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arxiv: 2111.10880 · v2 · submitted 2021-11-21 · 🧮 math.CV · math.FA

Bohr radius for Banach spaces on simply connected domains

Vasudevarao Allu , Himadri Halder This is my paper

Pith reviewed 2026-05-24 13:04 UTC · model grok-4.3

classification 🧮 math.CV math.FA
keywords Bohr radiusBanach spacesimply connected domainanalytic functionCesàro operatorBernardi operatorHilbert spacepower series
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The pith

A parameterized Bohr radius is defined and bounded for bounded analytic functions from simply connected domains into Banach spaces.

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

The paper defines the quantity R_{p,q,φ}(Ω,X) as the largest r such that the weighted p-norm of the constant term plus q-norm of the tail of coefficient norms stays at most φ_0(r) for every bounded holomorphic map f from Ω into a Banach space X. It determines the infimum of this quantity over the unit ball of H^∞(Ω,X) for arbitrary X and for certain Hilbert spaces. The authors also prove that the operator-valued Cesàro and Bernardi operators obey the corresponding weighted coefficient inequality inside this radius. A reader would care because the construction lifts the classical scalar Bohr phenomenon to vector-valued functions on non-disk domains while retaining explicit control via the weight sequence φ.

Core claim

We introduce R_{p,q,φ}(f,Ω,X) as the supremum of r ≥ 0 such that ||x_0||^p φ_0(r) + (∑_{n=1}^∞ ||x_n|| φ_n(r))^q ≤ φ_0(r) holds for the Taylor coefficients of f, then set R_{p,q,φ}(Ω,X) to be the infimum of this value over all f with ||f||_{H^∞(Ω,X)} ≤ 1. We obtain explicit estimates and sharp values of this radius when X is an arbitrary complex Banach space and when X is a Hilbert space; we further show that the Cesàro and Bernardi operators satisfy the Bohr inequality with this radius when acting on H^∞(Ω,X).

What carries the argument

The Bohr radius R_{p,q,φ}(Ω,X), defined as the infimum over the unit ball of the largest r for which the φ-weighted coefficient-norm inequality holds.

If this is right

  • The radius remains positive for every proper simply connected Ω containing the disk and every Banach space X.
  • When X is Hilbert, the inner-product structure yields strictly larger lower bounds than the general Banach-space case.
  • The Cesàro operator applied to any unit-ball function satisfies the defining inequality of R_{p,q,φ}(Ω,X) on the same interval.
  • The Bernardi operator likewise maps the unit ball into functions obeying the Bohr inequality up to the same radius.

Where Pith is reading between the lines

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

  • Specializing φ_n(r) = r^n recovers the classical Bohr radius on the disk when X is the scalars.
  • The same definition can be applied directly to other linear operators such as multiplication by a fixed function or composition with an automorphism.
  • Numerical verification of the radius on the unit disk for low-dimensional Banach spaces X would provide concrete test cases for the general bounds.

Load-bearing premise

Ω must be a proper simply connected domain containing the unit disk so that every bounded analytic function admits a power series expansion centered at zero, and the sequence φ must satisfy φ_0(r) ≤ 1 with locally uniform convergence of the sum.

What would settle it

An explicit function f in the unit ball of H^∞(Ω,X) for which the weighted coefficient inequality first fails at an r strictly larger than the claimed value of R_{p,q,φ}(Ω,X) would show the radius is not sharp.

Figures

Figures reproduced from arXiv: 2111.10880 by Himadri Halder, Vasudevarao Allu.

Figure 1
Figure 1. Figure 1: The graph of G1,γ(r) and G1.3,γ(r) in (0, 1) when γ = 0, 0.2, 0.4, 0.6, 0.8, 1. 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 r -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 G1.6,γ(r) 0.2588 0.2799 0.2982 0.3141 0.3284 0.3412 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.0 r 0.5 1.0 G2,γ(r) 0.2848 0.3064 0.3250 0.3412 0.3555 0.3684 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph of G1.6,γ(r) and G2,γ(r) in (0, 1) when γ = 0, 0.2, 0.4, 0.6, 0.8, 1. 2. Bohr inequality for Cesáro operator In this section, we study the Bohr inequality for the operator-valued Cesáro operator. For α ∈ C with Re α > −1, we have 1 (1 − z) α+1 = X∞ k=0 C α k z k where C α k = (α + 1). . .(α + k) k! [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graph of Cγ,0(r) and Cγ,10(r) in (0, 1) when γ = 0, 0.3, 0.5, 0.7, 0.9, 1. 0.992 0.994 0.996 0.998 r -40 -20 0 20 40 60 Cγ,20(r) 0.995 0.996 0.997 0.998 r -40 -20 0 20 40 60 Cγ,30(r) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph of Cγ,20(r) and Cγ,30(r) in (0, 1) when γ = 0, 0.3, 0.5, 0.7, 0.9, 1. From [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph of B0,1,γ(r) and B0,2,γ(r) in (0, 1) when γ = 0, 0.2, 0.4, 0.6, 0.8, 0.9, 1. 0.4 0.5 0.6 0.7 r -0.5 0.0 0.5 1.0 B1,2,γ(r) 0.40 0.45 0.50 0.55 0.60 0.65 0.00 r 0.05 0.10 0.15 0.20 B4,0,γ(r) [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The graph of B1,2,γ(r) and B4,0,γ(r) in (0, 1) when γ = 0, 0.2, 0.4, 0.6, 0.8, 0.9, 1. From [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
read the original abstract

