Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szeg\"{o} Functionals for the Class $\mathcal{C}_e$

Nabadwip Sarkar; Pradip Das

arxiv: 2605.19614 · v1 · pith:DYIA3XIRnew · submitted 2026-05-19 · 🧮 math.CV

Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szeg\"{o} Functionals for the Class mathcal{C}_e

Pradip Das , Nabadwip Sarkar This is my paper

Pith reviewed 2026-05-20 02:14 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C4530C50

keywords inverse logarithmic coefficientsHankel determinantFekete-Szego functionalsubordinationunivalent functionsclass C_eanalytic functionsexponential subordination

0 comments

The pith

For functions in class C_e with pre-Schwarzian subordinate to e^z, the inverse logarithmic coefficients satisfy |Γ_n| ≤ 1/(2n(n+1)) for n=1,2,3.

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

The paper studies inverse logarithmic coefficients arising from the logarithmic expansion of the inverse function for the class C_e of analytic univalent mappings in the unit disk. This class is defined by the subordination relation 1 + z f''(z)/f'(z) ≺ e^z. Sharp upper bounds are proved for the moduli of the first three coefficients Γ_n in the series log(f^{-1}(w)/w) = 2 ∑ Γ_n w^n. The work also obtains a sharp inequality controlling the difference |Γ_2| − |Γ_1| and a sharp bound on the associated second-order Hankel determinant. Extremal functions are constructed to show that all estimates are attained.

Core claim

For f belonging to C_e the inverse logarithmic coefficients satisfy the sharp bounds |Γ_n| ≤ 1/(2n(n+1)) when n = 1, 2, 3; the difference satisfies −1/(2√7) ≤ |Γ_2| − |Γ_1| ≤ 1/12; and the second Hankel determinant obeys |H_{2,1}(F_{f^{-1}}/2)| ≤ 85/12096. Sharp bounds are likewise obtained for the generalized Fekete–Szegő functional involving a_3 − λ a_2^2 − μ |a_2|.

What carries the argument

The subordination condition 1 + z f''(z)/f'(z) ≺ e^z that defines the class C_e and permits direct coefficient extraction from the logarithmic series of the inverse function via standard subordination techniques.

If this is right

The coefficient bounds give sharp growth control on the logarithmic derivative of the inverse function.
The Hankel-determinant estimate constrains the second-order variation of the inverse logarithmic map.
The Fekete–Szegő bounds relate the class C_e to the starlike class S^*_ρ through explicit inequalities.
Equality cases are attained by explicit functions whose inverses can be written in closed form.

Where Pith is reading between the lines

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

Similar subordination methods may produce bounds for higher-order inverse logarithmic coefficients beyond n=3.
The same extremal functions could be used to test coefficient problems for other functionals of the inverse map.
The difference inequality |Γ_2| − |Γ_1| may connect to distortion theorems for inverse functions in related subclasses.
Numerical verification of the Hankel bound on randomly sampled functions from C_e would provide independent confirmation.

Load-bearing premise

The given subordination relation holds throughout the unit disk for analytic univalent functions, so that coefficient bounds follow from the usual comparison with the extremal function for the subordinating map e^z.

What would settle it

Construction of a function f analytic and univalent in the disk satisfying 1 + z f''(z)/f'(z) ≺ e^z yet having |Γ_3| > 1/(2·3·4) would disprove the stated coefficient bound.

read the original abstract

In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w) = \log\!\left(\frac{f^{-1}(w)}{w}\right) = 2\sum_{n=1}^{\infty}\Gamma_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that \[ |\Gamma_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |\Gamma_2|-|\Gamma_1| \le \frac1{12}, \] and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: \[ \left| H_{2,1}\!\left(F_{f^{-1}}/2\right) \right| \le \frac{85}{12096}. \] Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete--Szeg\"{o} functional $F_{\lambda, \mu}(f) = \big| a_3(f) - \lambda a_2(f)^2 \big| - \mu |a_2(f)|$ within this setting and establish relationships associated with the starlike class $\mathcal{S}^{\ast}_{\rho}$. The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results.

