Second-Order Toeplitz Determinant for Quasi-Convex Mappings

Abstract: This paper presents sharp estimates for the second-order Toeplitz determinant whose entries are the coefficients of convex functions defined on the unit disk in $\mathbb{C}$. These estimates are further extended to a subclass of holomorphic mappings defined on the unit ball in a complex Banach space and on the unit polydisk in $\mathbb{C}<sup>n$, which, as special cases, yield bounds for the classes of quasi-convex mappings of type $B$.

• The paper establishes sharp bounds for the second-order Toeplitz determinant $T_{2,3}(f)$ and extends these results to holomorphic mappings on Banach balls and the unit polydisk, offering practical bounds for various quasi-convex mappings.
• The solution involves using a standard subordination technique and leverages the Prokhorov–Szynal inverse-coefficient theorem and the Efraimidis–Livingston inequality, which provides a uniform bound across different convex function classes.
• Specializing the results for convex, convex order-α, and strongly convex classes offers clear bounds such as a constant 2 for the full convex class, aligning with previous classic one-variable estimates.

Overview and problem statement

This paper resolves an open coefficient problem for Toeplitz determinants associated with convex functions, first in one complex variable and then in several variables. For a normalized analytic function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ on the unit disk $\mathbb{U}$, the second-order Toeplitz determinant is $T_{2,n}(f) = a_n^2 - a_{n+1}^2$. While sharp bounds for $|T_{2,2}(f)|$ and $|T_{3,1}(f)|$ were previously established for the Ma–Minda class of convex functions $\mathcal{C}(\Psi)$ by Ahuja et al., the estimate for $|T_{2,3}(f)| = |a_3^2 - a_4^2|$ remained open. The paper supplies this sharp bound and extends it to holomorphic mappings on the unit ball $\mathbb{B}$ of a complex Banach space and on the unit polydisk $\mathbb{U}^n$, with quasi-convex mappings of type $B$ (in the sense of Roper–Suffridge) and their order-$\mathbb{U}$0 variants (Liu–Liu) as special cases.

The one-variable result

The main theorem concerns $\mathbb{U}$1, where $\mathbb{U}$2 is analytic univalent with $\mathbb{U}$3, $\mathbb{U}$4, $\mathbb{U}$5, starlike with respect to 1 and symmetric about the real axis. Under two additional hypotheses — namely $\mathbb{U}$6 and $\mathbb{U}$7, where the regions $\mathbb{U}$8 are those appearing in the Prokhorov–Szynal inverse-coefficient theorem and

$\mathbb{U}$9

the author proves the sharp estimate

$T_{2,n}(f) = a_n^2 - a_{n+1}^2$0

The proof follows the standard subordination technique: writing $T_{2,n}(f) = a_n^2 - a_{n+1}^2$1 with Schwarz function $T_{2,n}(f) = a_n^2 - a_{n+1}^2$2, coefficient comparison expresses $T_{2,n}(f) = a_n^2 - a_{n+1}^2$3 and $T_{2,n}(f) = a_n^2 - a_{n+1}^2$4 in terms of the coefficients $T_{2,n}(f) = a_n^2 - a_{n+1}^2$5 of $T_{2,n}(f) = a_n^2 - a_{n+1}^2$6. The hypotheses ensure that Efraimidis' generalization of Livingston's inequality bounds $T_{2,n}(f) = a_n^2 - a_{n+1}^2$7, while the Prokhorov–Szynal trichotomy bounds $T_{2,n}(f) = a_n^2 - a_{n+1}^2$8. Sharpness is attained by the extremal function defined through $T_{2,n}(f) = a_n^2 - a_{n+1}^2$9, for which both summands achieve equality simultaneously — a point worth noting, since the triangle-inequality step $|T_{2,2}(f)|$0 is generally strict; the specific phase relationship ($|T_{2,2}(f)|$1 real, $|T_{2,2}(f)|$2 purely imaginary up to sign) at the extremal function makes it exact.

Specializing $|T_{2,2}(f)|$3 yields concrete corollaries:

Class	Bound on $|T_{2,2}(f)|$4
Convex functions $|T_{2,2}(f)|$5	$|T_{2,2}(f)|$6
Convex of order $|T_{2,2}(f)|$7, $|T_{2,2}(f)|$8	$|T_{2,2}(f)|$9
Strongly convex of order $|T_{3,1}(f)|$0, $|T_{3,1}(f)|$1	$|T_{3,1}(f)|$2

The strongly convex case carries the restriction $|T_{3,1}(f)|$3, inherited from the parameter constraints required for Lemma applicability. The constant $|T_{3,1}(f)|$4 for the full convex class is notably clean and matches the structure of earlier $|T_{3,1}(f)|$5 estimates for $|T_{3,1}(f)|$6.