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain $\Omega$ containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ into a complex Banach space $X$ with $\norm{f}_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q<\infty$, we denote \begin{equation*} R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \norm{x_{0}}^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \norm{x_{n}}\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\} \end{equation*} and define the Bohr radius associated with $\phi$ by $$R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \norm{f}_{H^{\infty}(\Omega,X)} \leq 1\right\}.$$ In this article, we extensively study the Bohr radius $R_{p,q,\phi}(\Omega,X)$, when $X$ is an arbitrary Banach space and $X$ is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Ces\'{a}ro operator and Bernardi operator.

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 paper defines the generalized Bohr radius R_{p,q,φ}(Ω,X) for the unit ball of H^∞(Ω,X), where Ω is a proper simply connected domain containing the unit disk and X is a complex Banach space. The radius is the infimum over all such f of the supremum r satisfying the majorant inequality ||x_0||^p φ_0(r) + (∑ ||x_n|| φ_n(r))^q ≤ φ_0(r), with φ a sequence satisfying φ_0(r) ≤ 1 and local uniform convergence of the sum. The authors study this quantity for arbitrary Banach spaces and certain Hilbert spaces, and establish the corresponding Bohr inequality for the operator-valued Cesàro and Bernardi operators.

Significance. If the derivations hold, the parameterized construction unifies several existing Bohr-radius generalizations by incorporating p, q and the majorant sequence φ while extending the setting to Banach-space-valued functions on non-disk domains. The explicit treatment of operator-valued Cesàro and Bernardi operators adds a concrete application layer. The definition is direct from the coefficient majorant and contains no free parameters beyond the given φ.

minor comments (3)
  1. The abstract (and likely the introduction) should state the principal explicit bounds or asymptotic behavior obtained for R_{p,q,φ}(Ω,X) rather than only announcing that the radius is studied.
  2. Notation: the dependence of φ on the specific operators (Cesàro, Bernardi) should be clarified in the statements of the operator-valued results.
  3. A brief comparison paragraph with prior work on vector-valued Bohr radii (e.g., on the disk) would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, including the summary of the generalized Bohr radius definition and the significance of the parameterized construction and applications to operator-valued Cesàro and Bernardi operators. We note the recommendation for minor revision and will address any editorial or minor improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines R_{p,q,φ}(f,Ω,X) explicitly as the supremum r satisfying the majorant inequality on coefficients and then takes the infimum over the unit ball to obtain R_{p,q,φ}(Ω,X). This is a direct definition, not a derivation that reduces to fitted inputs or self-citations. Extensions to Banach/Hilbert spaces and operator inequalities are obtained by applying the same coefficient-norm construction; no load-bearing step collapses to a prior result by the authors or renames an input as a prediction. The setup assumptions on Ω and φ are required for the power series to be well-defined and do not create self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the new definition of the radius using the given inequality and the standard theory of analytic functions in Banach spaces; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Properties of bounded analytic functions in Banach spaces, including power series expansion valid in the domain.
    Invoked in the definition of H^∞(Ω,X) and the coefficients x_n.
  • domain assumption Simply connected domain containing the unit disk allows the stated power series and radius definitions.
    Stated explicitly in the setup for Ω.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bohr operator on opertor valued polyanalytic functions on simply connected domains

    math.CV 2021-11 unverdicted novelty 4.0

    Derives Bohr radii for operator-valued polyanalytic functions of the form sum conjugate(z)^l f_l(z) where the leading term is subordinate to operator-valued convex or starlike biholomorphic functions.

Reference graph

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