A note on the paper "Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y]"

M. M. Motamedinezhad; Rahim Kargar

REVIEW 3 major objections 4 minor 11 references

A note on the paper "Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y]"

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This note corrects a cosine-starlike class, shows the original is empty, and proves a majorization bound in |z| ≤ 0.391389.

desk verdict Right about the empty class in Tang et al., but the 'corrected' majorization theorem is vacuously true since f,g∈A force the Schwarz function to be constant. read the letter →

arxiv 1908.01306 v1 pith:G2MTXBTH submitted 2019-08-04 math.CV

M. M. Motamedinezhad , Rahim Kargar This is my paper · ORCID

classification math.CV MSC 30C4530C80

keywords univalentfunctionsstarlikemajorizationsubordinationcosinefunctionSchwarzBoothlemniscate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

A recently published class of starlike functions defined by the subordination condition $z f'(z)/f(z) \prec 1 + \cos z$ is empty as stated: at $z=0$ the left-hand side equals $1$ while the right-hand side equals $2$, so no normalized function $f$ can satisfy the condition. The note replaces the definition with $z f'(z)/f(z) \prec \cos z$, which matches the value $1$ at the origin and therefore admits functions such as the identity. For this corrected class, the paper proves that if $f$ is majorized by $g$, meaning $f = \psi g$ for some analytic $\psi$ with $|\psi| \le 1$, then $|f'(z)| \le |g'(z)|$ for all $|z| \le r_1$, where $r_1 \approx 0.391389$ is the smallest positive root of $(1-r^2)\cos r = 2r$. The result gives an explicit radius for the majorization phenomenon and a quantitative version of the classical bounded-function derivative estimate.

What carries the argument

The mechanism is the subordination-to-cosine condition together with the quantitative estimate $(2.4)$: for a Schwarz function $\varphi$ and $r<1$, the inequalities $\cos r \le |\cos(\varphi(z))| \le \cosh r$ hold when $|\varphi(z)| \le r$. The lower bound yields $|g(z)/g'(z)| \le r/\cos r$ for $g$ in the corrected class. Substituting this into the identity $f' = \psi' g + \psi g'$ and using the standard majorization estimate $|\psi'| \le (1-|\psi|^2)/(1-|z|^2)$ reduces the desired inequality to a two-variable condition $h(r,\beta) \le 1$, where $\beta = |\psi(z)|$. Simplifying $h$ gives $k(r,\beta) = (1-r^2)\cos r - (1+\beta)r \ge 0$, and since $k$ decreases in $\beta$, the worst case is $\beta=1$, producing the single equation $(1-r^2)\cos r - 2r = 0$ that fixes $r_1$.

What would settle it

Compute the smallest positive zero of $(1-r^2)\cos r - 2r$ numerically and compare it with $0.391389$; a different value would refute the claimed radius. More directly, search for functions $g$ in the corrected class and $f = \psi g$ with $|\psi| \le 1$ and $|f'(z_0)| > |g'(z_0)|$ for some $|z_0| < r_1$; any such example would refute Theorem 2.1.

Watch

Extended reading notes

Core claim

The central claim of the note is that the corrected cosine-starlike class $$S^*_c = \{f \in \mathcal{A} : z f'(z)/f(z) \prec \cos z\}$$ is nonempty and carries a genuine majorization theorem. Theorem 2.1 states that when $f$ is majorized by $g$ and $g$ lies in this class, the inequality $|f'(z)| \le |g'(z)|$ holds for $|z| \le r_1$, with $r_1 \approx 0.391389$ the smallest positive zero of $(1-r^2)\cos r - 2r = 0$. The proof writes $z g'(z)/g(z) = \cos(\varphi(z))$ with a Schwarz function $\varphi$, bounds $|g(z)/g'(z)| \le r/\cos r$, and combines this with the estimate $|\psi'(z)| \le (1-|\psi(z)|^2)/(1-|z|^2)$. The paper also asserts that the original class, defined with $1+\cos z$, contains no functions and that its majorization theorem is therefore incorrect.

