The Third-Order Hermitian Toeplitz Determinant for Alpha-Convex Functions

Sharp lower and upper bounds of the second- and third-order Hermitian Toeplitz determinants for the class of α-convex functions were found. The symmetry properties of the arithmetic mean underlying the definition of α-convexity and the symmetry properties of Hermitian matrices were used.


Introduction
Let H be the class of all analytic functions in D := {z ∈ C : |z| < 1} and A be its subclass of functions f of the form: f (z) = ∞ ∑ n=1 a n z n , a 1 = 1, z ∈ D.
(1) Let S be the subclass of A of all univalent functions. For α ∈ [0, 1], let M α denote the subclass of A of all functions f satisfying the condition: The class M α was introduced by Mocanu [1] and its elements are called α-convex (see also [2] [Volume I, pp. 142-147]). For α = 0, the condition (2) describes the class of starlike functions denoted by S * introduced by Alexander [3] ( [4], as can also be seen in [2] [Volume I, Chapter 8]). For α = 1, the condition (2) specifies the class of convex functions denoted by S c defined by the study [5] (see also [2] [Volume I, Chapter 8]). Thus, the classes M α create a "continuous passage" on α ∈ [0, 1] from the set of convex functions M 1 to the set of starlike functions M 0 . One can see that the range of α can be extended to the real axis as well to the complex plane. In [1], Mocanu presented a geometrical interpretation of functions in the class M α . In [6], it was observed that M α ⊂ M 0 for every α ∈ [0, 1]. This result can be found in the paper due to Sakaguchi [7] that was published before the advent of the α-convexity concept (cf. [2] [Volume I. pp. 142-143]). Furthermore, in [6], the authors have shown that M α 1 ⊂ M α 2 for every 0 ≤ α 2 ≤ α 1 ≤ 1. The class M α plays an important role in the geometric function theory and was studied by various authors (e.g., [8,9] [Chapter 7] with further references).
For q, n ∈ N, by T q,n ( f ), with f ∈ A of the form (1), define the matrix: , where a k := a k . In case, a n is a real number, T q,n ( f ) is the Hermitian Toeplitz matrix. Thus, the matrix T q,1 ( f ) is like this. In particular: In recent years, many papers have been devoted to the estimation of determinants whose entries are 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., [10][11][12], with further references), and the symmetric Toeplitz matrices (see [13]), are of particular interest.
For this reason, considering the interest of specialists, in [14][15][16], the estimation of the determinants of the Hermitian Toeplitz matrices T q,1 ( f ) on the class A or its subclasses was started. Hermitian Toeplitz matrices play an important role in functional analysis, applied mathematics as well as in physics and technical sciences. Let us mention that only a few papers have been published that concern the estimation of Hermitian Toeplitz determinants in the basic subclasses of univalent functions. This is a new issue and it is important to find such estimates for the class of α-convex functions, which are among the most important in geometric function theory. This paper was dedicated to finding the sharp lower and upper bounds of the second-and third-order Hermitian Toeplitz determinants for the class M α .
Let P be the class of all p ∈ H of the form: which have a positive real part.
In the proof of the main result, we will use the following lemma which contains the Carathéodory result for c 1 [17], and the well-known formula for c 2 (e.g., [18] [p. 166]).
Observe that for each θ ∈ R, det T q, Recall now the following observation [15].

Main Results
We now compute the sharp bounds of det T 2,1 ( f ) and det T 3,1 ( f ) in the class of αconvex functions.
1. If f ∈ S * , then: 2. If f ∈ S c , then: All inequalities are sharp.
Now, we will compute the upper and lower bounds of det T 3,1 ( f ).
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.