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arxiv: 2506.23831 · v2 · submitted 2025-06-30 · 🧮 math.CV · math.FA

A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges

Javad Mashreghi , Annika Moucha , Ryan O'Loughlin , Thomas Ransford , Oliver Roth This is my paper

Pith reviewed 2026-05-19 07:28 UTC · model grok-4.3

classification 🧮 math.CV math.FA
keywords Schwarz-Jack lemmacircularly symmetric domainsconformal mapsnumerical rangespectral setCrouzeix theoremholomorphic functions2x2 matrix
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The pith

A symmetry-restricted Schwarz-Jack lemma implies monotonicity and convexity for conformal maps of symmetric domains and a new proof of Crouzeix's theorem.

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

The paper proves a Schwarz-Jack lemma for holomorphic functions on the unit disk that attain their maximum modulus on each circle at the positive real axis. This lemma establishes monotonicity properties for conformal maps of circularly symmetric domains and convexity properties for maps of bi-circularly symmetric domains. These properties deliver a new proof that the numerical range of any 2 by 2 matrix is a 2-spectral set for the matrix. The argument avoids any explicit formula for the conformal map from an ellipse to the unit disk. A reader would care because the result connects classical complex analysis to operator theory and matrix approximations without relying on special-case mappings.

Core claim

We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we establish monotonicity and convexity properties of conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, we give a new proof of Crouzeix's theorem that the numerical range of any 2×2 matrix is a 2-spectral set for the matrix. Unlike other proofs, our approach does not depend on the explicit formula for the conformal mapping of an ellipse onto the unit disk.

What carries the argument

The Schwarz-Jack lemma for holomorphic functions whose maximum modulus on each circle centered at the origin is attained on the positive real axis; it supplies the monotonicity and convexity statements for the conformal maps of the symmetric domains.

Load-bearing premise

The holomorphic functions or conformal maps under study must have their maximum modulus on each circle about the origin attained at a point on the positive real axis.

What would settle it

A holomorphic function on the unit disk that attains its maximum modulus on each circle at the positive real axis yet violates the stated conclusion of the Schwarz-Jack lemma would falsify the central result.

Figures

Figures reproduced from arXiv: 2506.23831 by Annika Moucha, Javad Mashreghi, Oliver Roth, Ryan O'Loughlin, Thomas Ransford.

Figure 1
Figure 1. Figure 1: Examples of bi-circularly symmetric domains: (a) Ellipse x 2 + 4y 2 < 1 (b) Example with R(θ) = p 1/4 + 1/(1 + 50θ 2) 2 for θ ∈ [0, π 2 ] (c) Image of D under z + z 3/4 − z 5/20 (see Proposition 4.4) (d) A non-star-shaped example In establishing this result, the following elementary lemma will be helpful. Lemma 4.3. Let h be a continuous, strictly increasing, convex function defined on an interval I. Then … view at source ↗
read the original abstract

We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we establish monotonicity and convexity properties of conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, we give a new proof of Crouzeix's theorem that the numerical range of any $2\times 2$ matrix is a $2$-spectral set for the matrix. Unlike other proofs, our approach does not depend on the explicit formula for the conformal mapping of an ellipse onto the unit disk.

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 paper proves a Schwarz-Jack lemma for holomorphic functions f on the unit disk such that, for every r, the maximum of |f(z)| on |z|=r is attained at a point on the positive real axis. This lemma is used to derive monotonicity and convexity properties of the conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, the authors give a new proof of Crouzeix's theorem that the numerical range of any 2×2 matrix is a 2-spectral set, without relying on the explicit formula for the conformal mapping of an ellipse onto the unit disk.