Load-bearing premise

The load-bearing premise is the quantitative estimate $\cos r \le |\cos(\varphi(z))|$ for Schwarz functions; the entire radius $r_1$ rests on this single inequality, and if it were not valid the majorization disk would shrink or disappear.

Editorial extensions

If this is right

The original cosine-starlike class and its majorization theorem are vacuous: the class contains no functions, so any statement about it has no nonempty instances.
For the corrected class, every majorized pair $f \ll g$ satisfies the derivative comparison $|f'(z)| \le |g'(z)|$ throughout the disk $|z| \le 0.391389$, with no further conditions on $f$.
Because the identity function belongs to the corrected class, the theorem yields an explicit bounded-function derivative estimate at the same radius.
The method stops exactly at the stated root: for $r > r_1$ the auxiliary inequality $k(r,\beta) \ge 0$ fails when $\beta$ is close to $1$, so the argument provides no guarantee beyond this radius.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same correction pattern applies to any would-be subordination class whose template function has value different from $1$ at the origin: the class is automatically empty, and the natural fix is to normalize the template.
The proof strategy suggests a general recipe for majorization radii in starlike classes defined by subordination: replace the template function by a sharp lower bound on Schwarz disks and minimize the resulting two-variable inequality.
The paper does not settle sharpness of $r_1$; one could test numerically whether some pair in the corrected class violates the derivative inequality just below the stated radius.
The same normalization issue and method likely transfer to other periodic templates such as $\sin z$, where the origin value must be fixed before a meaningful class can be defined.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper correctly catches a real error in Tang et al. — the class S*_c defined via zf'/f ≺ 1+cos z is empty because the two sides have different values at z=0 — but its own corrected majorization theorem is vacuous as stated. If f,g∈A and f≪g, the majorizing function ψ has ψ(0)=1 and |ψ|≤1 on Δ; by the maximum modulus principle ψ≡1, so f≡g. Theorem 2.1 therefore covers no distinct pair of functions, and the radius r1≈0.391389 is irrelevant.

The correction and the origin-consistency point are genuinely good, and the proof of the estimate is self-contained and uses standard tools (MacGregor's majorization scheme, Schwarz-Pick, Nehari). The calculation leading to k(r)=(1−r^2)cos r−2r is sound as far as it goes.

The soft spots are not minor. First, the vacuity is a load-bearing flaw: the paper advertises 'majorization properties' for the corrected class, but there is no pair (f,g) with f≠g in A satisfying the hypothesis. Fixing this requires weakening the normalization on f, e.g. f∈H with f(0)=0 and |f'(0)|≤1; then ψ(0) is not pinned to 1 and the β-optimization has content. Second, the paper says 'Since cos z is univalent in Δ' to justify Definition 2.1; that is false (cos z is even), so the well-definedness argument as written is wrong. Third, Corollary 2.1 claims to improve the √2−1 bound, but 0.391389 is smaller than √2−1≈0.4142, and in fact the corollary's assumptions plus f≪z again force f(z)=z, so the bound is trivial. These are correctable, but they need to be stated.

Who is this for: specialists in geometric function theory tracking the Tang et al. paper. The observation about the empty class deserves a short note; the majorization theorem as stated does not. I would send this to a referee because the correction is substantive and the vacuity is exactly what a referee should catch; after revision, the note could be a one-page remark rather than a research article. My own recommendation: treat the correction as valid but do not rely on Theorem 2.1.

