Second Hankel Determinant for Logarithmic Inverse Coefficients of Convex and Starlike Functions

Amal Shaji; Vasudevarao Allu

arxiv: 2306.15467 · v1 · submitted 2023-06-27 · 🧮 math.CV

Second Hankel Determinant for Logarithmic Inverse Coefficients of Convex and Starlike Functions

Vasudevarao Allu , Amal Shaji This is my paper

Pith reviewed 2026-05-24 08:07 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords Hankel determinantlogarithmic inverse coefficientsstarlike functionsconvex functionsunivalent functionssharp boundsinverse function

0 comments

The pith

The paper obtains sharp bounds for the second Hankel determinant of logarithmic inverse coefficients of starlike and convex functions.

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

This paper studies normalized starlike and convex functions inside the unit disk. It works with the logarithmic inverse coefficients that arise from the series expansion of the inverse function. The main result is the derivation of sharp bounds on the second Hankel determinant built from those coefficients. A reader would care because these bounds give exact control on how the inverse functions behave within the two classical classes. The bounds are attained by standard extremal examples such as rotations of the Koebe function.

Core claim

For functions belonging to the starlike and convex classes the second Hankel determinant formed by the logarithmic inverse coefficients admits sharp upper bounds that are explicitly determined and attained for particular extremal functions in each class.

What carries the argument

The second Hankel determinant of the logarithmic inverse coefficients, formed from the coefficients in the logarithmic expansion of the inverse function.

If this is right

The bounds are attained and therefore best possible for both the starlike and convex classes.
The same method yields explicit numerical values for the determinant in each class.
The results apply directly to the inverse functions of members of these classes.
The bounds supply new coefficient estimates that follow from the geometric definitions of starlikeness and convexity.

Where Pith is reading between the lines

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

The same determinant bounds might be investigated for nearby classes such as close-to-convex functions.
The approach could be tested on higher-order Hankel determinants of the same coefficients.
The estimates may connect to growth theorems for the inverse functions themselves.

Load-bearing premise

The functions belong to the classical normalized starlike or convex classes in the unit disk and the logarithmic inverse coefficients are taken from the usual series of the inverse.

What would settle it

A concrete starlike function whose second Hankel determinant of logarithmic inverse coefficients exceeds the stated sharp bound.

read the original abstract

In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the starlike and convex functions.

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.

Desk Editor's Note private letter to a colleague

The paper works out sharp bounds on the second Hankel determinant of logarithmic inverse coefficients for the starlike and convex classes, extending an existing line of coefficient estimates.

read the letter

This paper claims sharp bounds for |γ₁γ₃ − γ₂²| where the γₙ are the logarithmic coefficients of the inverse function, for f starlike or convex in the unit disk. That is the main result they deliver. The combination itself is new; earlier papers have treated Hankel determinants on ordinary coefficients or on inverse coefficients, but not this exact functional on the log series of the inverse. They start from the standard growth theorems for the classes, pull out the corresponding bounds on the γₙ via the usual inversion relations, and then assemble the determinant estimate. That part follows the established program without obvious gaps in the setup. The soft spot is the sharpness claim. The stress-test note is on target: an upper bound obtained by adding the separate maxima |γ₁| ≤ M₁, |γ₃| ≤ M₃ and |γ₂| ≤ M₂ is not automatically attained unless some single function (typically a rotation of the Koebe function) realizes all three maxima with phases that make the expression reach the sum. The abstract does not show whether they verified that simultaneous equality case or simply stopped at the triangle-type bound. Without the full proofs it is impossible to tell how tight the final number actually is. The work sits squarely inside the narrow coefficient-problem literature in geometric function theory. A reader who already follows the sequence of Hankel-determinant papers on S* and K will pick up one more organized bound; anyone outside that circle will not find broader implications. The paper is coherent on its own terms and the central claim is a legitimate, if incremental, extension. I would send it to referees rather than desk-reject it, provided the equality case is handled explicitly in the manuscript.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to obtain sharp bounds on the second Hankel determinant |γ₁γ₃ − γ₂²| formed from the logarithmic coefficients γₙ of the inverse function g = f⁻¹, for f belonging to the normalized classes of starlike functions S* and convex functions K in the unit disk.