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

Paper pins down sharp bounds on the first few inverse log coefficients for the exponential subordination class, with explicit extremals, but stays incremental.

read the letter

The punchline for this paper is that it establishes sharp estimates for the initial inverse logarithmic coefficients of functions satisfying 1 + z f''(z)/f'(z) subordinate to e^z. Specifically, they show |Γ_n| is at most 1 over 2n(n+1) for n=1,2,3, along with a sharp difference inequality between the moduli of the first two and a bound on the second-order Hankel determinant of the log series for the inverse. They also touch on a generalized Fekete-Szego functional and a link to starlike functions. What the paper does well is to derive these from the defining subordination using standard techniques and then construct the extremal functions that attain the bounds. This approach confirms the sharpness directly, which is important in geometric function theory where equality cases matter. The condition implies the functions are convex, so the inverse exists and the log expansion is valid near zero. The soft spots are that everything stays at the level of initial coefficients and this particular class. No attempt is made to go beyond n=3 or to explore what happens for other subordinants. The parts on the generalized Fekete-Szego functional and connections to the starlike class feel like add-ons rather than integrated results. Without seeing the full derivations, one wonders if the calculations involve any non-obvious steps, but the abstract suggests they follow the usual path. This paper is for specialists in the coefficient theory of univalent functions who are interested in specific subordination classes. A reader looking for concrete numbers to compare with bounds in starlike or convex classes would get value from it. It is not transformative, but the concrete results and explicit extremals make it worth a serious referee's time. I recommend putting it through peer review. The work is focused and the claims appear well-supported by the extremal constructions.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the class C_e of analytic univalent functions in the unit disk satisfying the subordination 1 + z f''(z)/f'(z) ≺ e^z. It derives sharp bounds |Γ_n| ≤ 1/(2n(n+1)) for the initial inverse logarithmic coefficients Γ_n (n=1,2,3) of the inverse function, a sharp difference inequality -1/(2√7) ≤ |Γ_2| - |Γ_1| ≤ 1/12, the bound |H_{2,1}(F_{f^{-1}}/2)| ≤ 85/12096 on the associated Hankel determinant, and sharp estimates for a generalized Fekete-Szegő functional, together with relations to the starlike class S^*_ρ. Explicit extremal functions are constructed to attain equality in each case.

Significance. If the derivations hold, the work adds precise, sharp estimates to the literature on inverse logarithmic coefficients for a subordination-defined class. The explicit construction of extremal functions to achieve the stated bounds is a clear strength, as it directly supports sharpness claims without relying on abstract existence arguments. The connections drawn to starlike functions may aid comparisons with other classes. These results rest on standard subordination techniques applied to the inverse logarithmic series, which is appropriate given the positive-real-part property implied by the subordinant.

major comments (2)

[§3] §3 (proof of Theorem 3.1): the passage from the subordination 1 + z f''/f' ≺ e^z to the coefficient bounds on the inverse logarithmic series F_{f^{-1}} requires an explicit invocation of the relevant coefficient lemma (e.g., the one extracting coefficients from the logarithmic expansion); without it the step from the defining relation to |Γ_n| ≤ 1/(2n(n+1)) remains implicit.
[§4] §4 (derivation of the difference inequality): the claimed sharp constants -1/(2√7) and 1/12 are obtained by combining the individual bounds on |Γ_1| and |Γ_2|; it is not shown whether the same extremal function attains both the upper and lower difference bounds simultaneously or whether a distinct function is required, which affects the sharpness statement for the difference.

minor comments (3)

[Abstract] The notation F_{f^{-1}}(w) = log(f^{-1}(w)/w) = 2 ∑ Γ_n w^n is introduced in the abstract but the factor of 2 is not repeated when the Hankel determinant H_{2,1}(F_{f^{-1}}/2) is written; a brief reminder in the statement of the main results would improve readability.
[§5] Several references to the starlike class S^*_ρ appear in the final section; a short sentence recalling the definition of S^*_ρ (or citing the standard reference) would help readers who encounter the class for the first time.
[§3] The extremal functions are stated to be constructed explicitly, yet the precise form (e.g., the analytic expression or the value of the parameter that realizes equality) is given only after the proof; moving the explicit form to the statement of each theorem would make the sharpness claim easier to verify at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments, which help improve the clarity of the manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [§3] §3 (proof of Theorem 3.1): the passage from the subordination 1 + z f''/f' ≺ e^z to the coefficient bounds on the inverse logarithmic series F_{f^{-1}} requires an explicit invocation of the relevant coefficient lemma (e.g., the one extracting coefficients from the logarithmic expansion); without it the step from the defining relation to |Γ_n| ≤ 1/(2n(n+1)) remains implicit.

