pith. sign in

arxiv: 2403.10031 · v1 · submitted 2024-03-15 · 🧮 math.CV

Moduli difference of inverse logarithmic coefficients of univalent functions

Vasudevarao Allu , Amal Shaji This is my paper

Pith reviewed 2026-05-24 03:55 UTC · model grok-4.3

classification 🧮 math.CV
keywords univalent functionsinverse logarithmic coefficientsmoduli differenceclass Ssharp boundssubclassesstarlike functionsconvex functions
0
0 comments X

The pith

Sharp upper and lower bounds hold for |Γ₂| − |Γ₁| of inverse logarithmic coefficients in the class S of univalent functions and its subclasses.

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

The paper defines the inverse logarithmic coefficients Γ_n of a normalized univalent function f through the relation log(F(w)/w) = 2 ∑ Γ_n w^n, where F is the local inverse of f. It then establishes the best possible constants that bound the modulus difference |Γ₂| − |Γ₁| from above and from below. The bounds are obtained both for the full class S and for standard subclasses such as the starlike and convex functions. A reader cares because these coefficients capture the initial distortion of the inverse mapping, which governs local invertibility and growth near the origin for conformal maps.

Core claim

For f in S the difference |Γ₂| − |Γ₁| satisfies explicit sharp inequalities; the same holds when f is restricted to important subclasses of univalent functions, with the inverse logarithmic coefficients defined by log(F(w)/w) = 2 ∑ Γ_n w^n on |w| < 1/4.

What carries the argument

The inverse logarithmic coefficients Γ_n extracted from the expansion log(F(w)/w) = 2 ∑ Γ_n w^n, used to bound the modulus difference |Γ₂| − |Γ₁| via growth and subordination estimates.

If this is right

  • The difference |Γ₂| − |Γ₁| is controlled uniformly across the entire class S.
  • The same modulus-difference bounds apply inside each listed subclass with possibly different extremal constants.
  • The estimates are attained by specific functions that serve as extremals in each class.
  • The bounds supply quantitative control on the initial coefficients of the inverse mapping.

Where Pith is reading between the lines

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

  • The same technique could be applied to obtain bounds on |Γ₃| − |Γ₂| or higher consecutive differences.
  • The results may extend to close-to-convex or typically real functions by analogous subordination arguments.
  • One could check whether the bounds remain valid when the univalent function is composed with a fixed Möbius transformation.

Load-bearing premise

The inverse function admits a power series around zero and the logarithmic expansion is valid on a disk of positive radius so that the coefficients Γ_n are well-defined.

What would settle it

Direct computation of |Γ₂| − |Γ₁| for the Koebe function or its rotations; if the value lies strictly outside the stated upper or lower bound, the claimed sharpness is refuted.

read the original abstract

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0) = 0$, and $f'(0) = 1$. Let $F$ be the inverse function of $f$, given by $F(w)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some disk $|w|\le r_0(f)$. The inverse logarithmic coefficients $\Gamma_n$, $n \in \mathbb{N}$, of $f$ are defined by the equation $ \log(F(w)/w)=2\sum_{n=1}^{\infty}\Gamma_{n}w^{n},\,|w|<1/4.$ In this paper, we find the sharp upper and lower bounds for moduli difference of second and first inverse logarithmic coefficients, {\em i.e.,} $|\Gamma_2|-|\Gamma_1|$ for functions in class $\mathcal{S}$ and for functions in some important subclasses of univalent 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.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to establish sharp upper and lower bounds on the modulus difference |Γ₂| − |Γ₁| of the inverse logarithmic coefficients defined via log(F(w)/w) = 2 ∑ Γ_n w^n (with F the inverse of f ∈ S) for the full class S and for several standard subclasses of univalent functions.

Significance. If the claimed sharpness can be rigorously verified, the results would supply new explicit estimates on joint coefficient behavior for inverse logarithmic coefficients, extending classical growth and subordination techniques in univalent function theory. The absence of free parameters or ad-hoc constants in the stated bounds would be a positive feature.

major comments (1)
  1. [Main theorems (e.g., Theorems 2.1–2.3 or equivalent statements for S and the subclasses)] The central claim of sharpness for |Γ₂| − |Γ₁| rests on the assumption that the individual extremal values of |Γ₂| and |Γ₁| are attained simultaneously. The manuscript must explicitly identify the extremal functions (or rotations thereof) and confirm that the difference bound is realized; otherwise the asserted sharpness reduces to a possibly non-sharp estimate obtained by subtracting separate modulus bounds.
minor comments (2)
  1. [Abstract] The abstract states the definition of Γ_n on |w| < 1/4 but does not indicate the analytic continuation or radius used in the coefficient extractions; a single clarifying sentence would improve readability.
  2. [Introduction / Section 1] Notation for the inverse function F and the radius r₀(f) is introduced without cross-reference to the Koebe 1/4 theorem; adding the standard citation would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting the need to rigorously confirm simultaneous attainment of the individual bounds. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Main theorems (e.g., Theorems 2.1–2.3 or equivalent statements for S and the subclasses)] The central claim of sharpness for |Γ₂| − |Γ₁| rests on the assumption that the individual extremal values of |Γ₂| and |Γ₁| are attained simultaneously. The manuscript must explicitly identify the extremal functions (or rotations thereof) and confirm that the difference bound is realized; otherwise the asserted sharpness reduces to a possibly non-sharp estimate obtained by subtracting separate modulus bounds.

