On Third-Order Determinant Bounds for the class $\mathcal{S}^*_{B}$

Abstract: This paper deals with sharp bounds for the third-order Hankel, Toeplitz and Hermitian-Toeplitz determinant of functions belonging to the class $\mathcal{S}<sup>*_{B}$ of starlike functions associated with a balloon-shaped domain, given by [ \mathcal{S}^{\ast}_{B}= \left{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log (1+z)} :=B(z), \quad z \in \mathbb{D} \right}. ] By applying coefficient inequalities and properties of these functions, we obtain sharp bounds for these determinants. The sharpness of the results is verified by constructing suitable extremal functions.

• The paper establishes sharp bounds for the balloon-domain class \(\mathcal{S}^*_{B}\), proving \(|H_{3,1}|\le 1/9\), \(|T_{3,1}|\le 1\), and \(-1/16\le H^T_{3,1}\le 1\).
• The authors reduce coefficient estimates to Carathe9odory parameters and optimize the resulting polynomial expressions, with explicit extremal functions attaining every stated bound.
• The results show that the balloon-shaped geometry imposes tighter Hermitian-Toeplitz constraints than broader starlike classes, although parts of the Hankel maximization rely on numerical verification.

This paper establishes sharp bounds for three third-order coefficient determinants—Hankel, Toeplitz, and Hermitian-Toeplitz—for the class $\mathcal{S}^*_{B}$ of starlike functions associated with a balloon-shaped domain (2603.10513). The class is defined by subordination

$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$

and its image domain is $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$, a balloon-shaped region. The work extends the Ma–Minda framework $\mathcal{S}^*(\varphi)$ to this particular symbol and contributes to the growing literature on sharp third-order determinant estimates, which are substantially harder than their second-order counterparts because $H_{3,1}$ couples the first five Taylor coefficients.

Main results

The paper proves three sharp theorems for $f \in \mathcal{S}^*_{B}$:

Determinant	Sharp bound	Extremal function
$H_{3,1}(f)$	$\lvert H_{3,1}(f)\rvert \le 1/9$	$$f_1(z) = z\exp\!\left(\int_0^z \frac{\log(1+t^3)}{t(1-\log(1+t^3))}\,dt\right)$$
$T_{3,1}(f)$	$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$0	$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$1
$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$2	$$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$3	upper: $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$4 (with $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$5); lower: $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$6

The Hankel bound of $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$7 matches the known sharp values for $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$8, $$\frac{zf'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} =: B(z),$$9, and $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$0, while being larger than the bound $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$1 for $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$2. The Hermitian-Toeplitz interval $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$3 is narrower than the corresponding ranges for $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$4 ($$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$5) and considerably narrower than for the full class $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$6 ($$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$7), reflecting the restrictive geometry of the balloon-shaped domain. Notably, the lower bound $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$8 is attained by the same extremal function that saturates the Toeplitz bound, indicating a common extremal configuration at $$B(\mathbb{D}) = \{w \in \mathbb{C}\setminus\{0\} : |\exp(1 - 1/w) - 1| < 1\}$$9, $\mathcal{S}^*(\varphi)$0.

Methodology

The proofs follow the standard reduction via the Carathéodory class. Writing $\mathcal{S}^*(\varphi)$1 with $\mathcal{S}^*(\varphi)$2, $\mathcal{S}^*(\varphi)$3, the initial coefficients $\mathcal{S}^*(\varphi)$4 are expressed in terms of $\mathcal{S}^*(\varphi)$5. The authors then invoke the classical parametric representation of Kwon–Lecko–Sim and Libera–Złotkiewicz expressing $\mathcal{S}^*(\varphi)$6 through parameters $\mathcal{S}^*(\varphi)$7, $\mathcal{S}^*(\varphi)$8 with moduli bounded by one.

For the Hankel determinant, substitution yields an expression linear in $\mathcal{S}^*(\varphi)$9 and quadratic in $H_{3,1}$0. After bounding by absolute values, the problem reduces to maximizing an explicit function $H_{3,1}$1 over the cuboid $H_{3,1}$2. The proof proceeds by exhaustive analysis: no interior critical point exists (the critical-point conditions force two inequalities that cannot hold simultaneously), each face is examined separately—with several faces admitting no interior extremum, verified partly by numerical computation—and all twelve edges are treated as single-variable optimization problems. The maximum over the boundary is $H_{3,1}$3, attained along the edge $H_{3,1}$4, $H_{3,1}$5 at $H_{3,1}$6; competing edge maxima such as $H_{3,1}$7 and $H_{3,1}$8 remain strictly below $H_{3,1}$9.

The Toeplitz proof is short: after substituting $f \in \mathcal{S}^*_{B}$0, the determinant factors into terms controlled directly by Carathéodory coefficient inequalities, giving $f \in \mathcal{S}^*_{B}$1 with equality at $f \in \mathcal{S}^*_{B}$2. The Hermitian-Toeplitz argument reduces the determinant to a two-variable polynomial $f \in \mathcal{S}^*_{B}$3 on $f \in \mathcal{S}^*_{B}$4, whose extrema are computed explicitly: $f \in \mathcal{S}^*_{B}$5 at $f \in \mathcal{S}^*_{B}$6 and $f \in \mathcal{S}^*_{B}$7 at $f \in \mathcal{S}^*_{B}$8.

Limitations and open questions

Several aspects of the Hankel proof warrant attention. The non-existence of interior critical points and of solutions to certain face systems is asserted "by numerical analysis" rather than by rigorous algebraic or analytic arguments; similarly, the claim that the graph of $f \in \mathcal{S}^*_{B}$9 remains negative on $H_{3,1}(f)$0 is supported only by a plotted figure. These steps leave the maximization argument dependent on computational evidence rather than formal verification, which is a recognized weakness in this genre of determinant-bound papers. Additionally, the extremal function $H_{3,1}(f)$1 has zero coefficients except at degrees congruent to $H_{3,1}(f)$2, so the sharpness construction exploits a sparse-coefficient structure; whether the same techniques transfer to higher-order determinants $H_{3,1}(f)$3 with $H_{3,1}(f)$4, as suggested in the conclusion, remains unverified. The paper also does not address the second-order Hankel determinant or radius problems for $H_{3,1}(f)$5, leaving those as natural open questions within the same framework.

Conclusion

The paper delivers sharp, extremal-attained bounds for the third-order Hankel, Toeplitz, and Hermitian-Toeplitz determinants of starlike functions subordinate to $H_{3,1}(f)$6: $H_{3,1}(f)$7, $H_{3,1}(f)$8, and $H_{3,1}(f)$9. The results situate the balloon-shaped class among other Ma–Minda subclasses with comparable determinant behavior and provide explicit extremal functions, while the rigor of parts of the Hankel maximization rests on numerical rather than symbolic verification—an issue that subsequent work could resolve formally.