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arxiv: 1907.03944 · v1 · pith:YXAFGYLAnew · submitted 2019-07-09 · 🧮 math.FA

More accurate numerical radius inequalities (II)

Hamid Reza Moradi , Mohammad Sababheh This is my paper

Pith reviewed 2026-05-25 00:21 UTC · model grok-4.3

classification 🧮 math.FA
keywords numerical radiusconvex functionsHilbert space operatorsCartesian decompositionoperator inequalitiesKittaneh inequality
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The pith

Convex functions bound the numerical radius of Hilbert-space operators via an integral of their Cartesian parts.

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

The paper establishes refined numerical radius inequalities by showing that for a bounded linear operator A on a Hilbert space with Cartesian decomposition A = B + iC and convex function f, the operator norm of f applied to the average of A*A and AA* is at most the norm of the integral from 0 to 1 of f applied to the convex combination (1-t)B² + t C², which is itself at most f of the square of the numerical radius of A. This chain complements earlier work by the same authors and generalizes known results such as Kittaneh's inequality through the application of convexity. A sympathetic reader would care because the integral provides an interpolating expression that yields sharper estimates than direct comparisons to the numerical radius alone.

Core claim

When A is a bounded linear operator on a Hilbert space having the Cartesian decomposition A = B + iC and f is a convex function, the inequality ||f((A*A + AA*)/4)|| ≤ ||∫ from 0 to 1 f((1-t)B² + t C²) dt|| ≤ f(w²(A)) holds, providing refined and generalized forms of known numerical radius inequalities.

What carries the argument

The integral ∫_0^1 f((1-t)B² + t C²) dt that uses convexity of f to connect the averaged operator (A*A + AA*)/4 to the numerical-radius bound w²(A).

If this is right

  • The inequality refines Kittaneh's bound by inserting an integral interpolation step.
  • It extends previous numerical-radius results to arbitrary convex functions rather than specific choices.
  • The same convexity argument yields analogous bounds when the numerical radius is replaced by other operator radii.
  • Direct consequences include improved estimates for the norm of functions of the real and imaginary parts of A.

Where Pith is reading between the lines

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

  • The integral form may admit numerical approximation schemes for computing bounds on finite matrices.
  • The technique could extend to other operator decompositions such as polar or singular-value forms.
  • Applications in quantum information might follow if the Cartesian parts correspond to observable operators.

Load-bearing premise

The function f must be convex on the non-negative reals.

What would settle it

Explicit computation on a 2-by-2 matrix A = B + iC and a convex f where the norm of the left-hand side exceeds the norm of the integral term.

read the original abstract

In a recent work of the authors, we showed some general inequalities governing numerical radius inequalities using convex functions. In this article, we present results that complement the aforementioned inequalities. In particular, the new versions can be looked at as refined and generalized forms of some well known numerical radius inequalities. Among many other results, we show that \[\left\| f\left( \frac{{{A}^{*}}A+A{{A}^{*}}}{4} \right) \right\|\le \left\| \int_{0}^{1}{f\left( \left( 1-t \right){{B}^{2}}+t{{C}^{2}} \right)dt} \right\|\le f\left( {{w}^{2}}\left( A \right) \right),\] when $A$ is a bounded linear operator on a Hilbert space having the Cartesian decomposition $A=B+iC.$ This result, for example, extends and refines a celebrated result by kittaneh.

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 / 0 minor

Summary. The manuscript presents refined numerical radius inequalities for bounded linear operators A on Hilbert spaces, using convex functions f. Building on prior work, it establishes the chain ||f((A*A + AA*)/4)|| ≤ ||∫_0^1 f((1-t)B² + t C²) dt|| ≤ f(w²(A)) where A = B + iC is the Cartesian decomposition, and claims this extends and refines Kittaneh's inequality among other results.