    Authors: We agree that explicit identification of the extremal functions is required to substantiate the sharpness claim for the difference |Γ₂| − |Γ₁|. In the proofs for the class S and the listed subclasses, the bounds are attained by suitable rotations of the Koebe function k(z) = z/(1−z)² (and its rotations), for which both |Γ₁| and |Γ₂| simultaneously reach the values that realize the stated upper and lower bounds on the difference. We will add a dedicated remark (or subsection) in the revised manuscript that explicitly names these functions, verifies the simultaneous attainment, and confirms that the difference bounds are realized. This addresses the concern directly without altering the stated results. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the inverse logarithmic coefficients Γ_n via the standard expansion log(F(w)/w)=2∑Γ_n w^n on |w|<1/4, which follows from the Koebe 1/4 theorem and is an external convention in univalent function theory rather than a self-referential definition. Bounds on |Γ₂|−|Γ₁| are sought via growth or subordination arguments that are independent of the target quantity. No fitted parameters are renamed as predictions, no self-citation chains are load-bearing, and no ansatz or uniqueness result is smuggled in. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard definitions of univalent functions and their inverses without introducing new entities or free parameters visible in the abstract.

axioms (2)
  • standard math The inverse function F has the series expansion F(w) = w + sum A_n w^n on |w| ≤ r0(f)
    Standard for analytic inverses of normalized univalent functions.
  • domain assumption The log expansion log(F(w)/w) = 2 sum Γ_n w^n holds for |w|<1/4
    Given directly in the definition of inverse logarithmic coefficients.

pith-pipeline@v0.9.0 · 5708 in / 1373 out tokens · 33405 ms · 2026-05-24T03:55:50.284927+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

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

    work page 2017

  2. [2]

    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)

    work page 2022

  3. [3]

    M. F. Ali and A. V asudevarao, On Coefficient Estimates of Negative Powers and Inverse Coeffi- cients for Certain Starlike Functions, Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017), 449–462

    work page 2017

  4. [4]

    Arora , Initial Successive coefficients for certain classes of univalent fun ctions, Lobachevskii J Math 43 (2022), 2080–2091

    V. Arora , Initial Successive coefficients for certain classes of univalent fun ctions, Lobachevskii J Math 43 (2022), 2080–2091

    work page 2022

  5. [5]

    Arora, S

    V. Arora, S. Ponnusamy , and S. K. Sahoo , Successive coefficients for spirallike and related functions, Rev. R. Acad. Cienc. Exactas F ´ ıs. Nat. Ser. A Mat. 113(4) (2019), 2969–2979

    work page 2019

  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]

    Brannan and W.E

    D.A. Brannan and W.E. Kirwan, On some classes of bounded univalent functions, J. Lond. Math. Soc. 2(1) (1969), 431–443

    work page 1969

  8. [8]

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

    work page 1983

  9. [9]

    Fekete and G

    M. Fekete and G. Szeg ¨o, Eine Bemerkung ber ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85–89

    work page 1933

  10. [10]

    G. M. Goluzin, On distortion theorems and coefficients of univalent functions, Mat. Sb. 19 (61) (1946), 183–202 (in Russian)

    work page 1946

  11. [11]

    A. Z. Grinspan, Improved bounds for the difference of adjacent coefficients of un ivalent functions (Russian), Questions in the mordern theory of functions (Novosib irsk), Sib. Inst. Mat. 38 (1976), 41–45

    work page 1976

  12. [12]

    W. K. Hayman, On successive coefficients of univalent functions, J. London. Math. Soc. 38 (1963), 228–243

    work page 1963

  13. [13]

    Kumar and N.E

    V. Kumar and N.E. Cho, Moduli difference of successive inverse and logarithmic coefficients f or a class of close-to-convex functions, Asian-Eur. J. Math. 16 (2023). https://doi.org/10.1007/s11587- 021- 00682-1

    work page doi:10.1007/s11587- 2023

  14. [14]

    Lecko and D

    A. Lecko and D. Partyka, Successive Logarithmic Coefficients of Univalent Functions, Compu- tational Methods and Function Theory (2023), https://doi.org/10.1007/s40315-023-00500-9

    work page doi:10.1007/s40315-023-00500-9 2023

  15. [15]

    Leung, Successive coefficients of starlike functions, Bull

    Y. Leung, Successive coefficients of starlike functions, Bull. Lond. Math. Soc. 10 (1978), 193–196. 16 V ASUDEV ARAO ALLU AND AMAL SHAJI

    work page 1978

  16. [16]

    Li and T

    M. Li and T. Sugawa , A note on successive coefficients of convex functions, Comput. Methods Funct. Theory 17(2) (2017), 179–193

    work page 2017

  17. [17]

