Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$

H. Mahzoon; R. Kargar

REVIEW 2 major objections 3 minor 7 references

Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Bi-starlike functions of order alpha have pre-Schwarzian norm at most 6, 4, or 2, depending on alpha — a corrected bound for an earlier wrong one.

desk verdict The critique of Rahmatan et al. is useful, but the paper's own Theorem 2.1 is unproved because the subordination step (2.5)→(2.6) requires f(Δ)⊂Δ, which fails for the paper's own example. read the letter →

arxiv 1908.01397 v1 pith:PGEFVWAA submitted 2019-08-04 math.CV

H. Mahzoon , R. Kargar This is my paper · ORCID

classification math.CV MSC 30C45

keywords univalentfunctionsbi-univalentstarlikeoforderalphapre-SchwarzianderivativesubordinationSchwarz-Picklemmanormestimate

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

The paper studies the pre-Schwarzian norm $\|f\|=\sup_{z\in\Delta}(1-|z|^2)\left|f''(z)/f'(z)\right|$ on bi-univalent functions that are starlike of order $\alpha$. It claims that an earlier published bound for this class is wrong and presents a corrected piecewise estimate: $\|f\|\le 6$ at $\alpha=0$, $\|f\|\le \min\{6-4\alpha,\,4(1-\alpha)/\alpha\}$ for $0<\alpha<1/2$, $\|f\|\le 4$ at $\alpha=1/2$, and $\|f\|\le 2$ for $1/2<\alpha<1$. The argument splits into the classical starlike-order bound and a new bound obtained by rewriting the inverse starlikeness condition as a subordination and applying the Schwarz–Pick lemma. On the related class $V_\Sigma(\alpha)$, the paper argues that the earlier proof remains invalid and leaves the sharp bound open.

What carries the argument

The load-bearing object is the subordination chain for the inverse function. From $\operatorname{Re}(wg'(w)/g(w))>\alpha$, the paper gets $wg'(w)/g(w)\prec(1+(1-2\alpha)w)/(1-w)$, and with $g=f^{-1}$ rewrites it as $f(z)/(zf'(z))\prec(1+(1-2\alpha)z)/(1-z)$. This identity turns the inverse-side hypothesis into an explicit expression for $f''/f'$ in terms of a Schwarz function $\varphi$ and its derivative; the Schwarz–Pick lemma, the pointwise derivative bound for holomorphic self-maps of the disk, controls that expression. The norm $\|f\|=\sup_{z\in\Delta}(1-|z|^2)|f''/f'|$ converts the pointwise bound into the piecewise constants of Theorem 2.1.

What would settle it

Between (2.5) and (2.6), the proof substitutes $w=f(z)$ into a subordination defined only for $w\in\Delta$. For the paper's own bi-univalent example $f(z)=z/(1-z)$, the point $z=1/2$ gives $f(1/2)=1\notin\Delta$, so the composite Schwarz function is not defined there; checking this one transition for this function settles that the equivalence the inverse-side bound depends on is not valid as stated.

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Extended reading notes

Core claim

The central claim is Theorem 2.1: every $f\in S^*_\Sigma(\alpha)$ obeys the piecewise norm bound above. The first input is the sharp starlike-order estimate $\|f\|\le 6-4\alpha$. The second input comes from the inverse condition: because $\operatorname{Re}(wg'(w)/g(w))>\alpha$, the ratio $wg'(w)/g(w)$ is subordinate to the half-plane map $(1+(1-2\alpha)w)/(1-w)$; the paper rewrites this as $f(z)/(zf'(z))\prec(1+(1-2\alpha)z)/(1-z)$ and derives a formula for $f''(z)/f'(z)$ in terms of a Schwarz function $\varphi$ and its derivative. The Schwarz–Pick lemma bounds $\varphi'$, and the norm supremum produces the second constant in each case. For the class $V_\Sigma(\alpha)$, the paper states that the earlier result and its proof are incorrect, and that completing the estimate would require a currently missing bound for $|f(z)/z|$.

Load-bearing premise

The proof needs $f(\Delta)\subset\Delta$ to rewrite the inverse subordination in the variable $z$, a containment the paper never states or proves and which its own example $f(z)=z/(1-z)$ fails; if that containment is false the key equivalence between (2.5) and (2.6) is not established.

