The second and third Hankel determinants for certain classes of functions

Abstract: Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f&#39;(0)=1$ in the unit disk $\mathbb{D}:={z \in \mathbb{C}: |z|&lt;1}.$ In this paper, we consider $\mathcal{S}^*(\varphi) := \left{ f \in \mathcal{A} : zf&#39;(z)/f(z) \prec \varphi(z):=(1+z/2)² \right}$, a subclass of starlike functions and we compute the sharp second and third Hankel determinants for the functions in $\mathcal{S}<sup>*(\varphi)$. Furthermore, we determine the extremal functions for the coefficient bounds of the functions belonging to $\mathcal{S}<sup>*(\varphi)$.

• The paper establishes sharp coefficient bounds and identifies extremal functions achieving |H2(2)| ≤ 1/4 and |H3(1)| ≤ 1/9 for the studied starlike class.
• It employs Carathéodory parameterizations, Schwarz functions, and the Bernstein basis method to rigorously maximize nonlinear dependencies among Taylor coefficients.
• The results advance geometric function theory by providing explicit determinant estimates and a framework for extending these techniques to higher-order problems.

Sharp Hankel Determinants for a Subclass of Starlike Functions Defined by Subordination to $\varphi(z) = (1 + z/2)^2$

Introduction and Problem Statement

This paper systematically investigates the sharp bounds of the second and third Hankel determinants for a prescribed subclass of starlike functions on the unit disk, specifically those subordinate to the univalent, symmetric Ma–Minda function $\varphi(z) = (1+z/2)^2$. The class under consideration is

$$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$

where $\mathcal{A}$ comprises analytic functions normalized by $f(0)=0$, $f'(0)=1$. The main contributions include exact coefficient bounds, sharp Hankel determinant estimates for $H_2(2)$ and $H_3(1)$, and identification of extremal (i.e., sharpness-attaining) functions.

Structure and Properties of $\mathcal{S}^*(\varphi)$

The function $\varphi$ is verified to fulfill all the Ma–Minda criteria (analyticity, univalence, normalization, positive real part, symmetry, and being starlike with respect to $\varphi(z) = (1+z/2)^2$0). The class $\varphi(z) = (1+z/2)^2$1 generalizes the classical starlike family and admits an explicit integral representation: $\varphi(z) = (1+z/2)^2$2 where $\varphi(z) = (1+z/2)^2$3 is a Schwarz function, analytic in $\varphi(z) = (1+z/2)^2$4 with $\varphi(z) = (1+z/2)^2$5 and $\varphi(z) = (1+z/2)^2$6.

Via this structure, an inclusion result is provided: every function in the Janowski class $\varphi(z) = (1+z/2)^2$7 with parameters satisfying

$\varphi(z) = (1+z/2)^2$8

is subordinate to $\varphi(z) = (1+z/2)^2$9.

The paper also provides radius results for geometric inclusion in the convexity class $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$0 using differential subordination methods.

Coefficient Estimates

For $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$1 in $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$2, sharp coefficient bounds are established as follows:

• $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$3
• $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$4
• $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$5

These estimates are sharp and extremal functions are explicitly constructed (e.g., $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$6, attaining the $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$7 and $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$8 extremals; a specific Möbius-type Schwarz function for $$\mathcal{S}^*(\varphi) := \{ f \in \mathcal{A} : z f'(z)/f(z) \prec \varphi(z) \}$$9).

Second and Third Hankel Determinants

The Hankel determinants, which encode nontrivial nonlinear dependencies among the Taylor coefficients, serve as higher-order functionals quantifying the nonlinearity and growth properties of univalent function classes.

Second Hankel Determinant $\mathcal{A}$0

For $\mathcal{A}$1, it is shown: $\mathcal{A}$2 Sharpness is exhibited by $\mathcal{A}$3, for which the equality is achieved. The derivation involves explicit coefficient relations, Carathéodory parameterizations, and maximization over the relevant set, utilizing the representation of Carathéodory and Schwarz functions and known bounds (Libera-Zlotkiewicz and Prokhorov-Szynal lemmas).

Third Hankel Determinant $\mathcal{A}$4

The third order determinant is bounded as

$\mathcal{A}$5

with the sharp example realized by $\mathcal{A}$6 and corresponding extremal function. The proof requires intricate conversion between coefficients in the Schwarz and Carathéodory representations, and a careful maximization over a 3D compact parameter set. Due to the combinatorial complexity and high degree (up to 6 in $\mathcal{A}$7 and 4 in $\mathcal{A}$8), a subdivision technique and the Bernstein basis for bivariate polynomials are employed, buttressed by positivity considerations, to certify the global maximum.

Extremal Functions and Methodological Aspects

The extremal functions achieving equality in both $\mathcal{A}$9 and $f(0)=0$0 are of explicit exponential/logarithmic form derived using monomial Schwarz components—a notable technical feature as this reveals the nontrivial role of higher-degree terms in maximizing these nonlinear functionals in the Ma–Minda context. The computational strategy for bounding the determinants, particularly for the third order case, is constructive and rigorous: subdivision of the domain, computation of Bernstein coefficients, and local estimates near the corners.

Implications and Future Directions

The sharp determination of $f(0)=0$1 and $f(0)=0$2 over $f(0)=0$3 stands as a precise answer to long-standing coefficient extremal problems in modern geometric function theory, especially for starlike classes defined by functional subordination. The extremal functions identified expand the catalog of known sharp examples and underscore the effectiveness of the Ma–Minda subordination approach for systematically generating and studying univalent subclasses.

Future research directions include:

• Extending the determinant estimates to higher Hankel orders and generalized Ma–Minda classes with different analytic targets $f(0)=0$4.
• Systematic classification of extremal points for other nonlinear coefficient functionals (e.g., higher-order logarithmic coefficients, inverse coefficient problems).
• Applications of these sharp inequalities in function-theoretic extremal problems, Loewner chains, and connections to classical conjectures (e.g., those related to growth, covering, and rotation theorems).
• Development and automation of computational approaches (such as subdivision schemes coupled with Bernstein positivity analysis) for bounding nonlinear functionals of analytic functions.

Conclusion

The paper establishes optimal bounds for the second and third Hankel determinants in the class $f(0)=0$5, with explicit characterization of extremal functions. The results affirm the sharpness of classical extremal methods, demonstrate the utility of the Ma–Minda paradigm for geometric function classes, and provide a technical blueprint for tackling similar problems in analytic function theory. These advances have both intrinsic mathematical value in coefficient theory and potential applications in complex analysis and related fields.