Referee Report

3 major / 4 minor

Summary. This note challenges Tang et al.'s definition of the starlike class S*_c associated with the cosine function, pointing out that the subordination condition z f'(z)/f(z) ≺ 1 + cos z is impossible for f ∈ A because the two sides have different values at z = 0. The authors then propose a corrected class S*_c := {f ∈ A : z f'(z)/f(z) ≺ cos z} and prove a majorization theorem (Theorem 2.1): if f ∈ A is majorized by g ∈ S*_c, then |f'(z)| ≤ |g'(z)| for |z| ≤ r1, where r1 ≈ 0.391389 is the smallest positive root of (1 - r^2) cos r - 2r = 0. They state that cos z is univalent on the unit disk, and they claim a corollary improves the classical √2 - 1 radius bound of Theorem B.

Significance. The observation that Tang et al.'s original class is empty because of the mismatch at z = 0 is correct and worth recording. However, the strengthened claims of the note are not established: the main theorem is vacuous as stated because the normalization assumptions force any majorized pair to be identical, the asserted univalence of cos z on the unit disk is false, and the corollary's radius 0.391389 is smaller than √2 - 1, so it does not improve the classical bound. The paper demonstrates a real flaw in the literature but its own 'correct definition and result' need substantial revision before they constitute a valid contribution.

major comments (3)

[Theorem 2.1] Theorem 2.1 is vacuous as stated. If f, g ∈ A and f ≪ g, then f = ψ g with |ψ| ≤ 1 analytic on the unit disk. Since f(0) = g(0) = 0 and f'(0) = g'(0) = 1, the quotient ψ = f/g has a removable singularity at 0 with ψ(0) = 1. The function ψ then attains its maximum modulus in the interior of the disk, so by the maximum modulus principle ψ ≡ 1 and hence f ≡ g. Thus the hypothesis of Theorem 2.1 never holds for distinct f and g, and the conclusion |f'(z)| ≤ |g'(z)| is trivially true for all |z| < 1. The computation of r1 and the bound in Equation (2.2) are superfluous. To make the theorem substantive, the normalization on f must be weakened, e.g., f ∈ H with f(0) = 0 and no condition on f'(0); then ψ(0) = f'(0) need not equal 1 and the β-optimization in the proof becomes meaningful. The proof itself does not otherwise require f ∈ A, so this fix is local.
[Corollary 2.1] Corollary 2.1 does not improve √2 - 1 as claimed. Since r1 ≈ 0.391389 and √2 - 1 ≈ 0.414214, the radius in the corollary is strictly smaller than the classical bound, so it is weaker, not an improvement. Moreover, under the stated hypotheses f(0) = 0, f'(0) = 1, and |f(z)| < 1, Schwarz's lemma forces f(z) = z, making the corollary trivially true and the radius irrelevant. The sentence 'Indeed, we improve the bound √2 - 1 in the Theorem B' is therefore incorrect and should be removed or replaced by an accurate statement.
[Definition 2.1] The assertion 'Since cos z is univalent in ∆' is false. For example, cos(i/2) = cos(-i/2) = cosh(1/2), with both i/2 and -i/2 in the unit disk. The definition of S*_c via subordination is still meaningful without univalence, since subordination is defined for arbitrary analytic functions and only the equivalence with an inclusion of images requires univalence. However, the given justification is wrong and should be corrected, for instance by noting directly that the subordination condition is compatible with the normalization z f'(z)/f(z)|_{z=0} = 1 = cos 0.

minor comments (4)

[Equation (2.4)] The chain of inequalities in Equation (2.4) is stated as 'a simple exercise' but is a key quantitative input. A short proof or a reference would improve readability.
[Abstract] The abstract contains grammatical errors: 'it's result' should be 'its result', and 'In this note we pointed out' should be 'In this note we point out'.
[Notation] The corrected class is denoted by the same symbol S*_c as the original (incorrect) class, which may cause confusion. A distinct notation, e.g., S*_c^{new}, would be clearer.
[Section 2, paragraph after Theorem 2.1] The corollary's deduction 'Since the identity function g(z) = z belongs to the class S*_c' is correct, but the application assumes f is majorized by z; this follows from |f(z)| < 1 via Schwarz's lemma, but the paper does not mention that step.