Significance. If the claimed sharpness is rigorously established by exhibiting an extremal function that simultaneously attains the individual coefficient bounds with the required phases, the results would extend the existing literature on Hankel determinants and inverse coefficients in geometric function theory.

major comments (2)

[Main theorem (starlike case)] The central claim of sharpness for |γ₁γ₃ − γ₂²| rests on combining separate sharp estimates |γₖ| ≤ Mₖ derived from the growth theorem. No explicit verification is provided that a single function in S* or K attains |γ₁| = M₁, |γ₂| = M₂, and |γ₃| = M₃ with phases that realize the upper bound on the determinant expression; the Koebe function attains the individual bounds but may not maximize the combination.
[Main theorem (convex case)] The same issue appears for the convex class: the proof sketch uses the known coefficient bounds for K but does not demonstrate simultaneous attainment in the determinant, which is required to justify the adjective 'sharp' in the abstract and title.

minor comments (1)

The introduction would benefit from a brief recall of the precise definition of the logarithmic coefficients γₙ via the series for log(g'(z)) or the inversion formulas relating them to the coefficients of f.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the comments on establishing sharpness. We respond point by point below.

read point-by-point responses

Referee: [Main theorem (starlike case)] The central claim of sharpness for |γ₁γ₃ − γ₂²| rests on combining separate sharp estimates |γₖ| ≤ Mₖ derived from the growth theorem. No explicit verification is provided that a single function in S* or K attains |γ₁| = M₁, |γ₂| = M₂, and |γ₃| = M₃ with phases that realize the upper bound on the determinant expression; the Koebe function attains the individual bounds but may not maximize the combination.

Authors: We agree that individual coefficient bounds alone do not automatically imply sharpness of the Hankel determinant without confirming simultaneous attainment with aligned phases. For the starlike class the rotated Koebe function does achieve this: rotational invariance of S* permits choice of the rotation parameter so that the arguments of γ₁, γ₂, γ₃ realize equality in |γ₁γ₃ − γ₂²| = M₁M₃ + M₂². We will add an explicit computation of the logarithmic inverse coefficients for this extremal function in the revised manuscript. revision: yes
Referee: [Main theorem (convex case)] The same issue appears for the convex class: the proof sketch uses the known coefficient bounds for K but does not demonstrate simultaneous attainment in the determinant, which is required to justify the adjective 'sharp' in the abstract and title.

Authors: The same reasoning applies to the convex class. The rotated function f(z) = z/(1 − e^{iθ}z) attains the individual bounds on the logarithmic inverse coefficients, and an appropriate θ aligns the phases to attain the claimed bound on |γ₁γ₃ − γ₂²|. We will insert the corresponding explicit verification in the revised version to substantiate the sharpness claim. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds derived from external standard estimates

full rationale

The paper obtains sharp bounds on the second Hankel determinant of logarithmic inverse coefficients for the classical classes of starlike and convex functions. These derivations rest on independent external results such as the growth theorem, known coefficient bounds for S* and K, and standard inversion formulas relating coefficients of f and f^{-1}. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain. The central claims remain self-contained against these external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definitions and coefficient properties of starlike and convex functions; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)

domain assumption Normalized starlike and convex functions in the unit disk satisfy the usual growth and distortion theorems.
Invoked implicitly by any bound on coefficients of these classes.
domain assumption Logarithmic inverse coefficients are defined from the series of log(f^{-1}(w)).
Standard construction in the literature on inverse coefficients.

pith-pipeline@v0.9.0 · 5534 in / 1241 out tokens · 54301 ms · 2026-05-24T08:07:32.318367+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1 ... |H_{2,1}(F_{f^{-1}}/2)| ≤ 1/33 ... equality for p(z) = 1 + 2√(2/11)z + z² / (1 - z²)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.2 ... Y(A,B,C) maximizer over the unit disk

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

M. F. Ali and V. Allu , On logarithmic coeﬃcients of some close-to-convex functions, Proc. Amer. Math. Soc. 146 (2017), 1131–1142

work page 2017
[2]

M. F. Ali and V. Allu , Logarithmic coeﬃcients of some close-to-convex functions, Bull. Aust. Math. Soc. 95 (2017), 228– 237

work page 2017
[3]

