Third-Order Toeplitz Determinants

• Third-Order Toeplitz Determinants are functionals built from the Taylor coefficients of normalized univalent functions, providing key quantitative invariants in geometric function theory.
• They offer sharp bounds and drive extremal problems in classes such as starlike and convex functions, with explicit formulas derived through advanced analytical techniques.
• Methodologies like Carathéodory representation, Fekete–Szegö inequalities, and companion matrix diagonalization underpin these determinants, extending their use to Banach spaces and multidimensional settings.

A third-order Toeplitz determinant is a determinantal functional constructed from the Taylor coefficients of a holomorphic or univalent function, with broad generalizations to higher dimensions, Banach spaces, and function-theoretic subclasses. Such determinants play a central role in geometric function theory, extremal problems, and the analytic description of univalent mappings, providing sharp quantitative invariants that encode regularity and geometric constraints.

1. Definition and Explicit Formulae

For a normalized analytic function $f$ in the unit disk %%%%1%%%%,

$f(z) = z + \sum_{n=2}^\infty a_n z^n,$

the third-order Toeplitz determinant, $T_3(f)$, is defined by the $3 \times 3$ Toeplitz matrix with entries given by the coefficients of $f$:

$$T_{3,1}(f) = \det \begin{pmatrix} 1 & a_2 & a_3 \ a_2 & 1 & a_2 \ a_3 & a_2 & 1 \end{pmatrix} = 1 - 2a_2^2 + 2a_2^2 a_3 - a_3^2.$$

When considering Hermitian Toeplitz determinants, one often symmetrizes the coefficients:

$$T_3^{\mathrm{Herm}}(f) = \det \begin{pmatrix} 1 & a_2 & a_3 \ \overline{a_2} & 1 & a_2 \ \overline{a_3} & \overline{a_2} & 1 \end{pmatrix} = 2\mathrm{Re}(a_2 a_3) - 2|a_2|^2 - |a_3|^2 + 1.$$

This determinant serves as a non-linear functional that is sensitive to both modulus and argument of the Taylor coefficients and is fundamental for extremal coefficient problems and the geometry of image domains (Ali et al., 2017, Giri et al., 2022, Obradović et al., 2019).

2. Sharp Bounds in Classical and Generalized Settings

Classical Extremal Results. For the class $\mathcal{S}$ of normalized univalent functions in $\mathbb{U}$, the sharp bounds for $|T_3(f)|$ are:

$|T_3(f)| \leq 24,$

with equality achieved for rotated Koebe functions $f(z)=z/(1 - e^{i\theta}z)^2$ (Ali et al., 2017).

Invariants for Starlike and Convex Classes. For starlike functions $\mathcal{S}^*$ and convex functions $\mathcal{C}$:

• $\mathcal{S}^*$ and $\mathcal{C}$ share the $|T_3(f)| \leq 24$ bound (sharp).
• For convex functions, $|T_3(f)| \leq 4$ is attained by the appropriately rotated half-Koebe map (Ali et al., 2017, Obradović et al., 2019).

Generalization by Vanishing Order (Zero-Order Constraint). For $f(z)$ with $f(z) - z = O(z^{k+1})$ (i.e., all coefficients $a_2,...,a_k$ vanish), setting $a:=a_{k+1}$, $b:=a_{2k+1}$, the determinant specializes to

$T_3(f) = 1 - b^2 - 2a^2 + a^2b.$

The extremal sharp bound, due to Giri–Kumar, is

$$|T_3(f)| \le \begin{cases} 1 + \dfrac{8}{k^2} + \dfrac{(k+2)(6-k)}{k^4}, & 1 \leq k \leq 3, \ 1 + \dfrac{8}{k^2} + \dfrac{k+2}{k^3}, & k \geq 3. \end{cases}$$

Equality is achieved for explicit extremal functions $f(z) = z/(1 - i z^k)^{1/k}$ (Giri et al., 6 Jan 2026).

3. Proof Methods and Underlying Function-Theoretic Techniques

The extremal analysis across different settings employs the following workflow:

• Carathéodory Representation: The logarithmic derivative or related function is written as $z f'(z)/f(z) = p(z)$, with $\operatorname{Re} p > 0$. This links coefficient problems to the Carathéodory class and exploits the sharp coefficient bounds $|p_n| \leq 2$ (Giri et al., 6 Jan 2026, Giri et al., 2022).
• Coefficient Relations: For the vanishing-order constraint,

$$a_{k+1} = \frac{p_k}{k}, \quad a_{2k+1} = \frac{k p_{2k} + p_k^2}{2k^2}.$$

• Fekete–Szegö Inequalities: The two-term refinement provides control on $\left| a_{2k+1} - 2a_{k+1}^2 \right|$, crucial for optimizing $|T_3|$ (Giri et al., 6 Jan 2026).
• Triangle Inequality and Optimization: The determinant is rewritten as a function of $|a|, |b|$, and $|b-2a^2|$. The bounds for these quantities, together with quadratic optimization, yield the advertised sharp constants (Giri et al., 2022, Giri et al., 2022).
• Extremality via Explicit Construction: Achievement of equality is demonstrated by model functions such as the Koebe mapping or its generalized forms.