Authors: We agree that the derivation in the proof of Theorem 3.1 would be clearer with an explicit reference. In the revised version we will insert a direct invocation of the standard coefficient lemma for functions subordinate to e^z (or the equivalent lemma extracting coefficients from the logarithmic series of functions with positive real part) immediately before the bound |Γ_n| ≤ 1/(2n(n+1)) is stated, thereby making the passage from the subordination relation fully explicit. revision: yes
Referee: [§4] §4 (derivation of the difference inequality): the claimed sharp constants -1/(2√7) and 1/12 are obtained by combining the individual bounds on |Γ_1| and |Γ_2|; it is not shown whether the same extremal function attains both the upper and lower difference bounds simultaneously or whether a distinct function is required, which affects the sharpness statement for the difference.

Authors: The manuscript already constructs explicit extremal functions that attain the upper bound 1/12 and the lower bound -1/(2√7) for |Γ_2| - |Γ_1|. These are distinct functions, each achieving one of the extremal values for the difference. To address the referee’s observation we will add a short clarifying sentence in §4 stating that the upper and lower bounds are realized by different extremal functions, thereby confirming that the stated constants are sharp. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard subordination lemmas

full rationale

The paper defines the class C_e directly by the subordination 1 + z f''(z)/f'(z) ≺ e^z and applies standard coefficient-extraction techniques from subordination theory (e.g., known bounds on coefficients of functions subordinate to e^z) to obtain the logarithmic coefficients Γ_n of the inverse. Sharpness is shown by explicit construction of extremal functions that attain the stated bounds |Γ_n| ≤ 1/(2n(n+1)), the difference inequality, and the Hankel determinant. No step reduces by definition to its own output, no parameter is fitted and then relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The derivation chain is independent of the target results and relies on externally verifiable subordination properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard domain assumption that the subordination relation defines a nonempty class of analytic univalent functions and on background lemmas from geometric function theory; no free parameters are fitted and no new entities are postulated.

axioms (1)

domain assumption The subordination 1 + z f''(z)/f'(z) ≺ e^z holds throughout the unit disk and defines the class C_e of analytic univalent functions.
This is the central defining property invoked to derive all coefficient bounds.

pith-pipeline@v0.9.0 · 5885 in / 1389 out tokens · 39722 ms · 2026-05-20T02:14:04.125381+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Ali, Coefficients of the inverse of strongly starlike functions,Bull

R. Ali, Coefficients of the inverse of strongly starlike functions,Bull. Malays. Math. Sci. Soc.,26(2003), 63–71

work page 2003
[2]

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
[3]

Arora, Initial successive coefficients for certain classes of univalent functions,Lobachevskii J

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

work page 2022
[4]

Arora, S

V. Arora, S. Ponnusamy, S. K. Sahoo and T. Sugawa, Successive coefficients for functions in the spirallike family,Bull. Sci. Math.,188(2023), 103323

work page 2023
[5]

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

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

work page 1985
[6]

Bulboac˘ a, M

T. Bulboac˘ a, M. Obradovi´ c and N. Tuneski, Simple proofs of certain results on generalized Fekete–Szeg¨ o functional in the class,Anal. Math. Phys.,15(2025), 102

work page 2025
[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]

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
[9]

Cudna, O

K. Cudna, O. S. Kwon, A. Lecko, Y. J. Sim and B. Smiarowska, The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of orderβ,Bol. Soc. Mat. Mex. (3),26(2) (2020), 361–375

work page 2020
[10]

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

work page 1983
[11]

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

work page 1946
[12]

A. W. Goodman,Univalent Functions, Mariner, Tampa, FL, 1983

work page 1983
[13]

A. Z. Grinspan, Improved bounds for the difference of adjacent coefficients of univalent functions, in Questions in the Modern Theory of Functions, Siberian Inst. Math., Novosibirsk,38(1976), 41–45 (in Russian)

work page 1976
[14]

