z into the three expressions and verify numerically that equality holds simultaneously at the radius r=1/√2 where the majorization becomes equality. This confirms that the constant is attained and is therefore sharp. revision: yes

  2. Referee: [§4] §4, improved Rogosinski inequality: the refinement over the classical constant is stated, but the derivation relies on the same majorization technique used earlier; a short remark confirming that the improvement is strict for non-constant functions would strengthen the central claim.

    Authors: We appreciate the suggestion. We have added a one-sentence remark immediately after the statement of the improved Rogosinski inequality: 'Equality holds if and only if f is a suitable rotation of the identity; hence the improvement is strict for all non-constant functions that do not attain this extremal form.' This remark follows directly from the equality case in the majorization step already present in the proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives sharp Bohr inequalities and an improved Rogosinski inequality by applying the standard Schwarz lemma hypotheses (analyticity, origin-fixing, disk-to-disk mapping) to multiple Schwarz functions, followed by explicit coefficient majorization or subordination for the bounded target function |f| ≤ 1. These steps are carried out directly in the proofs using classical complex-analysis tools, with extremal functions constructed to satisfy all hypotheses simultaneously. No load-bearing self-citations, self-definitional reductions, or fitted inputs renamed as predictions appear; the central claims remain independent of the paper's own equations or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of Schwarz functions and the assumption that the target functions are analytic and bounded in the unit disk; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Target functions are analytic and bounded by 1 in the unit disk
    Required to state the Bohr-type inequalities
  • standard math Schwarz functions are analytic, fix the origin, and map the disk into itself
    Standard definition used to invoke subordination or majorization

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Reference graph

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