Editorial extensions

If this is right

If Theorem 2.1 holds, the $\alpha=0$ case reproduces the sharp universal bound $\|f\|\le 6$, so bi-univalence adds no new restriction at order zero.
For $1/2<\alpha<1$, the theorem forces $\|f\|\le 2$, which is well inside the univalence criterion $\|f\|\le 1$; bi-starlike order above $1/2$ is therefore a strong normalization.
For $0<\alpha<1/2$, the inverse-condition constant $4(1-\alpha)/\alpha$ can be smaller than the starlike-order bound $6-4\alpha$, so the bi-univalence hypothesis genuinely improves the estimate in this range.
The paper's Section 3 remarks imply that the analogous norm problem for the class $V_\Sigma(\alpha)$ remains open, since the missing estimate for $|f(z)/z|$ is not yet available.

Reading between the lines

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

A natural next step is to apply the same subordination-plus-Schwarz–Pick template to other bi-univalent subclasses whose inverse condition admits a subordination form, such as bi-convex or bi-spiral-like functions.
If the $\alpha>1/2$ constant $2$ is sharp, the extremal functions would sit at the boundary of the subordination half-plane; a coefficient-based search over $S^*_\Sigma(\alpha)$ could test sharpness.
The open $|f(z)/z|$ bound for $V_\Sigma(\alpha)$ suggests that growth or radius-of-univalence theorems for that class would be the most direct route to completing the estimate.

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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

Colleague,

The paper has one genuinely useful contribution and one load-bearing mistake. The useful part is the critique of Rahmatan et al.: the old Theorem B proof is indeed flawed, and the note that |f(z)/z| is not controlled is fair. The authors also catch that the earlier Theorem A proof could not be right, and they try a new route via the inverse condition.

The new route, however, does not work as written. The equivalence between (2.5) and (2.6) is asserted by substituting w=f(z). That only makes sense if f(Δ) lies inside Δ, which is not part of bi-univalence. The paper's own example f1(z)=z/(1-z) is bi-starlike of order 1/2 and maps the disk onto a half-plane, so the substitution is illegitimate. For that example (2.6) is actually false: it would require the function -z/(1-z) to be a Schwarz function, which it is not. So Theorem 2.1 is not proved.

There is also a smaller arithmetic slip: at α=1/2 the claimed φ(α) is not infinite; it is 2. Taken at face value the proof would predict ‖f‖≤2 for α=1/2, and their own example f1 has norm 4, so something has to give. The paper actually states the correct bound 4 for α=1/2, but the proof as written cannot deliver it.

One more thing: in the case 0<α<1/2 the "min" is redundant, because 6−4α is always the smaller term. So the inverse-condition improvement only matters, if at all, for α>1/2, and the argument there is the part that collapses.

I want to give credit where it is due. The paper is not a restatement of old results; the piecewise bound is a real attempt, and identifying the published errors is a service. The style is clear and the references look appropriate. But the central theorem is unproven, and the arithmetic slip shows the proof was not checked carefully.

Who is this for? Specialists in geometric function theory working on bi-univalent functions. The critique part could be published as a short note. As it stands, the paper is not acceptable. A serious referee could sort out whether the α>1/2 bound can be rescued, but the current text should not appear unchanged.

I'd send it to review rather than desk-reject, because the error analysis of the published results deserves to be sorted out. But my own verdict would be: major revision or reject.

Referee Report

2 major / 3 minor

Summary. The paper studies the norm of the pre-Schwarzian derivative for functions in the bi-starlike class S*_Sigma(alpha). It states Theorem 2.1 with a piecewise bound, claiming ||f|| ≤ 6 for alpha=0, min{6-4alpha, 4(1-alpha)/alpha} for 0<alpha<1/2, 4 for alpha=1/2, and 2 for 1/2<alpha<1. The proof uses Yamashita's estimate ||f|| ≤ 6-4alpha from the direct starlike condition and attempts to derive an additional estimate from the subordination (2.5) associated with the inverse condition, obtaining the bound phi(alpha)=4(1-alpha)/(1-|1-2alpha|). Section 3 gives remarks arguing that the prior proof of Theorem B by Rahmatan et al. used an invalid identity and states that estimating |f(z)/z| for V_Sigma(alpha) remains an open problem.