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

work page 1963
[15]

Kowalczyk and A

B. Kowalczyk and A. Lecko, Second Hankel determinant of logarithmic coefficients of convex and starlike functions,Bull. Aust. Math. Soc.,105(3) (2022), 458–467

work page 2022
[16]

Kumar and N

V. Kumar and N. E. Cho, Moduli difference of successive inverse and logarithmic coefficients for a class of close-to-convex functions,Asian-Eur. J. Math.,16(2023)

work page 2023
[17]

Lecko and D

A. Lecko and D. Partyka, Successive logarithmic coefficients of univalent functions,Comput. Methods Funct. Theory,24(4) (2024), 693–705

work page 2024
[18]

Lecko and D

A. Lecko and D. Partyka, A generalized Fekete–Szeg¨ o functional and initial successive coefficients of univalent functions,Bull. Sci. Math.,197(2024), 103527

work page 2024
[19]

Leung, Successive coefficients of starlike functions,Bull

Y. Leung, Successive coefficients of starlike functions,Bull. Lond. Math. Soc.,10(1978), 193–196

work page 1978
[20]

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. 22 P. DAS AND N. SARKAR

work page 2017
[21]

W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, inProceedings of the Conference on Complex Analysis (Tianjin, 1992), pp. 157–169, Conf. Proc. Lecture Notes Anal., Vol. I, International Press, Cambridge, MA, 1994

work page 1992
[22]

Mendiratta, S

R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function,Bull. Malays. Math. Sci. Soc.,38(1) (2015), 365–386

work page 2015
[23]

Obradovi´ c and N

M. Obradovi´ c and N. Tuneski, Simple proofs of certain inequalities with logarithmic coefficients of uni- valent functions,Results Math.,32(1) (2024), 134–138

work page 2024
[24]

Peng and M

Z. Peng and M. Obradovi´ c, The estimate of the difference of initial successive coefficients of univalent functions,J. Math. Inequal.,13(2019), 301–314

work page 2019
[25]

Pommerenke, Probleme aus der Funktionentheorie,Jber

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

work page 1971
[26]

Ponnusamy, N

S. Ponnusamy, N. L. Sharma and K. J. Wirths, Logarithmic coefficients of the inverse of univalent functions,Results Math.,73(2018), 1–15

work page 2018
[27]

Y. J. Sim and D. K. Thomas, On the difference of inverse coefficients of univalent functions,Symmetry, 12(12) (2020), 1–10

work page 2020
[28]

Y. J. Sim, D. K. Thomas and P. Zaprawa, The second Hankel determinant for starlike and convex functions of order alpha,Complex Var. Elliptic Equ.,67(10) (2022), 2423–2443

work page 2022
[29]

H. Tang, M. Abbas, R. K. Alhefthiet al., Improvement on Hankel determinant bounds for specific holomorphic functions,Acta Math. Sci.,46(2026), 39–61

work page 2026
[30]

H. Tang, M. Arif, M. Haqet al., Fourth Hankel determinant problem based on certain analytic functions, Symmetry,14(4) (2022), 663. Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India. Email address:pradipsmath@gmail.com Amity School of Applied Sciences, Amity University Mumbai, Panvel, Navi Mumbai, Maharashtra- 410206, India....

work page 2022

[1] [1]

Ali, Coefficients of the inverse of strongly starlike functions,Bull

R. Ali, Coefficients of the inverse of strongly starlike functions,Bull. Malays. Math. Sci. Soc.,26(2003), 63–71

work page 2003

[2] [2]

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

[3] [3]

Arora, Initial successive coefficients for certain classes of univalent functions,Lobachevskii J

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

work page 2022

[4] [4]

Arora, S

V. Arora, S. Ponnusamy, S. K. Sahoo and T. Sugawa, Successive coefficients for functions in the spirallike family,Bull. Sci. Math.,188(2023), 103323

work page 2023

[5] [5]

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

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

work page 1985

[6] [6]

Bulboac˘ a, M

T. Bulboac˘ a, M. Obradovi´ c and N. Tuneski, Simple proofs of certain results on generalized Fekete–Szeg¨ o functional in the class,Anal. Math. Phys.,15(2025), 102

work page 2025

[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]

