Toeplitz Determinants of Inverse Functions

• Toeplitz determinants of inverse functions are structured matrices formed from the coefficients of the inverse of a normalized univalent function, encoding geometric distortion.
• They provide sharp quantifications of nonlinearity and deviation in geometric function theory, with precise bounds established for classes like starlike, convex, and Ma-Minda subclasses.
• Advanced techniques, including subordination, Schwarz lemma, and coefficient inequalities, are employed to derive exact constants and improve extremal analysis.

A Toeplitz determinant is a structured polynomial in the coefficients of a power series, with entries aligned so that each descending diagonal is constant. In geometric function theory, Toeplitz and symmetric Toeplitz determinants are constructed from the coefficients of the inverse $f^{-1}(w) = w + \sum_{n\geq2} b_n w^n$ of a normalized univalent function $f(z) = z + \sum_{n\geq2} a_n z^n$ in the unit disk. These determinants offer quantifications of nonlinearity and geometric deviation, and their sharp bounds enable fine classification of function-theoretic subclasses and extremal phenomena. Recent developments have elucidated both classical and modern analytic families, notably the Ma-Minda classes, with exact constants recovered through advanced coefficient inequalities and subordination techniques (Obradović et al., 24 Dec 2025, Giri, 16 May 2025, Mandal et al., 2023).

1. Core Definitions and Variants

For $n \geq 1$, $p \geq 1$, the $n \times n$ Toeplitz determinant of the inverse coefficients $b_k$ is: $$T_{n,p}(f^{-1}) = \det\left[b_{i-j+p}\right]_{i,j=1}^{n},$$ with $b_k = 0$ for $k < 1$. The symmetric Toeplitz determinant is

$$T^s_{n,p}(f^{-1}) = \det\left[b_{|i-j|+p}\right]_{i,j=1}^{n}.$$

For low orders:

• $f(z) = z + \sum_{n\geq2} a_n z^n$0
• $f(z) = z + \sum_{n\geq2} a_n z^n$1
• $f(z) = z + \sum_{n\geq2} a_n z^n$2, etc.

The determinants encode geometric information about the mapping and its inverse, with symmetric forms providing refined sensitivity to the underlying structure (Obradović et al., 24 Dec 2025).

2. Determinant Estimates in Classical and Ma-Minda Classes

For the normalized univalent class $f(z) = z + \sum_{n\geq2} a_n z^n$3, and its subclasses:

• Bounded-turning ($f(z) = z + \sum_{n\geq2} a_n z^n$4): $f(z) = z + \sum_{n\geq2} a_n z^n$5
• Starlike ($f(z) = z + \sum_{n\geq2} a_n z^n$6): $f(z) = z + \sum_{n\geq2} a_n z^n$7
• Convex ($f(z) = z + \sum_{n\geq2} a_n z^n$8): $f(z) = z + \sum_{n\geq2} a_n z^n$9

Sharp bounds for Toeplitz determinants are:

Class	$n \geq 1$0	$n \geq 1$1	$n \geq 1$2
$n \geq 1$3	29	221	24
$n \geq 1$4	$n \geq 1$5	$n \geq 1$6	$n \geq 1$7
$n \geq 1$8	29	221	416
$n \geq 1$9	2	2	4

All bounds are attained by explicit extremal inverse functions, such as the Koebe map and certain rotations (Obradović et al., 24 Dec 2025).

For Ma-Minda generalizations, where subclasses are defined by subordination to an analytic function $p \geq 1$0, the sharp bounds for second-order Toeplitz determinants can be expressed in terms of the coefficients $p \geq 1$1 of $p \geq 1$2 (Giri, 16 May 2025):

$p \geq 1$3

$p \geq 1$4

These bounds apply immediately to Janowski, strongly starlike/convex, and other analytic subclasses.

3. Proof Techniques and Coefficient Relations

The determinant bounds are established through precise algebraic manipulations:

• Formal composition yields inverse coefficients

$p \geq 1$5

• Class-specific bounds are derived for the original coefficients ($p \geq 1$6), frequently using
  • Schwarz subordination,
  • The Prokhorov–Szynal lemma,
  • Fekete–Szegö inequalities.

For subclasses defined by subordination ($p \geq 1$7), the power series expansion of $p \geq 1$8 provides explicit relations between function and inverse coefficients. Sharp estimates utilize triangle inequalities and maximization over the parameter domain determined by Schwarz-lemmatype bounds ($p \geq 1$9), reducing the problem to real polynomial optimization (Obradović et al., 24 Dec 2025, Mandal et al., 2023, Giri, 16 May 2025).

4. Logarithmic Coefficients and Associated Toeplitz Determinants

For the logarithmic inverse function

$n \times n$0

the Toeplitz determinant computed over $n \times n$1 is

$n \times n$2

with

$n \times n$3

for the inverse series coefficients $n \times n$4. Sharp inequalities are established:

• For starlike with respect to symmetric points ($n \times n$5): $n \times n$6
• For convex with respect to symmetric points ($n \times n$7): $n \times n$8

Extremal examples are constructed by specialized choices of Schwarz function, with corresponding coefficients saturating the inequalities (Mandal et al., 2023).

5. Comparison With Prior Results and Error Correction

Recent work (Obradović et al., 24 Dec 2025) clarifies key discrepancies in previous estimates, notably correcting numerical errors and demonstrating the necessity of employing advanced coefficient inequalities. Notable outcomes include:

• Verification that, for low orders, $n \times n$9 for $b_k$0, further confirming the symmetry in geometric distortion between $b_k$1 and its inverse.
• Identification that symmetric Toeplitz bounds for $b_k$2 are generally larger than for $b_k$3, reflecting the increased growth of inverse coefficients.
• Correction of earlier claims, notably those in [Hadi et al., Math. 2025], using the Prokhorov–Szynal approach to improve sharpness.

6. Illustrative Examples and Explicit Extremals

Explicit examples illustrate both calculation process and sharpness of results. For the convex function $b_k$4, the coefficients of its inverse enable direct computation:

• $b_k$5
• $b_k$6
• $b_k$7
• $b_k$8 (absolute value 4)

These explicit calculations confirm the attainability of previously stated bounds and provide templates for direct evaluation in higher orders and more general classes (Obradović et al., 24 Dec 2025, Mandal et al., 2023).

7. Extensions and Higher-Order Determinants

The methodology for sharp bounding of Toeplitz determinants for inverse functions extends naturally to higher orders. For an $b_k$9 Toeplitz matrix,

$$T_{n,p}(f^{-1}) = \det\left[b_{i-j+p}\right]_{i,j=1}^{n},$$0

obtaining sharp bounds requires generalized inequalities for combinations of Schwarz function coefficients and multilinear forms in $$T_{n,p}(f^{-1}) = \det\left[b_{i-j+p}\right]_{i,j=1}^{n},$$1 (Giri, 16 May 2025). This area remains active, with recent work developing fourth-order Hermitian–Toeplitz estimates and other generalizations, especially within Ma-Minda-type subordination frameworks.

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Toeplitz determinants of inverse functions serve as precise invariants for the study of univalent functions and their subclasses, providing both sharp constants and structural understanding via algebraic and analytic techniques. The landscape continues to expand through subordination theory, advanced extremal analysis, and systematic generalization to higher orders (Obradović et al., 24 Dec 2025, Mandal et al., 2023, Giri, 16 May 2025).