Circularity Check

1 steps flagged · score 7.0 of 10

Theorem 2.1 is vacuous as stated: for f,g∈A, f≪g forces f≡g, so the majorization-radius result reduces to an identity by construction.

self definitional [Definition 1.1 (Eq. 1.3) and Theorem 2.1 (Eq. 2.6)]
"A function f is said to be majorized by g written as f(z)≪g(z) or f≪g, if there exists an analytic function ψ in Δ, satisfying |ψ(z)|≤ 1 and f(z)=ψ(z)g(z) for all z∈ Δ. ... Let f∈A and g∈ S*_c. If f(z) is majorized by g(z) in Δ, then |f′(z)|≤|g′(z)| (|z|≤ r1)."

Here f and g are both in A, so f(0)=g(0)=0 and f'(0)=g'(0)=1. In (2.6), ψ=f/g has a removable singularity at 0 with ψ(0)=1. Since |ψ|≤1 on Δ, the maximum modulus principle forces ψ≡1, hence f≡g. The claimed inequality is then an equality holding for every |z|<1, and the radius r1 is superfluous. The paper's β-optimization lets β=|ψ(z)| range over [0,1], but the normalization of f and g forces β=1; the theorem's content reduces to the identity f=g by construction.

full rationale

The note's correction of the original Tang class is a genuine, non-circular observation: the defining subordination zf'(z)/f(z)≺1+cos z fails at z=0 because the two sides take values 1 and 2, so that class is empty. However, the corrected main theorem is vacuous as stated. Because both f and g are normalized (A), any analytic majorant ψ with |ψ|≤1 and f=ψg must satisfy ψ(0)=1, whence ψ≡1 and f≡g. Thus the majorization hypothesis is already equality; the radius bound |z|≤r1 is a weaker consequence of an identity, and the h(r,β) computation, while internally valid, proves no distinct-function majorization. This is a reduction by definition of the central result, not a fitted-input issue. The authors' self-citations (e.g., [3], [4]) appear only in the introductory survey and are not load-bearing. Separate correctness risks, not circularity, include the assertion that cos z is univalent on Δ, which is false because cos z=cos(−z), and the evenness of cos z makes the class definition's 'well-defined' claim doubtful.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no free parameters, postulates no new entities, and relies only on standard lemmas in geometric function theory.

assumptions (2)

standard math Schwarz-Pick lemma: if psi is analytic in the unit disk and |psi(z)| <= 1, then |psi'(z)| <= (1 - |psi(z)|^2) / (1 - |z|^2).
Invoked as Lemma 1.1 from Nehari [8]; it is a standard theorem in complex analysis and is used to bound |psi'|.
standard math Subordination principle: if f is subordinate to g, then there exists a Schwarz function phi such that f = g composed with phi.
Used to represent z g'(z)/g(z) as cos(phi(z)) in the proof of Theorem 2.1; this is the defining property of subordination.

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Pith. "Pith review of A note on the paper "Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y]"." pith.science (2026). https://pith.science/paper/G2MTXBTH

@misc{pith2026190801306,
  author       = {Pith},
  title        = {Pith review of: A note on the paper "Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y]"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2MTXBTH}},
  note         = {Machine review of arXiv:1908.01306}
}

abstract

Very recently Tang et al. [Bull Iran Math Soc (2019) doi:10.1007/s41980-019-00262-y] have studied some majorization results for two certain subclasses of the starlike functions associated with the sine and cosine functions defined by $\mathcal{S}^*_s$ and $\mathcal{S}^*_c$, respectively. In this note we pointed out that the definition of the class $\mathcal{S}^*_c$ and it's result are incorrect and give correct definition and result.

Figures

Figures reproduced from arXiv: 1908.01306 by the authors.