Significance. If Theorem 2.1 were correct, it would improve on the earlier estimate of Rahmatan et al. by combining the starlike-order bound of Yamashita with a bound obtained from the inverse condition; the claimed sharp values for alpha in (1/2,1) would be of interest to specialists in geometric function theory. However, the proof of the theorem has a fundamental gap, so the improvement is not established. The paper does make a useful critical observation about the earlier proof of Theorem B, and it is honest in stating that estimating |f(z)/z| for V_Sigma(alpha) remains open. No circular reasoning or computational artifacts are present; the failure is a mathematical error in a central step.

major comments (2)

[§2, Eqs. (2.5)–(2.6)] The equivalence between (2.5) and (2.6) is invalid. Substituting w=f(z) into (2.5) yields f(z)/(z f'(z)) = F(phi(f(z))), where phi is the Schwarz function from (2.5). For this to be a subordination in z, the map phi∘f must be a Schwarz function, which requires f(Delta)⊂Delta. The class S*_Sigma(alpha) does not imply this inclusion, and the authors' own example f1(z)=z/(1-z), which lies in S*_Sigma(1/2), maps Delta to {Re w>-1/2}, not into Delta. For this example, (2.5) holds for g1(w)=w/(1+w), but (2.6) would assert 1-z ≺ 1/(1-z), i.e. that phi(z)=-z/(1-z) is a Schwarz function, which is false. Consequently, the bound 4(1-alpha)/alpha and the cases alpha=1/2 and alpha>1/2 in Theorem 2.1 are not established; the only bound that follows from the proof is ||f||≤6-4alpha from (2.2).
[§2, Case 1 of proof of Theorem 2.1] The value of phi(alpha) at alpha=1/2 is computed incorrectly. The displayed formula gives phi(1/2)=4(1-1/2)/(1-|1-2*1/2|)=2, not infinity. The case analysis in Case 1 is therefore wrong, and the stated theorem's value 4 for alpha=1/2 is inconsistent with the proof's own formula. Moreover, the incorrect value is not harmless: f1(z)=z/(1-z) belongs to S*_Sigma(1/2) and has pre-Schwarzian norm sup_{|z|<1} 2(1-|z|^2)/|1-z| = 4, so no bound of 2 can hold for the class at alpha=1/2. This supports the conclusion that the derivation of (2.6)–(2.8) is not sound.

minor comments (3)

[§2, Case 2] In Case 2, the sentence '6 - 4alpha in (2,4) when alpha in (1/2,1)' is outside the case under consideration (0<alpha<1/2); for that range one has 6-4alpha in (4,6).
[§3, Eq. (3.4)] The displayed formula for f''(z)/f'(z) has ambiguous parentheses; the term should be written as (f(z)/z)( (1+(1-2alpha)phi(z))/(z(1-phi(z))) - 1 ).
[Abstract and throughout] There are many typographical errors, e.g. 'pre-Schwarzi an' in the abstract and 'i s' in §2; a careful proofreading is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Theorem 2.1 follows from Yamashita's external starlike bound plus Schwarz-Pick estimates, not from its own conclusion.

full rationale

The paper's central derivation is self-contained against external benchmarks. The starlike-order bound 6-4alpha is imported from Yamashita [7], an external theorem for the classical class S*(alpha), not from the bi-univalent class under study. The additional piecewise bounds are obtained by combining subordination from the inverse condition with the standard Schwarz-Pick lemma and the Schwarz lemma from Duren [3]; these are parameter-free external results. No fitted parameter is renamed as a prediction, no definition is framed in terms of the norm being estimated, and the only citations to prior work are to the original incorrect paper [6] and to standard textbooks. The mathematical gap identified by the reviewer—the asserted equivalence between (2.5) and (2.6) requires f(Delta) contained in Delta, which may fail—is a correctness issue in the proof, not a circularity issue: the bound is not assumed as an input but derived from subordination assumptions and external estimates. Accordingly, the paper contains no significant circular dependence, and the correct overall circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard tools listed, plus one ad hoc equivalence that is not justified. No free parameters or invented entities appear. The unproved equivalence is the main fragility.

assumptions (5)

domain assumption Definition 1.1: f and g=f^{-1} are both starlike of order alpha on Delta
This defines S*_Sigma(alpha) and implicitly requires g to be analytic and univalent on Delta, which is not guaranteed by the authors' examples. It is the domain on which the whole proof operates.
standard math Subordination criterion: Re h > alpha and h(0)=1 imply h is subordinate to (1+(1-2alpha)z)/(1-z)
Standard Caratheodory lemma, used at equations (2.4) and (2.5).
standard math Yamashita's norm bound: f in S*(alpha) implies ||f|| <= 6-4alpha
External result [7], used for the 6-4alpha part of the bound.
standard math Schwarz-Pick lemma: |phi'(z)| <= (1-|phi(z)|^2)/(1-|z|^2)
Used at equation (2.9) to bound the second term in (2.8).
ad hoc to paper Equivalence (2.5) implies (2.6): substituting w=f(z) preserves subordination
This is the load-bearing, unproved step; it requires f(Delta) to be contained in Delta or a new argument. The paper's own example f1(z)=z/(1-z) maps Delta outside Delta, so the step is not generally valid.