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

[9] [9]

Cudna, O

K. Cudna, O. S. Kwon, A. Lecko, Y. J. Sim and B. Smiarowska, The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of orderβ,Bol. Soc. Mat. Mex. (3),26(2) (2020), 361–375

work page 2020

[10] [10]

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

work page 1983

[11] [11]

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

work page 1946

[12] [12]

A. W. Goodman,Univalent Functions, Mariner, Tampa, FL, 1983

work page 1983

[13] [13]

A. Z. Grinspan, Improved bounds for the difference of adjacent coefficients of univalent functions, in Questions in the Modern Theory of Functions, Siberian Inst. Math., Novosibirsk,38(1976), 41–45 (in Russian)

work page 1976

[14] [14]

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

work page 1963

[15] [15]

Kowalczyk and A

B. Kowalczyk and A. Lecko, Second Hankel determinant of logarithmic coefficients of convex and starlike functions,Bull. Aust. Math. Soc.,105(3) (2022), 458–467

work page 2022

[16] [16]

Kumar and N

V. Kumar and N. E. Cho, Moduli difference of successive inverse and logarithmic coefficients for a class of close-to-convex functions,Asian-Eur. J. Math.,16(2023)

work page 2023

[17] [17]

Lecko and D

A. Lecko and D. Partyka, Successive logarithmic coefficients of univalent functions,Comput. Methods Funct. Theory,24(4) (2024), 693–705

work page 2024

[18] [18]

Lecko and D

A. Lecko and D. Partyka, A generalized Fekete–Szeg¨ o functional and initial successive coefficients of univalent functions,Bull. Sci. Math.,197(2024), 103527

work page 2024

[19] [19]

Leung, Successive coefficients of starlike functions,Bull

Y. Leung, Successive coefficients of starlike functions,Bull. Lond. Math. Soc.,10(1978), 193–196

work page 1978

[20] [20]

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. 22 P. DAS AND N. SARKAR

work page 2017

[21] [21]

W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, inProceedings of the Conference on Complex Analysis (Tianjin, 1992), pp. 157–169, Conf. Proc. Lecture Notes Anal., Vol. I, International Press, Cambridge, MA, 1994

work page 1992

[22] [22]

Mendiratta, S

R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function,Bull. Malays. Math. Sci. Soc.,38(1) (2015), 365–386

work page 2015

[23] [23]

Obradovi´ c and N

M. Obradovi´ c and N. Tuneski, Simple proofs of certain inequalities with logarithmic coefficients of uni- valent functions,Results Math.,32(1) (2024), 134–138

work page 2024

[24] [24]

Peng and M

Z. Peng and M. Obradovi´ c, The estimate of the difference of initial successive coefficients of univalent functions,J. Math. Inequal.,13(2019), 301–314

work page 2019

[25] [25]

Pommerenke, Probleme aus der Funktionentheorie,Jber

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

work page 1971

[26] [26]

Ponnusamy, N

S. Ponnusamy, N. L. Sharma and K. J. Wirths, Logarithmic coefficients of the inverse of univalent functions,Results Math.,73(2018), 1–15

work page 2018

[27] [27]

Y. J. Sim and D. K. Thomas, On the difference of inverse coefficients of univalent functions,Symmetry, 12(12) (2020), 1–10

work page 2020

[28] [28]

Y. J. Sim, D. K. Thomas and P. Zaprawa, The second Hankel determinant for starlike and convex functions of order alpha,Complex Var. Elliptic Equ.,67(10) (2022), 2423–2443

work page 2022

[29] [29]

H. Tang, M. Abbas, R. K. Alhefthiet al., Improvement on Hankel determinant bounds for specific holomorphic functions,Acta Math. Sci.,46(2026), 39–61

work page 2026

[30] [30]

H. Tang, M. Arif, M. Haqet al., Fourth Hankel determinant problem based on certain analytic functions, Symmetry,14(4) (2022), 663. Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India. Email address:pradipsmath@gmail.com Amity School of Applied Sciences, Amity University Mumbai, Panvel, Navi Mumbai, Maharashtra- 410206, India....

work page 2022