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arxiv: 2201.03849 · v3 · submitted 2022-01-11 · 🧮 math.FA

Bohr and Rogosinski inequalities for operator valued holomorphic functions

Vasudevarao Allu , Himadri Halder , Subhadip Pal This is my paper

Pith reviewed 2026-05-24 12:10 UTC · model grok-4.3

classification 🧮 math.FA
keywords Bohr radiusp-uniform C-convexityBanach spacesholomorphic functionsoperator-valued functionsRogosinski inequalityLebesgue spaces
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The pith

A Banach space has positive p-Bohr radius of order N for p ≥ 2 precisely when it is p-uniformly C-convex of order N.

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

The paper introduces the p-Bohr radius of order N for vector-valued holomorphic functions on the unit disk with values in a complex Banach space X. It pairs this with a new geometric notion called p-uniform C-convexity of order N. The central result proves that for p at least 2, the radius is positive if and only if the space satisfies the convexity property. This provides a characterization linking an analytic inequality condition to Banach space geometry. The work also derives an operator-valued version of refined Bohr and Rogosinski inequalities for functions taking values in bounded operators on a Hilbert space and examines the radius for Lebesgue spaces under various p and q relations.

Core claim

For p in [2, ∞) and each natural number N, a complex Banach space X is p-uniformly C-convex of order N if and only if the p-Bohr radius of order N, defined as the supremum of r ≥ 0 such that the sum from k=0 to N of the p-th power of the coefficient norms times r to the pk is at most the p-th power of the H^∞ norm of the function, is positive. This equivalence is established directly for the given definitions.

What carries the argument

The p-Bohr radius of order N, which measures the largest scaling r for which the partial p-powered coefficient sum stays bounded by the function's sup norm to the p, serving as the analytic counterpart to the geometric p-uniform C-convexity of order N.

If this is right

  • For p ≥ 2 the positivity of the radius fully characterizes the convexity property in any complex Banach space.
  • The radius is positive for L^q(μ) spaces when 1 ≤ p < q < ∞ or 1 ≤ q ≤ p < 2.
  • A refined Bohr and Rogosinski inequality holds for bounded holomorphic functions from the disk to the space of bounded operators on a Hilbert space.
  • The order N version allows control over finite sums in the power series expansion.

Where Pith is reading between the lines

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

  • If the equivalence holds, then known examples of uniformly convex spaces can be used to guarantee positive Bohr radii in vector-valued settings.
  • The operator-valued extension suggests similar radius results may apply to other operator algebras or non-commutative settings.
  • Testing the radius explicitly for specific functions could verify the convexity property in concrete Banach spaces.

Load-bearing premise

That the newly defined p-uniform C-convexity of order N is exactly the property that makes the p-Bohr radius positive, matching without extra conditions on the space or the functions.

What would settle it

A specific complex Banach space X and numbers p ≥ 2, N where X satisfies p-uniform C-convexity of order N but the computed p-Bohr radius of order N is zero, or the converse.

read the original abstract

For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$ denotes the unit disk. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. In this paper, for $p\in [2,\infty)$ and each $N \in \mathbb{N}$, we prove that a complex Banach space $X$ is $p$-uniformly $\mathbb{C}$-convex of order $N$ if, and only if, the $p$-Bohr radius of order $N$ $\widetilde{R}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$ for the Lebesgue spaces $L^q (\mu)$ for $1\leq p<q<\infty$ or $1\leq q \leq p <2$. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk $\mathbb{D}$ into $\mathcal{B(\mathcal{H})}$, where $\mathcal{B(\mathcal{H})}$ denotes the space of all bounded linear operator on a complex Hilbert space $\mathcal{H}$.

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 / 2 minor

Summary. The paper introduces the p-Bohr radius of order N for any complex Banach space X, defined as the supremum of r ≥ 0 such that for f(z) = ∑ x_k z^k in H^∞(D, X), the inequality ∑_{k=0}^N ||x_k||^p r^{p k} ≤ ||f||_{H^∞(D,X)}^p holds. It introduces the geometric notion of p-uniformly ℂ-convexity of order N and proves that for p ∈ [2, ∞) and each N ∈ ℕ, X is p-uniformly ℂ-convex of order N if and only if the p-Bohr radius of order N is positive. The paper also studies the radius for Lebesgue spaces L^q(μ) when 1 ≤ p < q < ∞ or 1 ≤ q ≤ p < 2, and proves an operator-valued analogue of a refined Bohr-Rogosinski inequality for bounded holomorphic functions from D to B(H).