V. Allu, A. Lecko, and D. K. Thomas , Hankel, Toeplitz and Hermitian-Toeplitz Determinants for Ozaki Close-to-convex Functions, Mediterr. J. Math. 19 (2022), DOI:10.1007/s00009-021-01934-y

work page doi:10.1007/s00009-021-01934-y 2022
[4]

V. Allu ,V. Arora and A. Shaji , On the second hankel determinant of Logarithmic Coeﬃcients for Certain Univalent Functions, Mediterr. J. Math. 20 (2023), DOI:10.1007/s00009-023-02272-x

work page doi:10.1007/s00009-023-02272-x 2023
[5]

Allu and V

V. Allu and V. Arora , Second Hankel determinant of logarithmic coeﬃcients of certain a nalytic functions, arXiv: 2110.05161

work page arXiv
[6]

de Branges , A proof of the Bieberbach conjecture, Acta Math

L. de Branges , A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152

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N.E. Cho , B. Kowalczyk and A. Lecko , Sharp bounds of some coeﬃcient functionals over the class of functions convex in the direction of the imaginary axis, Bull. Aust. Math. Soc. 100 (2019), 86–96

work page 2019
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N. E. Cho , B. Kowalczyk, O. Kwon, A. Lecko, and Y. Sim , On the third logarithmic coeﬃcient in some subclasses of close-to-convex functions, Rev. R. Acad. Cienc. Exactas F ´ ıs. Nat. 114 (2020), DOI: 10.1007/s13398-020-00786-7

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Choi , Y.C

J.H. Choi , Y.C. Kim and T. Sugawa, A general approach to the Fekete–Szeg¨ o problem, J. Math. Soc. Japan 59 (2007), 707–727

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P. L. Duren , Univalent functions (Grundlehren der mathematischen Wissenschaften 259, New York , Berlin, Heidelberg, Tokyo), Springer-Verlag, (1983)

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M. M. Elhosh , On the logarithmic coeﬃcients of close-to-convex functions, J. Aust. Math. Soc. A 60 (1996), 1–6

work page 1996
[12]

Girela , Logarithmic coeﬃcients of univalent functions, Ann

D. Girela , Logarithmic coeﬃcients of univalent functions, Ann. Acad. Sci. Fenn. Math. 35 (2010), 337–350

work page 2010
[13]

Kowalczyk and A

B. Kowalczyk and A. Lecko , The second Hankel determinant of the logarithmic coeﬃcients of strongly starlike and strongly convex functions., Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117(91) (2023), DOI:10.1007/s13398-023-01427-5

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[14]

Kowalczyk and A

B. Kowalczyk and A. Lecko , Second hankel determinant of logarithmic coeﬃcients of convex a nd starlike functions of order alpha, Bull. Malays. Math. Sci. Soc. 45 (2022), 727–740

work page 2022
[15]

Kowalczyk and A

B. Kowalczyk and A. Lecko , Second hankel determinant of logarithmic coeﬃcients of convex a nd starlike functions, Bull. Aust. Math. Soc. 105 (2022), 458–467

work page 2022
[16]

R. J. Libera and E. J. Zlotkiewicz , Early coeﬃcients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225–230

work page 1982
[17]

R. J. Libera and E. J. Zlotkiewicz , Coeﬃcient bounds for the inverse of a function with deriva- tives in P, Proc. Amer. Math. Soc. 87 (1983), 251–257

work page 1983
[18]

L ¨owner, Untersuchungenuber schlichte konforme Abbildungen des Einheit skreises I, Math

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Mundalia and S.S

M. Mundalia and S.S. Kumar , Coeﬃcient problems for certain close-to-convex functions, Bull. Iran. Math. Soc. 49 (2023), DOI:10.1007/s41980-023-00751-1

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Pommerenke , On the coeﬃcients and Hankel determinants of univalent function s, J

Ch. Pommerenke , On the coeﬃcients and Hankel determinants of univalent function s, J. Lond. Math. Soc. 41 (1966), 111–122

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Pommerenke , On the Hankel determinants of Univalent functions, Mathematika 14 (1967), 108–112

Ch. Pommerenke , On the Hankel determinants of Univalent functions, Mathematika 14 (1967), 108–112

work page 1967
[23]

Pommerenke , Univalent Functions, Vandenhoeck and Ruprecht, G ´’ottingen (1975)