This combination of analytic subordination, coefficient bounds, and quadratic optimization is standard across the Toeplitz determinant literature.

4. Extensions to Several Complex Variables and Banach Spaces

Starlike Mappings on the Unit Ball in Banach Spaces. The Toeplitz determinant generalizes naturally by replacing the coefficients with particular normalized derivatives:

$$a_{k+1} = \frac{\ell_z(D^{k+1}F(0)(z^{k+1}))}{(k+1)! \|z\|^{k+1}}, \quad a_{2k+1} = \frac{\ell_z(D^{2k+1}F(0)(z^{2k+1}))}{(2k+1)! \|z\|^{2k+1}}$$

where $\ell_z$ is a Hahn–Banach functional (Giri et al., 6 Jan 2026, Giri et al., 2022).

Bounded Starlike Circular Domains $\Omega \subset \mathbb{C}^n$. The Minkowski functional $\rho(z)$ characterizes starlikeness, and the sharp Toeplitz determinant bounds persist in the same form as in the Banach ball case by reduction to one-variable analytic function theory.

In both cases, the proofs rely on diagonalization to a one-dimensional subordinate function, and the sharp estimates rely on the same Carathéodory and Fekete–Szegö machinery as in the classical context (Giri et al., 2022).

5. Connections with Hermitian Toeplitz Determinants and Subclasses

Hermitian Toeplitz Determinants. For subclasses such as the Ma–Minda, Sakaguchi–starlike, and convex types, analogous determinant formulas are evaluated, for example:

$$T_3[f] = 1 - 2|a_2|^2 + 2\operatorname{Re}(a_2^2 \overline{a_3}) - |a_3|^2$$

with sharp upper and lower bounds depending on the Taylor coefficients of the defining function $\varphi(z) = 1 + B_1 z + B_2 z^2 + \dots$:

• Upper bound: $T_3[f] \leq 1$ for all $|B_2| \leq B_1$;
• Lower bound: Piecewise expressions depending on $B_1$, $B_2$, and explicit rational functions provide the sharp constants for the various classes (Giri et al., 2022, Giri et al., 2022).

The method of proof in these cases leverages subordination, Carathéodory parameterization, and a real-variable maximization over the parameter region to establish the extremal results.

6. Algorithmic and Structural Results

A general $n \times n$ banded Toeplitz matrix with bandwidth $k$ admits a determinant expression in terms of the $k \times k$ companion matrix $C$:

$\det T_n = (-1)^{n-1} a_1^{n-3} (C^n)_{11}$

where $C$ is explicitly constructed from the Toeplitz coefficients (Cinkir, 2011). Furthermore, there exists an $O(k^2 \log n + k^3)$ algorithm for computing such determinants using binary powering and companion matrix diagonalization, with the $k=3$ case being directly relevant for third-order Toeplitz determinants.

Worked examples in (Cinkir, 2011) demonstrate the explicit symbolic formula, connection to eigenvalues, and numerical evaluation for small $n$.

7. Comparative Summary and Impact

Setting	Explicit Bound for $|T_3(f)|$	Extremal Function Type
$\mathcal{S}$ univalent, $k=1$	$24$	Rotated Koebe map
Starlike/Convex (Ma–Minda)	Depends on $B_1,B_2$; often $1$ or sharper	Exponential of $\varphi$
Vanishing order $k$ ($f(z)-z=O(z^{k+1})$)	$1 + \frac{8}{k^2} + \frac{(k+2)(6-k)}{k^4}$, $k\le 3$<br>$1 + \frac{8}{k^2} + \frac{k+2}{k^3}$, $k \ge 3$	Generalized Koebe-type
Banach ball or $\mathbb{C}^n$, $k$	Same as above (structurally identical piecewise)	Generalized radial mapping

These results unify the classical sharp extremal inequalities for third-order Toeplitz determinants and extend them to highly structured subclasses, Banach spaces, and multidimensional starlike domains. The analytical principles—Carathéodory representation, Fekete–Szegö estimates, and subordination—are consistently fundamental across all dimensions and settings (Giri et al., 6 Jan 2026, Giri et al., 2022, Giri et al., 2022, Cinkir, 2011, Obradović et al., 2019).