    R. J. Libera Univalent α-spiral functions, Canad. J. Math. 19 (1967) 249–456

    work page 1967

  18. [18]

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

    work page 1977

  19. [19]

    Obradovi ´c, S

    M. Obradovi ´c, S. Ponnusamy, and K.-J. Wirths, Disk of univalence of the ratio of two analytic functions, Complex Var. Elliptic Equ. 60(3) (2015), 392–404

    work page 2015

  20. [20]

    Obradovi ´c and N

    M. Obradovi ´c and N. Tuneski Simple proofs of certain inequalities with logarithmic coefficients of univalent functions submitted. arXiv:2311.09901

    work page arXiv

  21. [21]

    Ozaki, On the theory of multivalent functions

    S. Ozaki, On the theory of multivalent functions. II, Sci. Rep. Tokyo Bunrika Daigaku. Sect. A. 4 (1941) 45–87

    work page 1941

  22. [22]

    J. A. Pfaltzgraff, Univalence of the integral of f ′(z)λ, Bull. London Math. Soc. 7 (1975), 254–256

    work page 1975

  23. [23]

    J. A. Pfaltzgraff, M. O. Reade, and T. Umezawa, Sufficient conditions for univalence, Ann. Fac. Sci. Univ. Nat. Za ¨ ıre (Kinshasa) Sect. Math.-Phys. 2(2) (1976), 211–218

    work page 1976

  24. [24]

    Pommerenke , Probleme aus der Funktionentheorie, Jber

    Ch. Pommerenke , Probleme aus der Funktionentheorie, Jber. Deutsch. Math.-Verein. 73 (1971), 1–5

    work page 1971

  25. [25]

    Ponnusamy and S

    S. Ponnusamy and S. Rajasekaran, New sufficient conditions for starlike and univalent functions, Soochow J. Math. 21(2) (1995), 193–201

    work page 1995

  26. [26]

    Ponnusamy, S

    S. Ponnusamy, S. K. Sahoo, and H. Yanagihara, Radius of convexity of partial sums of func- tions in the close-to-convex family, Nonlinear Anal. 95 (2014), 219–228

    work page 2014

  27. [27]

    Ponnusamy , N.L

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

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

  28. [28]

    Ponnusamy and A

    S. Ponnusamy and A. V asudevarao, Region of variability of two subclasses of univalent functions, J. Math. Anal. Appl. 332(2) (2007), 1323–1334

    work page 2007

  29. [29]

    M. S. Robertson , Univalent functions f (z) for which zf ′(z) is spirallike, Michigan Math. J. 16 (1969), 97–101

    work page 1969

  30. [30]

    Y. J. Sim and D. K. Thomas , On the difference of inverse coefficients of univalent functions, Symmetry 12(12) (2020)

    work page 2020

  31. [31]

    Y. J. Sim and D. K. Thomas , A note on spirallike functions, Bull. Aust. Math. Soc. , https://doi.org/10.1017/S0004972721000198

    work page doi:10.1017/s0004972721000198

  32. [32]

    Singh and P

    V. Singh and P. N. Chichra , Univalent functions f (z) for which zf ′(z) is α-spirallike, Indian J. Pure Appl. Math. 8 (1977), 253–259

    work page 1977

  33. [33]

    ˇSpaˇcek Contributiona la theorie des fonctions univalentes, Casop Pest

    L. ˇSpaˇcek Contributiona la theorie des fonctions univalentes, Casop Pest. Mat.-Fys. 62 (1933) 12–19

    work page 1933

  34. [34]

    Stankiewicz , Quelques probl` emes extr´ emaux dans les classes des fonctions α-angulairement ´ etoil´ ees,Ann

    J. Stankiewicz , Quelques probl` emes extr´ emaux dans les classes des fonctions α-angulairement ´ etoil´ ees,Ann. Univ. Mariae Curie-Sk/suppress lodowska Sect. A.20 (1966), 59–75

    work page 1966

  35. [35]

    Stankiewicz , On a family of starlike functions, Ann

    J. Stankiewicz , On a family of starlike functions, Ann. Univ. Mariae Curie-Sk/suppress lodowska Sect. A. 22-24 (1968-1970), 175–181

    work page 1968

  36. [36]

    D. K. Thomas , On the logarithmic coefficients of close-to-convex functions, Proc. Amer. Math. Soc. 144 (2016), 1681–1687

    work page 2016

  37. [37]

    D. K. Thomas , N. Tuneski and A. V asudevarao Univalent Functions: A Primer , De Gruyter Studies in Mathematics 69, Berlin, Boston, (2018)

    work page 2018

  38. [38]

    Umezawa, Analytic functions convex in one direction, J

    T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 195–202. V asudevarao Allu, School of Basic Sciences, Indian Institu te of Technology Bhubaneswar, Odisha, India. Email address : avrao@iitbbs.ac.in Amal Shaji, School of Basic Sciences, Indian Institute of Te chnology Bhubaneswar, Bhubaneswar-752050, Odisha, India. Emai...

    work page 1952