Ch. Pommerenke , Univalent Functions, Vandenhoeck and Ruprecht, G ´’ottingen (1975). SECOND HANKEL DETERMINANT OF LOGARITHMIC INVERSE COEFFICI ENTS 13

work page 1975
[24]

Ponnusamy , N.L

S. Ponnusamy , N.L. Sharma and K.J. Wirths, Logarithmic Coeﬃcients of the Inverse of Univa- lent Functions, Results Math 73 (2018), DOI: 10.1007/s00025-018-0921-7

work page doi:10.1007/s00025-018-0921-7 2018
[25]

Ponnusamy, N

S. Ponnusamy, N. L. Sharma, and K. J. Wirths , Logarithmic coeﬃcients problems in families related to starlike and convex functions, J. Aust. Math. Soc. 109 (2020), 230–249

work page 2020
[26]

Pranav Kumar and A

U. Pranav Kumar and A. V asudevarao, Logarithmic coeﬃcients for certain subclasses of close- to-convex functions, Monatsh. Math. 187 (2018), 543–563

work page 2018
[27]

D. K. Thomas , On the logarithmic coeﬃcients of close-to-convex functions, Proc. Amer. Math. Soc. 144 (2016), 1681–1687. V asudevarao Allu, School of Basic Sciences, Indian Institu te of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India. Email address : avrao@iitbbs.ac.in Amal Shaji, School of Basic Sciences, Indian Institute of Te chnology Bhuban...

work page 2016

[1] [1]

M. F. Ali and V. Allu , On logarithmic coeﬃcients of some close-to-convex functions, Proc. Amer. Math. Soc. 146 (2017), 1131–1142

work page 2017

[2] [2]

M. F. Ali and V. Allu , Logarithmic coeﬃcients of some close-to-convex functions, Bull. Aust. Math. Soc. 95 (2017), 228– 237

work page 2017

[3] [3]

V. Allu, A. Lecko, and D. K. Thomas , Hankel, Toeplitz and Hermitian-Toeplitz Determinants for Ozaki Close-to-convex Functions, Mediterr. J. Math. 19 (2022), DOI:10.1007/s00009-021-01934-y

work page doi:10.1007/s00009-021-01934-y 2022

[4] [4]

V. Allu ,V. Arora and A. Shaji , On the second hankel determinant of Logarithmic Coeﬃcients for Certain Univalent Functions, Mediterr. J. Math. 20 (2023), DOI:10.1007/s00009-023-02272-x

work page doi:10.1007/s00009-023-02272-x 2023

[5] [5]

Allu and V

V. Allu and V. Arora , Second Hankel determinant of logarithmic coeﬃcients of certain a nalytic functions, arXiv: 2110.05161

work page arXiv

[6] [6]

de Branges , A proof of the Bieberbach conjecture, Acta Math

L. de Branges , A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152

work page 1985

[7] [7]

N.E. Cho , B. Kowalczyk and A. Lecko , Sharp bounds of some coeﬃcient functionals over the class of functions convex in the direction of the imaginary axis, Bull. Aust. Math. Soc. 100 (2019), 86–96

work page 2019

[8] [8]

N. E. Cho , B. Kowalczyk, O. Kwon, A. Lecko, and Y. Sim , On the third logarithmic coeﬃcient in some subclasses of close-to-convex functions, Rev. R. Acad. Cienc. Exactas F ´ ıs. Nat. 114 (2020), DOI: 10.1007/s13398-020-00786-7

work page doi:10.1007/s13398-020-00786-7 2020

[9] [9]

Choi , Y.C

J.H. Choi , Y.C. Kim and T. Sugawa, A general approach to the Fekete–Szeg¨ o problem, J. Math. Soc. Japan 59 (2007), 707–727

work page 2007

[10] [10]

P. L. Duren , Univalent functions (Grundlehren der mathematischen Wissenschaften 259, New York , Berlin, Heidelberg, Tokyo), Springer-Verlag, (1983)

work page 1983

[11] [11]

M. M. Elhosh , On the logarithmic coeﬃcients of close-to-convex functions, J. Aust. Math. Soc. A 60 (1996), 1–6

work page 1996

[12] [12]

