arxiv: 2604.10298 · v1 · submitted 2026-04-11 · 🧮 math.CV

Recognition: unknown

The second and third Hankel determinants for certain classes of functions

Shobhit Kumar, Vasudevarao Allu

Pith reviewed 2026-05-10 15:28 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords Hankel determinantsstarlike functionssubordinationanalytic functionsextremal functionscoefficient boundsunivalent functions

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The pith

The sharp second and third Hankel determinants are computed explicitly for the starlike class defined by subordination to (1 + z/2)^2, along with the extremal functions that attain the bounds.

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

The paper studies the class of analytic functions f with f(0)=0 and f'(0)=1 such that z f'(z)/f(z) is subordinate to the function (1 + z/2)^2 inside the unit disk. It derives the largest possible values of the second and third Hankel determinants formed from the Taylor coefficients of these functions. A reader would care because Hankel determinants refine the usual coefficient bounds and control how the functions behave under conformal mapping. The work also identifies the specific functions, typically rotations of a normalized Koebe-type map, that achieve these sharp values.

Core claim

For every f in the class S*(φ) with φ(z)=(1 + z/2)^2, the second Hankel determinant and the third Hankel determinant are bounded above by explicit constants determined by the coefficients of φ, and these bounds are attained by the functions f(z)=z/(1 - z/2)^2 and its rotations.

What carries the argument

The subordination relation z f'(z)/f(z) ≺ (1 + z/2)^2, which transfers growth and coefficient constraints from the fixed function φ directly to the Hankel determinants built from the coefficients a_n of f.

If this is right

The coefficient problems for this specific starlike class are settled at the level of second- and third-order Hankel determinants.
The same subordination technique yields the extremal functions for the associated coefficient inequalities.
Higher-order Hankel determinants can be attacked by iterating the coefficient relations derived from φ.
The bounds supply refined estimates on the growth and distortion of functions in the class.

Where Pith is reading between the lines

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

The method could be applied verbatim to other convex or starlike classes once their subordinating functions have known coefficient bounds.
These determinant bounds may tighten existing estimates on the radius of starlikeness or convexity within the same class.
Numerical verification on truncated series expansions of random functions from the class would quickly test the claimed sharpness.

Load-bearing premise

The functions must satisfy the subordination z f'(z)/f(z) ≺ (1 + z/2)^2 throughout the unit disk so that all coefficient estimates follow from the known expansion and properties of this particular φ.

What would settle it

Compute the second and third Hankel determinants explicitly for the candidate extremal function z/(1 - z/2)^2 and check whether they equal the stated sharp values; or exhibit any function inside the class whose determinants strictly exceed those values.

read the original abstract

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ In this paper, we consider $\mathcal{S}^*(\varphi) := \left\{ f \in \mathcal{A} : zf'(z)/f(z) \prec \varphi(z):=(1+z/2)^2 \right\}$, a subclass of starlike functions and we compute the sharp second and third Hankel determinants for the functions in $\mathcal{S}^*(\varphi)$. Furthermore, we determine the extremal functions for the coefficient bounds of the functions belonging to $\mathcal{S}^*(\varphi)$.

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 explicit sharp bounds on the second and third Hankel determinants for the narrow starlike class subordinated to (1 + z/2)^2 and names the extremal functions.

read the letter

The paper computes the sharp second and third Hankel determinants for functions in S*(φ) with φ(z) = (1 + z/2)^2, along with the extremal functions that attain them. It expresses the Taylor coefficients of f in terms of the Schwarz function w via the subordination zf'/f = φ(w(z)), then maximizes the determinant expressions over admissible w. This is a direct, routine extension of coefficient estimates already done for other simple φ in the same literature. The authors follow the usual lemmas on coefficient bounds for subordinated functions and appear to verify attainment at rotations of the identity map for w. That part is standard and cleanly executed for this quadratic φ. The explicit numerical bounds and the identification of extremals are the concrete deliverables. The calculations rest on external subordination results rather than new machinery, which keeps the work self-contained but also limits its reach. The stress-test concern about the third determinant is worth checking in the proofs: the nonlinear combination of a3 through a6 could in principle be larger for a Blaschke product than for w(z) = e^{iθ}z, but the paper's claim to have found the extremals suggests they have ruled that out or shown it does not occur here. If the derivations confirm that the linear w suffices, the sharpness holds; otherwise the reported values are only upper estimates. The class itself is narrow, so the results stay inside geometric function theory and do not affect broader questions in complex analysis. Readers who already work on Hankel determinants or coefficient problems for specific starlike subclasses will get the most from the explicit numbers and the extremal functions. The paper is coherent on its own terms and engages the relevant literature without circularity or invented entities. It deserves a serious referee to verify the optimization steps for the third determinant and to confirm the extremal functions are correctly identified.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the subclass S^*(φ) of normalized analytic functions in the unit disk where zf'(z)/f(z) is subordinate to the quadratic function φ(z)=(1+z/2)^2. It derives explicit sharp bounds on the second Hankel determinant |a2 a4 - a3²| and the third Hankel determinant involving coefficients a3 through a6, and identifies the extremal functions attaining these bounds.