Significance. If the lemma and the subsequent reductions hold, the work supplies a symmetry-driven route to spectral-set conclusions that bypasses explicit conformal mappings. This approach may extend to other classes of domains or matrices and clarifies how circular symmetry interacts with monotonicity/convexity in conformal mapping theory.

major comments (2)
  1. [§4] §4 (application to Crouzeix theorem): the reduction to the Schwarz-Jack lemma requires that, after rotation to align the elliptical numerical range, the associated conformal map still satisfies the positive-real-axis maximum-modulus condition on every circle. The manuscript states this can be arranged without loss of generality but does not supply an explicit verification that bi-circular symmetry survives the rotation or that the attainment point remains on the positive real axis; this step is load-bearing for the factor-2 bound.
  2. [§3.2] §3.2 (convexity for bi-circular domains): the convexity claim for the conformal map relies on the lemma being applicable after a preliminary normalization; if the normalization alters the location of the maximum-modulus point, the monotonicity inequality used to bound the numerical radius may not hold with the stated constant.
minor comments (3)
  1. [Definition 2.1] The precise statement of the symmetry hypothesis in the Schwarz-Jack lemma (Definition 2.1) would be clearer if written as an equation: max_{|z|=r} |f(z)| = |f(r)| for all r < 1.
  2. [Figure 1] Figure 1 (illustration of circularly symmetric domain) lacks a label for the positive real axis; adding it would help readers track the attainment condition.
  3. [Introduction] A short remark comparing the length of the new proof with the classical one that uses the explicit ellipse map would help readers assess the advantage claimed in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the need for more explicit verifications in the applications of the Schwarz-Jack lemma. We believe the core results hold and address the concerns below by providing additional arguments and committing to revisions that will include the requested details.

read point-by-point responses

  1. Referee: [§4] §4 (application to Crouzeix theorem): the reduction to the Schwarz-Jack lemma requires that, after rotation to align the elliptical numerical range, the associated conformal map still satisfies the positive-real-axis maximum-modulus condition on every circle. The manuscript states this can be arranged without loss of generality but does not supply an explicit verification that bi-circular symmetry survives the rotation or that the attainment point remains on the positive real axis; this step is load-bearing for the factor-2 bound.

    Authors: We agree that an explicit verification would strengthen the argument. In the revised version, we will add a detailed explanation showing that rotating the numerical range (which corresponds to considering e^{iθ}A for a suitable θ) preserves the bi-circular symmetry of the domain. Consequently, the Riemann mapping function from the unit disk to the rotated domain can be normalized so that it maps the positive real axis to itself, ensuring that the maximum of |f(z)| on each circle |z|=r is attained at a positive real point. This follows from the symmetry with respect to both the real and imaginary axes after alignment. We will include this verification to make the reduction rigorous. revision: yes

  2. Referee: [§3.2] §3.2 (convexity for bi-circular domains): the convexity claim for the conformal map relies on the lemma being applicable after a preliminary normalization; if the normalization alters the location of the maximum-modulus point, the monotonicity inequality used to bound the numerical radius may not hold with the stated constant.

    Authors: The preliminary normalization in §3.2 consists of a rotation that aligns one axis of symmetry with the positive real axis. Due to the bi-circular symmetry, this rotation ensures that the point of maximum modulus on each circle remains on the positive real axis after normalization. We will revise the manuscript to explicitly state this fact and explain why the normalization preserves the hypothesis of the Schwarz-Jack lemma. This will confirm that the monotonicity and convexity properties hold as stated. revision: yes

Circularity Check

0 steps flagged

No circularity: new lemma derived independently and applied to symmetric domains

full rationale

The paper establishes a Schwarz-Jack lemma under an explicit symmetry hypothesis (maximum modulus attained on the positive real axis for each circle), then derives monotonicity/convexity properties for conformal maps of circularly symmetric domains directly from that hypothesis. The Crouzeix application follows by arranging coordinates so the numerical-range ellipse satisfies the lemma's symmetry condition, without presupposing the target bound or using self-cited uniqueness results. No equation reduces to a fitted parameter renamed as prediction, no ansatz is smuggled via prior self-citation, and the derivation chain remains self-contained against external benchmarks such as the classical Schwarz lemma and standard properties of conformal mappings.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results of complex analysis and conformal mapping theory; no free parameters, ad-hoc constants, or newly invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard theorems on holomorphic functions, maximum modulus principle, and existence of conformal maps for simply connected domains.
    Invoked implicitly to define the setting and to guarantee the existence of the maps whose properties are studied.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

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