The fourth-order Hermitian Toeplitz determinant for convex functions

The sharp bounds for the fourth-order Hermitian Toeplitz determinant over the class of convex functions are computed.


Introduction
Let H be the class of analytic functions in D := {z ∈ C : |z| < 1}, A be its subclass normalized by f (0) := 0, f (0) := 1, that is, functions of the form f (z) = ∞ n=1 a n z n , a 1 := 1, z ∈ D, (1) and S be the subclass of A of univalent functions. Let S c denote the subclass of S of convex functions, that is, univalent functions f ∈ A such that f (D) is a convex domain in C. By the well-known result of Study [11] (see also [5, p. 42]), a function f is in S c if and only if Given q, n ∈ N, the Hermitian Toeplitz matrix T q,n ( f ) of f ∈ A of the form (1) is defined by where a k := a k . Let |T q,n ( f )| denote the determinant of T q,n ( f ). In particular, the third Toeplitz determinant |T 3,1 ( f )| is given by and the fourth Toeplitz determinant |T 4,1 ( f )| is given by In recent years a lot of papers has been devoted to the estimation of determinants built with using coefficients of functions in the class A or its subclasses. Hankel matrices i.e., square matrices which have constant entries along the reverse diagonal (see e.g., [3] with further references), and the symmetric Toeplitz determinant (see [1]) are of particular interest.
For this reason looking on the interest of specialists in [4] and [7] the study of the Hermitian Toeplitz determinants on the class A or its subclasses has begun. Hermitian Toeplitz matrices play an important role in functional analysis, applied mathematics as well as in physics and technical sciences.
In [4] the conjecture that the sharp inequalities 0 ≤ |T q,1 ( f )| ≤ 1 for all q ≥ 2, holds over the class S c was proposed and was confirmed for q = 2 and q = 3. The purpose of this paper is to prove this conjecture for q = 4.
Let P be the class of all p ∈ H of the form having a positive real part in D.
The key to the proof of the main result is the following lemma. It contains the well-known formula for c 2 (see e.g., [10, p. 166]) and the formula for c 3 due to Libera and Zlotkiewicz [8,9]. and for some ζ, η ∈ D := {z ∈ C : |z| ≤ 1}

Main result
In [4] the sharp bounds for the Hermitian-Toeplitz determinants of the second and third-order for the class of convex functions of order α were computed. In particular for α = 0, the results obtained are reduced for the class of convex functions, namely, Both results suggested the conjecture that 0 ≤ |T q,1 ( f )| ≤ 1 for every q ≥ 2. In this paper, this conjecture was confirmed for q = 4. For the sake of consistency, we also provide a short proof of the case q = 3.
Both inequalities are sharp with equalities attained by and by the identity, respectively.
It is clear that the function (9) and the identity make the results sharp.
We will now estimate the fourth-order Toeplitz determinant Both inequalities are sharp with equalities attained by the function (9) and the identity, respectively.
We will show that γ increasis. Note that is equivalent to where for t ∈ R, Indeed, as easy to see the function (0, x 1 (2)) x → H 2 (2, x) decreases. (vi) It remains to consider the interior of Δ 1 . Since the system of equations so h has no critical points in the interior of Δ 1 .
D2. Assume that y w ≥ 1, i.e., equivalently that (iv) On the side c = 2, is decreasing. Therefore (v) It remains to consider the interior of Δ 2 . The system of equations has the solution c = x = 0 evidently. Let x = 0 and x = 3. From the first equation we get which satisfies the inequality 0 ≤ c ≤ 2 only when x ∈ [x , 1), where x ≈ 0.81244. Now substituting (24) into the second equation of (23) we obtain the equation which has the unique solution in (0, 1), namely x ≈ 0.089756 < x . Thus g has no critical point in the interior of Δ 2 . Summarizing, from par D it follows that It is clear that equality for the upper bound in (12) holds for the identity function, and for the lower bound for the function (9).

Compliance with ethical standards
Conflict of interest The authors declare that there is no conflict of interest.
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