On the Third Hankel Determinant for Inverse Coefficients of Starlike Functions: A Bernstein Polynomial Approach

Published 25 Apr 2026 in math.CV | (2604.23220v1)

Abstract: Let $\mathcal{A}$ denote the class of normalized analytic functions $f$ in the open unit disk defined as $ \mathbb{D}:={z\in\mathbb{C}:|z|<1} $ with $f(0)=0$ and $f'(0)=1$. A function $f\in\mathcal{A}$ is said to be starlike if $f(\mathbb{D})$ is starlike domain. By using the Bernstein polynomial method to obtain the required maximum estimate, we establish sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of starlike univalent ({\it i.e.}, one-to-one) functions in the unit disk $\mathbb{D}$ .

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Summary

The paper establishes a sharp upper bound of 1 for the third Hankel determinant derived from inverse coefficients of starlike functions.
It employs a Bernstein polynomial approach combined with Carathéodory function parametrization to transform the extremal problem into an explicit, tractable calculation.
The results verify the extremality of the Koebe function, extending insights in geometric function theory and informing application in algorithmic constructions.

Third Hankel Determinant of Inverse Coefficients for Starlike Functions via Bernstein Polynomial Method

Background

This work addresses a sharp upper bound for the third Hankel determinant constructed from the inverse coefficients of starlike univalent functions defined in the open unit disk $\mathbb{D}$ . Hankel determinants $H(q,n)$ have a prominent role in geometric function theory, encoding information about the higher-order dependencies between Taylor coefficients of analytic, univalent function classes, such as starlike or convex functions. Estimates for the second and third Hankel determinants have been pivotal for the characterization of extremal functions and rigidity phenomena in the theory of univalent mappings.

The class of starlike functions $\mathcal{S}^*$ is characterized by $\operatorname{Re}(\frac{z f'(z)}{f(z)}) > 0$ , ensuring geometric starlikeness of the image domain. The local inverse $f^{-1}$ admits an expansion whose coefficients can be computed recursively in terms of the original $a_n$ . The determinant $H(3,1)(f^{-1})$ is substantially more intricate than its lower-order counterparts and has not been sharply bounded in terms of inverse coefficients prior to this work.

Methodology

The authors employ a combination of classical coefficient representations (Carathéodory function parametrizations, Libera–Złotkiewicz formulae) and a Bernstein polynomial approach. This latter method, effectively used for multivariate polynomial maximization over compact domains, enables reduction of the extremal problem for $H(3,1)(f^{-1})$ to an explicit, tractable calculation using Bernstein coefficients.

Key steps include:

Carathéodory Parametrization: The representation $z f'(z) = f(z) p(z)$ for $f \in \mathcal{S}^*$ , where $H(q,n)$ 0 and $H(q,n)$ 1, enables coefficient decomposition for $H(q,n)$ 2 and thus for the inverse coefficients $H(q,n)$ 3.
Inverse Coefficients Expression: The recursive relationship between $H(q,n)$ 4 and $H(q,n)$ 5 is laid out, facilitating explicit computation of $H(q,n)$ 6 in terms of $H(q,n)$ 7 (coefficients of $H(q,n)$ 8).
Normalization and Domain Reduction: By exploiting rotational invariance and suitable choice of parameters, the maximization problem is reduced to studying $H(q,n)$ 9 over the cuboid $\mathcal{S}^*$ 0.
Bernstein Polynomial Representation: $\mathcal{S}^*$ 1 is expressed as a Bernstein sum over $\mathcal{S}^*$ 2. Rigorous computation of all matrix entries for Bernstein coefficients confirms nonnegativity throughout the domain.

Main Results

The paper proves:

$\mathcal{S}^*$ 3

for all $\mathcal{S}^*$ 4, with equality attained for the Koebe function $\mathcal{S}^*$ 5. The proof is exhaustive—no larger value of the third inverse Hankel determinant is possible within the class, as established by explicit inspection of Bernstein coefficient matrices.

Numerical sharpness is demonstrated: by evaluating $\mathcal{S}^*$ 6 via the explicit inverse power series for the Koebe function, the value is verified as exactly $\mathcal{S}^*$ 7, attaining the bound.

Implications and Future Developments

This result constitutes a complete solution to the extremal problem for the third Hankel determinant in the context of inverse coefficients of starlike functions, adding to previously known bounds for the original coefficients and for convex subclasses.

Theoretical implications include:

Extending sharp coefficient bounds for inverse function expansions, contributing to a deeper understanding of nonlinearity and coefficient interdependence in geometric function classes.
Strong numerical validity for the extremality of the Koebe function across related determinant maximization problems.
Demonstrating the effectiveness of the Bernstein polynomial approach for multivariate extremal problems in function theory.

Practical implications:

Such determinant bounds provide constraints for algorithmic geometric function construction, informing design in areas such as conformal mapping and complex dynamical systems.

Future directions may focus on:

Generalizing the Bernstein polynomial maximization method for higher-order Hankel determinants and broader function classes (e.g., close-to-convex, subclasses defined by differential inequalities).
Applying the inverse coefficient determinant framework to study stability, rigidity, and extremal behaviors in related classes (logarithmic coefficients, typically-real functions, mappings with prescribed boundary behaviors).

Conclusion

The paper rigorously establishes a sharp bound for the third Hankel determinant of inverse coefficients for starlike functions, confirming that $\mathcal{S}^*$ 8 and achieving equality with the Koebe function. The combination of Carathéodory parametrization and Bernstein polynomial methodology offers a robust, generalizable framework for extremal coefficient problems in geometric function theory, with substantial avenues for theoretical advancement and algorithmic application.