Significance. If the central chain holds under the stated hypotheses, the results would provide a parameterized refinement of classical numerical-radius bounds via convex functions, with potential applications in operator inequalities. The manuscript does not appear to include machine-checked proofs or reproducible code.

major comments (2)
  1. [Abstract] Abstract, displayed inequality: the left-hand comparison ||f((A*A + AA*)/4)|| ≤ ||∫_0^1 f((1-t)B² + t C²) dt|| is asserted for merely convex f. When [B, C] ≠ 0 the path operators N(t) = (1-t)B² + t C² generally fail to commute, so the Jensen-type step f(∫ N(t) dt) ≤ ∫ f(N(t)) dt requires operator convexity of f rather than scalar convexity; no additional hypothesis, commutativity assumption, or norm-only argument is indicated to close this gap.
  2. [Abstract] Abstract, displayed inequality (right-hand side): the bound ||∫_0^1 f((1-t)B² + t C²) dt|| ≤ f(w²(A)) is stated to follow from convexity alone, yet the passage from the integral average to f(w(A)²) also invokes that f is non-decreasing (to obtain ||f(T)|| ≤ f(‖T‖) ≤ f(w(A)²)); this monotonicity hypothesis is not listed among the standing assumptions on f.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying two points where the hypotheses on f need to be stated more precisely. Both observations are valid, and we will revise the manuscript to incorporate the necessary assumptions on f (operator convexity together with monotonicity). The body of the paper already works under these stronger hypotheses; only the abstract statement was insufficiently precise. We address each comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, displayed inequality: the left-hand comparison ||f((A*A + AA*)/4)|| ≤ ||∫_0^1 f((1-t)B² + t C²) dt|| is asserted for merely convex f. When [B, C] ≠ 0 the path operators N(t) = (1-t)B² + t C² generally fail to commute, so the Jensen-type step f(∫ N(t) dt) ≤ ∫ f(N(t)) dt requires operator convexity of f rather than scalar convexity; no additional hypothesis, commutativity assumption, or norm-only argument is indicated to close this gap.

    Authors: We agree. The inequality ||f((A*A + AA*)/4)|| ≤ ||∫ f((1-t)B² + t C²) dt|| relies on the operator Jensen inequality, which holds when f is operator convex (not merely convex). The manuscript’s proofs in the body already assume f is operator convex; the abstract omitted this qualifier. We will revise the abstract to state explicitly that f is operator convex (and non-decreasing). revision: yes

  2. Referee: [Abstract] Abstract, displayed inequality (right-hand side): the bound ||∫_0^1 f((1-t)B² + t C²) dt|| ≤ f(w²(A)) is stated to follow from convexity alone, yet the passage from the integral average to f(w(A)²) also invokes that f is non-decreasing (to obtain ||f(T)|| ≤ f(‖T‖) ≤ f(w(A)²)); this monotonicity hypothesis is not listed among the standing assumptions on f.

    Authors: We agree. The right-hand inequality uses both convexity and monotonicity of f to pass from the norm of the integral to f(w(A)²). The body of the paper works under the standing assumption that f is non-decreasing, but the abstract does not record it. We will add the monotonicity hypothesis to the abstract statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and provided excerpts present the central inequality as a new result extending Kittaneh's theorem via standard convexity and numerical-radius properties. The mention of prior author work is contextual only and does not serve as the sole justification for the displayed chain. No equations reduce by construction to fitted parameters, self-definitions, or unverified self-citations; the derivation chain remains independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard assumption that f is convex together with the usual properties of the numerical radius and operator norm on Hilbert space; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption f is convex on [0, ∞)
    Convexity is required for the integral and norm inequalities to hold as stated.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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    Sababheh and H

    M. Sababheh and H. R. Moradi, More accurate numerical radius inequalities , Linear Multilinear Algebra., accepted. (H. R. Moradi) Department of Mathematics, Payame Noor Unive rsity (PNU), P.O. Box 19395-4697, Tehran, Iran. E-mail address: hrmoradi@mshdiau.ac.ir (M. Sababheh) Department of Basic Sciences, Princess Sumay a University For Technology, Al Juba...

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