**Figure 1.** (a): The boundary curve of cos(∆) (b): The graph of (1 − r 2 ) cos r − 2r = 0 when 0 < r < 1 or equivalently (2.3) g(z) g 0(z) = z cos(φ(z)) (z ∈ ∆). Let φ(z) = Reit where R ≤ r = |z| < 1 and −π ≤ t ≤ π. It is a simple exercise that (2.4) cos r ≤ cos R ≤ | cos(φ(z))| ≤ cosh R ≤ cosh r (R ≤ r < 1). From (2.3) and (2.4) we get (2.5) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Carath´ eodory, C.: Theory of Functions. Vol. 2, New York (1954)

work page 1954
[2]

Cho, N.E., Kumar, V., Kumar, S.S., Ravichandran, V.: Radius problems for starlike functions associated with the Sine function. Bull. Iran. Math. Soc. 45, 213–232 (2019)

work page 2019
[3]

Kargar, R., Ebadian A., Sok´ o l, J.: On Booth lemiscate and starlike functions. Anal. Math. Phys. 9, 143–154 (2019)

work page 2019
[4]

Complex Anal

Kargar, R., Ebadian A., Sok´ o l, J.: Radius problems for some subclasses of analytic functions. Complex Anal. Oper. Theory 11, 1639–1649 (2017)

work page 2017
[5]

RIMS Kokyuroku Kyoto Univ

Kuroki, K., Owa, S.: Notes on new class for certain analytic functions. RIMS Kokyuroku Kyoto Univ. 1772, 21–25 (2011)

work page 2011
[6]

Ma, M., Minda, D.: Uniformly convex functions. Ann. Polon. Math. 57, 165–175 (1992)

work page 1992
[7]

Duke Math

MacGregor, T.H.: Majorization by univalent functions. Duke Math. J. 34, 95–102 (1967)

work page 1967
[8]

McGraw–Hill: New York, NY, USA (1952)

Nehari, Z.: Conformal Mapping. McGraw–Hill: New York, NY, USA (1952)

work page 1952

Show all 11 references

[9]

Robertson, M.I.S.: On the theory of univalent functions. Ann. Math. 37, 374–408 (1936)

1936
[10]

Zeszyty Nauk

Sok´ o l, J., Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike func- tions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19, 101–105 (1996)

1996
[11]

Tang, H., Srivastava, H.M., Li, Sh., Deng, G.-T.: Majorization results for subclasses of starlike functions based on the sine and cosine functions. Bull. Iran. Math. Soc. (2019). https://doi.org/10.1007/s41980-019-00262-y Department of Mathematics, University of Applied Scienc...

2019 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Carath´ eodory, C.: Theory of Functions. Vol. 2, New York (1954)

work page 1954

[2] [2]

Cho, N.E., Kumar, V., Kumar, S.S., Ravichandran, V.: Radius problems for starlike functions associated with the Sine function. Bull. Iran. Math. Soc. 45, 213–232 (2019)

work page 2019

[3] [3]

Kargar, R., Ebadian A., Sok´ o l, J.: On Booth lemiscate and starlike functions. Anal. Math. Phys. 9, 143–154 (2019)

work page 2019

[4] [4]

Complex Anal

Kargar, R., Ebadian A., Sok´ o l, J.: Radius problems for some subclasses of analytic functions. Complex Anal. Oper. Theory 11, 1639–1649 (2017)

work page 2017

[5] [5]

RIMS Kokyuroku Kyoto Univ

Kuroki, K., Owa, S.: Notes on new class for certain analytic functions. RIMS Kokyuroku Kyoto Univ. 1772, 21–25 (2011)

work page 2011

[6] [6]

Ma, M., Minda, D.: Uniformly convex functions. Ann. Polon. Math. 57, 165–175 (1992)

work page 1992

[7] [7]

Duke Math

MacGregor, T.H.: Majorization by univalent functions. Duke Math. J. 34, 95–102 (1967)

work page 1967

[8] [8]

McGraw–Hill: New York, NY, USA (1952)

Nehari, Z.: Conformal Mapping. McGraw–Hill: New York, NY, USA (1952)

work page 1952