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Cite this review

Pith. "Pith review of Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$." pith.science (2026). https://pith.science/paper/PGEFVWAA

@misc{pith2026190801397,
  author       = {Pith},
  title        = {Pith review of: Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PGEFVWAA}},
  note         = {Machine review of arXiv:1908.01397}
}

abstract

In the present paper we estimate the norm of the pre-Schwarzian derivative of bi-starlike functions of order $\alpha$ where $\alpha\in[0,1)$. Initially this problem was handled by Rahmatan et al. in [Bull Iran Math Soc {\bf43}: 1037-1043, 2017]. We pointed out that the proofs and bounds by Rahmatan et al. are incorrect and present correct proofs and bounds.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Astala and F.W

K. Astala and F.W. Gehring, Injectivity, the BMO norm and the universal Teichm¨ uller sp ace, J. Anal. Math. 46 (1986), 16–57

work page 1986
[2]

Becker and Ch

J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete , J. Reine Angew. Math. 354 (1984), 74–94

work page 1984
[3]

Duren, Univalent Functions, Springer–Verlag, New York, 1983

P.L. Duren, Univalent Functions, Springer–Verlag, New York, 1983

work page 1983
[4]

Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mi t den schlichten Funktionen, Monatsh

H. Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mi t den schlichten Funktionen, Monatsh. Math. 73 (1969), 36–45

work page 1969
[5]

Obradovi´ c and S

M. Obradovi´ c and S. Ponnusamy, Radius of univalence of certain class of analytic functions , Filomat 27 (2013), 1085–1090

work page 2013
[6]

Rahmatan, Sh

H. Rahmatan, Sh. Najafzadeh and A. Ebadian, The norm of pre-Schwarzian derivatives on bi-univalent functions of order α , Bull. Iran. Math. Soc. 43 (2017), 1037–1043

work page 2017
[7]

Yamashita, Norm estimates for function starlike or convex of order alph a, Hokkaido Math

S. Yamashita, Norm estimates for function starlike or convex of order alph a, Hokkaido Math. J. 28 (1999), 217–230. Department of Mathematics, Islamic Azad University, West Teh ran Branch, Tehran, Iran E-mail address : hesammahzoon1@gmail.com Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ard- abil, Iran E-mail address : rkarga...

work page 1999

Pith tools

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

[1] [1]

Astala and F.W

K. Astala and F.W. Gehring, Injectivity, the BMO norm and the universal Teichm¨ uller sp ace, J. Anal. Math. 46 (1986), 16–57

work page 1986

[2] [2]

Becker and Ch

J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete , J. Reine Angew. Math. 354 (1984), 74–94

work page 1984

[3] [3]

Duren, Univalent Functions, Springer–Verlag, New York, 1983

P.L. Duren, Univalent Functions, Springer–Verlag, New York, 1983

work page 1983

[4] [4]

Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mi t den schlichten Funktionen, Monatsh

H. Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mi t den schlichten Funktionen, Monatsh. Math. 73 (1969), 36–45

work page 1969

[5] [5]

Obradovi´ c and S

M. Obradovi´ c and S. Ponnusamy, Radius of univalence of certain class of analytic functions , Filomat 27 (2013), 1085–1090

work page 2013

[6] [6]

Rahmatan, Sh

H. Rahmatan, Sh. Najafzadeh and A. Ebadian, The norm of pre-Schwarzian derivatives on bi-univalent functions of order α , Bull. Iran. Math. Soc. 43 (2017), 1037–1043

work page 2017

[7] [7]

Yamashita, Norm estimates for function starlike or convex of order alph a, Hokkaido Math

S. Yamashita, Norm estimates for function starlike or convex of order alph a, Hokkaido Math. J. 28 (1999), 217–230. Department of Mathematics, Islamic Azad University, West Teh ran Branch, Tehran, Iran E-mail address : hesammahzoon1@gmail.com Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ard- abil, Iran E-mail address : rkarga...

work page 1999