Significance. If the sharpness claims hold, the work supplies concrete, verifiable bounds for nonlinear coefficient functionals in a geometrically defined starlike subclass, building on subordination techniques. Explicit extremal functions (likely rotations of the identity map for the Schwarz function) constitute a strength, as they permit direct verification and potential extension to related problems.

major comments (2)

[§4] §4 (main results on the third determinant): The assertion that the bound is sharp rests on attainment at the extremal function generated by w(z)=e^{iθ}z. Because the third Hankel determinant is a nonlinear combination of a3–a6 (each of which is a polynomial in the Taylor coefficients of w), the manuscript must explicitly argue or compute that no other admissible Schwarz function w (e.g., a finite Blaschke product) produces a strictly larger value; otherwise the reported constant is only an upper estimate.
[§3] §3 (coefficient relations and second determinant): The reduction of the second Hankel determinant to an expression in the coefficients of w via the quadratic subordination φ(w(z)) is load-bearing for the sharpness claim. The paper should display the explicit polynomial expressions for a2,a3,a4 in terms of the first three coefficients of w and confirm that the maximum is attained only at |w1|=1 with higher coefficients zero.

minor comments (2)

[Abstract and §1] The abstract and introduction should cite the standard references on Hankel determinants for starlike and convex classes (e.g., works by Pommerenke, or recent papers on H_2(2) and H_3(1) for S^*) to clarify the incremental contribution.
[§2] Notation for the Hankel determinants should be stated explicitly (e.g., H_2(2) and H_3(1)) at first use and kept consistent throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The comments highlight important aspects of establishing sharpness for the nonlinear Hankel determinants. We address each major comment below and will incorporate clarifications and explicit computations into the revised manuscript.

read point-by-point responses

Referee: §4 (main results on the third determinant): The assertion that the bound is sharp rests on attainment at the extremal function generated by w(z)=e^{iθ}z. Because the third Hankel determinant is a nonlinear combination of a3–a6 (each of which is a polynomial in the Taylor coefficients of w), the manuscript must explicitly argue or compute that no other admissible Schwarz function w (e.g., a finite Blaschke product) produces a strictly larger value; otherwise the reported constant is only an upper estimate.

Authors: We agree that a complete justification of sharpness for the third Hankel determinant requires more than verification at w(z)=e^{iθ}z. In the subordination framework, the coefficients a_n of f are determined by the Taylor coefficients of w via the expansion of φ(w(z)) with φ quadratic. We will add an explicit argument in §4 showing that the functional is maximized when higher-order coefficients of w vanish. This follows from the coefficient bounds |w_k| ≤ 1 for Schwarz functions together with the specific algebraic form of the third Hankel determinant; any nonzero w_k for k≥2 contributes non-positively to the expression under the given φ. We will also include a brief computation for a sample Blaschke product of degree 2 to confirm it does not exceed the reported bound. This constitutes a partial revision: the bound itself remains unchanged, but the sharpness proof is strengthened. revision: partial
Referee: §3 (coefficient relations and second determinant): The reduction of the second Hankel determinant to an expression in the coefficients of w via the quadratic subordination φ(w(z)) is load-bearing for the sharpness claim. The paper should display the explicit polynomial expressions for a2,a3,a4 in terms of the first three coefficients of w and confirm that the maximum is attained only at |w1|=1 with higher coefficients zero.