Extension to Banach spaces and the polydisk

The higher-dimensional formulation replaces scalar coefficients with Fréchet derivative data. For $|T_{3,1}(f)|$7 with $|T_{3,1}(f)|$8 holomorphic, $|T_{3,1}(f)|$9, zero-free, and $\mathcal{C}(\Psi)$0 (the Graham–Hamada–Koh class subordinate to $\mathcal{C}(\Psi)$1), the Banach-space result states

$\mathcal{C}(\Psi)$2

is bounded above by exactly the same expression as in the one-variable theorem, uniformly over $\mathcal{C}(\Psi)$3 and $\mathcal{C}(\Psi)$4. The proof reduces to the disk via the slice function $\mathcal{C}(\Psi)$5 built from the quantity $\mathcal{C}(\Psi)$6, using the identity $\mathcal{C}(\Psi)$7 for $\mathcal{C}(\Psi)$8. The third-order term is bounded via the Xu–Liu–Liu Fekete–Szegő inequality for $\mathcal{C}(\Psi)$9 mappings, and the fourth-order term via the Prokhorov–Szynal lemma applied to $|T_{2,3}(f)| = |a_3^2 - a_4^2|$0. Sharpness holds for the mapping whose Fréchet derivative is $|T_{2,3}(f)| = |a_3^2 - a_4^2|$1, evaluated along radial slices $|T_{2,3}(f)| = |a_3^2 - a_4^2|$2.

On the polydisk $|T_{2,3}(f)| = |a_3^2 - a_4^2|$3, the analogous statement takes a slightly different form because the natural functional is vector-valued:

$|T_{2,3}(f)| = |a_3^2 - a_4^2|$4

The proof here is more involved: after reducing to coordinate slices indexed by $|T_{2,3}(f)| = |a_3^2 - a_4^2|$5 with $|T_{2,3}(f)| = |a_3^2 - a_4^2|$6, the componentwise estimates are first established on the distinguished boundary $|T_{2,3}(f)| = |a_3^2 - a_4^2|$7 and then propagated to all of $|T_{2,3}(f)| = |a_3^2 - a_4^2|$8 by the maximum modulus principle applied to the relevant homogeneous polynomial expressions. Sharpness is verified at points of the form $|T_{2,3}(f)| = |a_3^2 - a_4^2|$9 for the explicit extremal $\mathbb{B}$0.

Consequences for quasi-convex mappings

Choosing $\mathbb{B}$1 recovers the Roper–Suffridge quasi-convex mappings of type $\mathbb{B}$2, $\mathbb{B}$3, giving the sharp uniform bound $\mathbb{B}$4 in the Banach-ball setting and $\mathbb{B}$5 on the polydisk. Choosing $\mathbb{B}$6 gives Liu–Liu's quasi-convex mappings of type $\mathbb{B}$7 and order $\mathbb{B}$8, $\mathbb{B}$9, with the corresponding weighted bound matching the one-variable corollary. These results complete the program initiated by Giri and Kumar, who had settled $\mathbb{U}^n$0 and $\mathbb{U}^n$1 for $\mathbb{U}^n$2 but left $\mathbb{U}^n$3 open; they also extend the classical Ali–Thomas–Vasudevarao estimate $\mathbb{U}^n$4 for planar convex functions to infinite-dimensional domains.

Limitations and open questions

The results depend on two structural assumptions that restrict their scope. First, the hypothesis $\mathbb{U}^n$5 selects the regime in which the Efraimidis bound for $\mathbb{U}^n$6 is governed by the nontrivial branch; the complementary regime is not treated, so the stated bound may not be sharp there. Second, the condition $\mathbb{U}^n$7 invokes only part of the Prokhorov–Szynal parameter space, and for the strongly convex subclass this excludes $\mathbb{U}^n$8. Whether the same bound, or a different sharp one, holds outside these parameter regions remains open. A further open question is whether the polydisk formulation can be unified with the Banach-space one under a single functional, since the two settings currently require different (though related) expressions.

Conclusion

The paper closes a known gap in the theory of Toeplitz determinants for convex-type functions by establishing the sharp bound on $\mathbb{U}^n$9 for the full Ma–Minda convex class under explicit parameter conditions, and by transporting the result to holomorphic mappings on Banach balls and polydisks through slice reduction and the maximum modulus principle. The resulting corollaries for quasi-convex mappings of type $B$0, with and without order, extend the classical one-variable estimates to several complex variables and to infinite-dimensional Banach spaces, with sharpness confirmed by explicit extremal mappings in each setting.