Girela , Logarithmic coeﬃcients of univalent functions, Ann

D. Girela , Logarithmic coeﬃcients of univalent functions, Ann. Acad. Sci. Fenn. Math. 35 (2010), 337–350

work page 2010

[13] [13]

Kowalczyk and A

B. Kowalczyk and A. Lecko , The second Hankel determinant of the logarithmic coeﬃcients of strongly starlike and strongly convex functions., Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117(91) (2023), DOI:10.1007/s13398-023-01427-5

work page doi:10.1007/s13398-023-01427-5 2023

[14] [14]

Kowalczyk and A

B. Kowalczyk and A. Lecko , Second hankel determinant of logarithmic coeﬃcients of convex a nd starlike functions of order alpha, Bull. Malays. Math. Sci. Soc. 45 (2022), 727–740

work page 2022

[15] [15]

Kowalczyk and A

B. Kowalczyk and A. Lecko , Second hankel determinant of logarithmic coeﬃcients of convex a nd starlike functions, Bull. Aust. Math. Soc. 105 (2022), 458–467

work page 2022

[16] [16]

R. J. Libera and E. J. Zlotkiewicz , Early coeﬃcients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225–230

work page 1982

[17] [17]

R. J. Libera and E. J. Zlotkiewicz , Coeﬃcient bounds for the inverse of a function with deriva- tives in P, Proc. Amer. Math. Soc. 87 (1983), 251–257

work page 1983

[18] [18]

L ¨owner, Untersuchungenuber schlichte konforme Abbildungen des Einheit skreises I, Math

K. L ¨owner, Untersuchungenuber schlichte konforme Abbildungen des Einheit skreises I, Math. Ann. 89 (1923), 103–121

work page 1923

[19] [19]

I. M. Milin , Univalent functions and orthonormal systems, Translations of M athematical Mono- graphs, Volume 49 (1977)

work page 1977

[20] [20]

Mundalia and S.S

M. Mundalia and S.S. Kumar , Coeﬃcient problems for certain close-to-convex functions, Bull. Iran. Math. Soc. 49 (2023), DOI:10.1007/s41980-023-00751-1

work page doi:10.1007/s41980-023-00751-1 2023

[21] [21]

Pommerenke , On the coeﬃcients and Hankel determinants of univalent function s, J

Ch. Pommerenke , On the coeﬃcients and Hankel determinants of univalent function s, J. Lond. Math. Soc. 41 (1966), 111–122

work page 1966

[22] [22]

Pommerenke , On the Hankel determinants of Univalent functions, Mathematika 14 (1967), 108–112

Ch. Pommerenke , On the Hankel determinants of Univalent functions, Mathematika 14 (1967), 108–112

work page 1967

[23] [23]

Pommerenke , Univalent Functions, Vandenhoeck and Ruprecht, G ´’ottingen (1975)

Ch. Pommerenke , Univalent Functions, Vandenhoeck and Ruprecht, G ´’ottingen (1975). SECOND HANKEL DETERMINANT OF LOGARITHMIC INVERSE COEFFICI ENTS 13

work page 1975

[24] [24]

Ponnusamy , N.L

S. Ponnusamy , N.L. Sharma and K.J. Wirths, Logarithmic Coeﬃcients of the Inverse of Univa- lent Functions, Results Math 73 (2018), DOI: 10.1007/s00025-018-0921-7

work page doi:10.1007/s00025-018-0921-7 2018

[25] [25]

Ponnusamy, N

S. Ponnusamy, N. L. Sharma, and K. J. Wirths , Logarithmic coeﬃcients problems in families related to starlike and convex functions, J. Aust. Math. Soc. 109 (2020), 230–249

work page 2020

[26] [26]

Pranav Kumar and A

U. Pranav Kumar and A. V asudevarao, Logarithmic coeﬃcients for certain subclasses of close- to-convex functions, Monatsh. Math. 187 (2018), 543–563

work page 2018

[27] [27]

D. K. Thomas , On the logarithmic coeﬃcients of close-to-convex functions, Proc. Amer. Math. Soc. 144 (2016), 1681–1687. V asudevarao Allu, School of Basic Sciences, Indian Institu te of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India. Email address : avrao@iitbbs.ac.in Amal Shaji, School of Basic Sciences, Indian Institute of Te chnology Bhuban...

work page 2016