Authors: We accept this suggestion. Although the derivations in §3 proceed from the subordination relation zf'(z)/f(z) = φ(w(z)) and the resulting recursive coefficient formulas, we did not tabulate the expanded polynomials. In the revision we will insert the explicit expressions: a2 = (1/2)w1, a3 = (1/2)w2 + (1/4)w1² + (1/8)w1, a4 = (1/2)w3 + (1/2)w1 w2 + (1/8)w1³ + … (full polynomials up to the required order). Substituting these into |a2 a4 − a3²| yields a polynomial in w1,w2,w3 whose maximum modulus on the unit polydisk is attained precisely when |w1|=1 and w2=w3=0, which corresponds to w(z)=e^{iθ}z. The verification uses the standard growth estimates |w_k| ≤ 1−|w1|² for k≥2 and direct differentiation or calculus on the resulting real-valued function. These explicit forms and the maximizer confirmation will be added to §3. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper defines S*(φ) via subordination to the fixed φ(z)=(1+z/2)^2, expands coefficients a_k via the composition φ(w(z)) with w a Schwarz function, and maximizes the explicit Hankel expressions |a2 a4 - a3^2| and the third-order determinant over the admissible coefficient region for w. These steps use standard series manipulation and known bounds on |w_k| or extremal w, without any self-definition of quantities in terms of the target determinants, without fitting parameters to data then relabeling as predictions, and without load-bearing self-citations that close a loop. The extremal-function claims rest on external subordination lemmas and coefficient estimates rather than reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard complex-analysis background rather than new postulates. No free parameters are introduced. The work uses known subordination and coefficient lemmas for starlike functions.

axioms (2)

standard math Analytic functions in the unit disk satisfy the usual Taylor expansion and subordination properties.
Invoked throughout the definition of A and S*(φ).
domain assumption The function φ(z) = (1 + z/2)^2 maps the unit disk into a region compatible with starlikeness.
Used to define the subclass and to obtain coefficient bounds.

pith-pipeline@v0.9.0 · 5424 in / 1315 out tokens · 30767 ms · 2026-05-10T15:28:37.560247+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references

[1]

Allu and A

V. Allu and A. Shaji , Second Hankel determinant for logarithmic inverse coefficients of convex and starlike functions, Bull. Aust. Math. Soc. 111(1) (2025), 128--139

2025
[2]

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

L. de Branges , A proof of the Bieberbach conjecture, Acta Math. 154(1--2) (1985), 137--152

1985
[3]

J. H. Choi, Y. C. Kim and T. Sugawa , A general approach to the Fekete--Szeg o problem, J. Math. Soc. Japan 59(3) (2007), 707--727

2007
[4]

Goel and S

P. Goel and S. S. Kumar , Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43(1) (2020), 957--991

2020
[5]

Janowski , Extremal problems for a family of functions with positive real part and for some related families, Ann

W. Janowski , Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/71), 159--177

1970
[6]

S. S. Kumar and G. Kamaljeet , A cardioid domain and starlike functions, Anal. Math. Phys. 11(2) (2021), Paper No. 54, 34 pp

2021
[7]

O. S. Kwon, A. Lecko and Y. J. Sim , On the fourth coefficient of functions in the Carath\'eodory class, Comput. Methods Funct. Theory 18(2) (2018), 307--314

2018
[8]

R. J. Libera and E. J. Z otkiewicz , Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85(2) (1982), 225--230

1982
[9]

W. C. Ma and D. Minda , A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157--169

1992
[10]

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

2015
[11]

Pommerenke , On the coefficients and Hankel determinants of univalent functions, J

C. Pommerenke , On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111--122

1966
[12]

D. V. Prokhorov and J. Szynal , Inverse coefficients for $( , )$-convex functions, Ann. Univ. Mariae Curie-Sk odowska Sect. A 35 (1981), 125--143 (1984)

1981
[13]

R. K. Raina and J. Soko , Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353(11) (2015), 973--978

2015
[14]

Ravichandran and S

V. Ravichandran and S. S. Kumar , Argument estimate for starlike functions of reciprocal order, Southeast Asian Bull. Math. 35(5) (2011), 837--843

2011
[15]

Sharma, N

K. Sharma, N. K. Jain and V. Ravichandran , Starlike functions associated with a cardioid, Afr. Mat. 27(5--6) (2016), 923--939

2016
[16]

Soko and J

J. Soko and J. Stankiewicz , Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101--105

1996