--- title: Third-Order Toeplitz Determinants url: https://www.emergentmind.com/topics/third-order-toeplitz-determinants type: topic --- # Third-Order Toeplitz Determinants 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 $\mathbb{U}$, $$ f(z) = z + \sum_{n=2}^\infty a_n z^n, $$ 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 [1704.00657][2203.10429][1912.11719]. ## 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 $f(z)=z/(1 - e^{i\theta}z)^2$ [1704.00657]. **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 [1704.00657][1912.11719]. **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 $f(z) = z/(1 - i z^k)^{1/k}$ [2601.03039]. ## 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 $|p_n| \leq 2$ [2601.03039][2210.13158]. - **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 $|T_3|$ [2601.03039]. - **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 [2210.13158][2203.10429]. - **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 [2601.03039][2210.13158]. **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 [2210.13158]. ## 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 [2203.10429][2210.13170]. 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 [1102.0453]. 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 [1102.0453] 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 $k$ ($f(z)-z=O(z^{k+1})$) | $1 + \frac{8}{k^2} + \frac{(k+2)(6-k)}{k^4}$, $k\le 3$
$1 + \frac{8}{k^2} + \frac{k+2}{k^3}$, $k \ge 3$| 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 [2601.03039][2210.13158][2203.10429][1102.0453][1912.11719]. Source: https://www.emergentmind.com/topics/third-order-toeplitz-determinants