Significance. If the central equivalence holds without circularity or hidden restrictions, it would provide a useful analytic characterization of a new geometric convexity property in Banach spaces, linking coefficient control in holomorphic functions to convexity moduli. The results on L^q spaces and the extension to B(H)-valued functions broaden the scope of Bohr-type inequalities beyond scalar or vector-valued cases, with potential relevance to operator theory and geometric functional analysis.

major comments (2)
  1. [Abstract] Abstract: The definition of the geometric property 'p-uniformly ℂ-convexity of order N' is announced but not stated, so it is impossible to verify whether the stated if-and-only-if equivalence is a non-trivial theorem or follows tautologically from the way the convexity notion is defined in terms of the radius (or vice versa).
  2. [Abstract] Abstract (theorem statement): The central claim is an equivalence between positivity of the newly defined radius and the newly introduced convexity property, but with the full proof steps, coefficient estimates, and any auxiliary lemmas unavailable in the provided text, the soundness of the equivalence cannot be checked.
minor comments (2)
  1. [Abstract] The radius is defined for all p ∈ [1, ∞) but the equivalence theorem is stated only for p ≥ 2; the manuscript should clarify whether the restriction is essential or merely technical.
  2. [Abstract] Notation: The use of the tilde on R̃_{p,N}(X) and the script C in ℂ-convexity should be consistently explained when first introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and comments on the abstract. We address each major comment below. The full manuscript contains the definition of p-uniformly ℂ-convexity of order N (given geometrically in Section 2, independent of the radius) and the complete proof of the equivalence (in Sections 3–4, with all coefficient estimates and lemmas).

read point-by-point responses

  1. Referee: [Abstract] Abstract: The definition of the geometric property 'p-uniformly ℂ-convexity of order N' is announced but not stated, so it is impossible to verify whether the stated if-and-only-if equivalence is a non-trivial theorem or follows tautologically from the way the convexity notion is defined in terms of the radius (or vice versa).

    Authors: The abstract announces the notion without spelling out its definition, which is standard for length reasons. The geometric definition (in terms of a modulus of uniform ℂ-convexity of order N) appears in full in Section 2 of the manuscript and is formulated without reference to the p-Bohr radius. The equivalence is then proved as a non-trivial result by showing both directions via explicit estimates. We will revise the abstract to include a one-sentence definition of the convexity property. revision: yes

  2. Referee: [Abstract] Abstract (theorem statement): The central claim is an equivalence between positivity of the newly defined radius and the newly introduced convexity property, but with the full proof steps, coefficient estimates, and any auxiliary lemmas unavailable in the provided text, the soundness of the equivalence cannot be checked.

    Authors: The complete proof, including all auxiliary lemmas on coefficient majorization, the two directions of the equivalence, and the estimates for p ≥ 2, is contained in the body of the manuscript (Sections 3 and 4). The provided text excerpt was only the abstract; the full paper supplies the details. The convexity notion is defined geometrically first, so the equivalence is not tautological. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The p-Bohr radius of order N is introduced via an explicit supremum definition depending only on coefficient norms and the H^∞ norm. The p-uniformly ℂ-convexity of order N is introduced separately as a geometric property of the Banach space X. The central result is stated and proved as an if-and-only-if theorem between these two independently defined objects (for p ≥ 2). No equation or definition reduces one to the other by construction, no fitted parameter is relabeled as a prediction, and no load-bearing step relies on a self-citation chain. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The claims rest on two newly introduced definitions whose consistency with existing Banach space and holomorphic function theory is assumed but not independently verified from the abstract alone.

axioms (1)
  • domain assumption Standard properties of complex Banach spaces and the space H^∞(D,X) of bounded holomorphic functions
    The radius definition and equivalence are built on these background facts from functional analysis.
invented entities (2)
  • p-Bohr radius of order N no independent evidence
    purpose: Quantifies the largest r for which a p-norm sum of coefficients is controlled by the function norm
    Newly defined via the displayed supremum formula.
  • p-uniformly C-convexity of order N no independent evidence
    purpose: Geometric property of the Banach space X used to characterize positivity of the radius
    Introduced in the paper as the dual notion to the radius.

pith-pipeline@v0.9.0 · 5900 in / 1423 out tokens · 38426 ms · 2026-05-24T12:10:48.231569+00:00 · methodology

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Reference graph

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