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arxiv: 2111.10883 · v2 · submitted 2021-11-21 · 🧮 math.CV · math.FA

Bohr operator on opertor valued polyanalytic functions on simply connected domains

Vasudevarao Allu , Himadri Halder This is my paper

Pith reviewed 2026-05-24 13:00 UTC · model grok-4.3

classification 🧮 math.CV math.FA
keywords Bohr radiuspolyanalytic functionsoperator-valued functionssubordinationconvex biholomorphicstarlike biholomorphicsimply connected domainsself-analytic mappings
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The pith

Bohr radii are obtained for operator-valued polyanalytic functions whose leading term is subordinate to an operator-valued convex or starlike biholomorphic mapping.

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

The paper extends classical Bohr inequalities from scalar holomorphic functions to the operator-valued setting on Hilbert spaces. It focuses on polyanalytic functions F(z) = sum_{l=0}^{p-1} conjugate(z)^l f_l(z) defined on simply connected domains, with the leading holomorphic coefficient f_0 belonging to the subordination class S(f) where f is operator-valued and either convex biholomorphic or starlike biholomorphic. Subordination results are derived as direct analogues of known scalar-valued theorems, and a von Neumann-type inequality is proved for self-analytic mappings that fix the origin. The main contribution is the explicit determination of Bohr radii that guarantee the sum of the operator norms of the coefficients remains bounded by the norm of F inside a smaller disk.

Core claim

We obtain Bohr radius for the operator valued polyanalytic functions of the form F(z)= ∑_{l=0}^{p-1} conjugate(z)^l f_l(z), where f_0 is subordinate to an operator valued convex biholomorphic function, and operator valued starlike biholomorphic function in the unit disk D. Several subordination results are established for the class S(f) consisting of holomorphic functions subordinate to such an f, together with a von Neumann-type inequality for self-analytic mappings of the disk fixing the origin.

What carries the argument

The subordination class S(f) of operator-valued holomorphic functions subordinate to a fixed convex or starlike biholomorphic operator-valued function f, applied inside the polyanalytic expansion of F.

If this is right

  • Scalar subordination results lift to the operator-valued class S(f) on Hilbert-space-valued functions.
  • A von Neumann-type inequality holds for self-analytic mappings of the unit disk that fix the origin.
  • Bohr radii exist for the stated polyanalytic form when f_0 is subordinate to operator-valued convex biholomorphic mappings.
  • Bohr radii exist for the stated polyanalytic form when f_0 is subordinate to operator-valued starlike biholomorphic mappings.
  • The radii apply inside any proper simply connected domain in the plane.

Where Pith is reading between the lines

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

  • The same polyanalytic structure may allow Bohr radii to be computed when the subordination is with respect to other operator-valued univalent classes beyond convex and starlike.
  • The von Neumann-type inequality could be tested directly on finite-dimensional matrix approximations to the operators.
  • Relaxing the simply-connected assumption while keeping the polyanalytic form would require new majorant estimates.

Load-bearing premise

The leading term f_0 must belong to the subordination class S(f) generated by an operator-valued convex or starlike biholomorphic function while the domain stays simply connected.

What would settle it

An explicit operator-valued polyanalytic function with f_0 in S(f) for which the sum of coefficient norms exceeds the value of F inside the claimed Bohr radius would falsify the radius.

read the original abstract

In this article, we study the Bohr operator for the operator valued subordination class $S(f)$ consisting of holomorphic functions subordinate to $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$, where $f:\mathbb{D} \rightarrow \mathcal{B}(\mathcal{H})$ is holomorphic and $\mathcal{B}(\mathcal{H})$ is the algebra of bounded linear operators on a complex Hilbert space $\mathcal{H}$. We establish several subordination results, which can be viewed as the analogues of a couple of interesting subordination results from scalar valued settings. We also obtain a von Neumann-type inequality for the class of self-analytic mappings of the unit disk $\mathbb{D}$ which fix the origin. Furthermore, we extensively study Bohr inequalities for operator valued polyanalytic functions in certain proper simply connected domains in $\mathbb{C}$. We obtain Bohr radius for the operator valued polyanalytic functions of the form $F(z)= \sum_{l=0}^{p-1} \overline{z}^l \, f_{l}(z) $, where $f_{0}$ is subordinate to an operator valued convex biholomorphic function, and operator valued starlike biholomorphic function in the unit disk $\mathbb{D}$.

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

0 major / 3 minor

Summary. The manuscript studies the Bohr operator acting on the operator-valued subordination class S(f) of holomorphic functions from the unit disk D to B(H), where f is holomorphic and operator-valued. It derives several subordination results as operator-valued analogues of known scalar results, establishes a von Neumann-type inequality for self-analytic mappings of D that fix the origin, and obtains Bohr radii for operator-valued polyanalytic functions F(z) = sum_{l=0}^{p-1} conjugate(z)^l f_l(z) on certain proper simply connected domains in C, under the assumption that the leading coefficient f_0 belongs to S(f) for f operator-valued convex biholomorphic or starlike biholomorphic.

Significance. If the derivations hold, the work supplies explicit Bohr radii in the operator-valued polyanalytic setting, extending classical subordination and Bohr-phenomenon results from scalar holomorphic functions to this more general context. The von Neumann-type inequality for self-analytic mappings is presented as an auxiliary result and may be of separate interest in operator theory.

minor comments (3)
  1. The abstract states results for 'certain proper simply connected domains in C' but defines subordination and the polyanalytic decomposition explicitly with respect to the unit disk D; a clarifying sentence in the introduction or §4 on how the domain extension is handled would improve readability.
  2. Notation for the polyanalytic decomposition F(z) = sum conjugate(z)^l f_l(z) is introduced in the abstract but the precise range of the index l and the holomorphicity requirements on each f_l should be restated at the beginning of the polyanalytic section for self-contained reading.
  3. The title contains the typographical error 'opertor' (should be 'operator'); this does not affect the mathematics but should be corrected.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for recommending minor revision. The referee's description accurately reflects the scope of our work on Bohr radii for operator-valued polyanalytic functions and the auxiliary von Neumann-type inequality. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation obtains Bohr radii for operator-valued polyanalytic F by applying standard subordination of the leading holomorphic coefficient f0 to an operator-valued convex or starlike biholomorphic function, together with the polyanalytic decomposition and simply-connected domain. These are direct extensions of classical scalar subordination and Bohr results; the provided abstract and description contain no fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations that reduce the central claim to its own inputs by construction. The von Neumann-type inequality is listed as auxiliary and is not required for the radius statement.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard axioms of complex analysis (holomorphic functions, subordination, biholomorphic mappings) and operator theory on Hilbert spaces; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • standard math Holomorphic operator-valued functions on the unit disk satisfy the usual properties of complex differentiability and power-series expansion.
    Invoked for the definition of polyanalytic functions and subordination classes throughout the abstract.
  • domain assumption Subordination to convex or starlike biholomorphic mappings preserves the required growth and coefficient bounds.
    Central to the statement that f_0 is subordinate to such an f.

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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. Operator valued analogues of multidimensional Bohr's inequality

    math.FA 2021-11 unverdicted novelty 6.0

    The paper establishes sharp improved and refined operator-valued versions of Bohr's inequality on the unit disk together with their multidimensional analogues on complete circular domains in C^n.

Reference graph

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