Hermitian Toeplitz determinants for the class S of univalent functions

Introducing a new method, we give sharp estimates of the Hermitian Toeplitz determinants of third order for the class S of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class S.


Introduction
Let A be the class of functions f that are analytic in the open unit disc D = {z : |z| < 1} and normalized such that f (0) = f (0) − 1 = 0, and let S ⊂ A be the class of univalent functions in the unit disc D (functions that are analytic, one-on-one and onto).
For functions f ∈ A of the form f (z) = z + a 2 z 2 + a 3 z 3 + · · · and positive integers q and n, the Toeplitz matrix is defined by The concept of Toeplitz matrices plays an important role in functional analysis, applied mathematics as well as in physics and technical sciences (for more details, see [28]).
If a n is real, then the Toeplitz matrix T q,n (f ) is an Hermitian one, i.e., it is equal to its conjugate transpose: T q,n (f ) = [T q,n (f )] T . Determinants of Hermitian matrices are real numbers. Additionally, if n = 1, the determinant |T q,1 (f )| is rotationally invariant, i.e., for any real θ, the determinants |T q,1 (f )| and |T q,1 (f θ )| of the Hermitian Toeplitz matrices of functions f ∈ A and f θ (z) := e −iθ f (e iθ z) have the same values.
Recently, various problems regarding upper bounds, preferably sharp, of determinants involving coefficients of univalent functions, were rediscovered and attract significant interest. The highest focus is on the Hankel determinant and valuable references with overview of older results and the new ones are [2, 3, 6-9, 11, 18-25, 27-29].
Naturally rises the question of finding lower and upper bound estimates of the determinant of the Hermitian Toeplitz matrices for the class of univalent functions and its subclasses. This problem was initiated by Cudna et al. ( [4]) and Kowalczyk et al. ( [5]), and sharply solved in [4] for the classes of starlike and convex functions of order α, 0 ≤ α < 1, defined respectfully by For finding sharp estimates of the Hermitian Toeplitz determinant of second order, it is enough to know sharp estimate for the second coefficient. The same question for the third order determinant turns out to be more complicated.
In this paper, we introduce new method for obtaining estimates of the Hermitian Toeplitz determinants of third order and receive sharp result for the general class S of univalent functions.
We illustrate the new method also on the class of convex functions, receiving the same sharp result as in [4]. In a similar manner, we study classes and and "≺" denotes the ususal subordination. Class U(λ) is not included in the class of starlike functions S * := S * (0), nor vice versa (see [13,14]). Therefore, estimates for S * can not be transferred to the class U(λ). Sharp upper bound of the Hankel determinant of second and third order for the class U := U(1) are given in [19]. One can note that U = U s (1), since for all functions f from U, z/f (z) ≺ 1/(1 + z) 2 (see [12]), while the general implication 0 < λ < 1, was claimed in [15], but proven to be wrong in [10] by giving a counterexample.

Main results
We start with the following sharp estimates for the Hermitian Toeplitz determinants.
All inequalities are sharp.
In that sense we have two cases. The first one is 0 ≤ |a 2 | 2 ≤ 3, i.e., The result is sharp as the Koebe function k(z) shows.

Remark 1
(i) The same result as in Theorem 1 holds for the class S * = S * (0) (see Corollary 1 and Corollary 3 in [4]).
(ii) The same result as in Theorem 1 holds for the class U = U(1) since U ⊂ S and both extremal functions f 1 and k belong to U.
Remark 2 It is evident that for applying the method used in the proof of Theorem 1 on other classes of univalent functions, it is enough to know the sharp estimates for |a 2 |, |a 3 | and |a 3 − a 2 2 | and apply them on and on where ϕ(t) = −t 2 + 2|a 2 | 2 t − 2|a 2 | 2 + 1 and t = |a 3 |.
In the sense of Remark 2, for the class U s (λ), using the sharp estimates (estimate |a 2 | ≤ 1 + λ is sharp on the whole class U(λ)) given in [16], [17] and [10], we receive the following theorem.
Theorem 2 If f ∈ U(λ), then and if additionally f ∈ U s (λ), then where λ 0 = 0.44762 . . . is the positive real root of the equation All inequalities are sharp.
Proof. The estimates for the second Hermitian Toeplitz determinant follow directly from (3) and (5), and they are sharp due to the functions f 3 (z) = z and For the lower bound of the third Hermitian Toeplitz determinant, from (4) and (5), we have with sharpness for the function f 2 (z) = z/(1 − z + λz 2 ) = z + z 2 + (1 − λ)z 3 + · · · . Function f 2 is analytic on D since 1 − z + λz 2 equals zero on the unit disk only when λ = 0 and λ = −2.
For the upper bound of |T 3,1 (f )| we consider two cases. In the first one, when 0 ≤ |a 2 | 2 ≤ 1 + λ + λ 2 , the vertex of the parabola ϕ(t) is obtained for t = |a 2 | 2 and lies in the range of t = |a 3 |. Thus, Similarly, in the second case, 1 + λ + λ 2 ≤ |a 2 | 2 ≤ (1 + λ) 2 , we have that the vertex lies on the right of the range of t = |a 3 |. Thus, By using all these facts, we conclude that where λ 0 = 0.44762 . . . is the positive real root of the equation The upper bound of the third order determinant is also sharp with extremal function f 3 when 0 < λ ≤ λ 0 and f 4 when λ 0 ≤ λ ≤ 1.
For λ = 1, we receive the following corollary with the same estimates as for the class S already discussed in Remark 1(ii).
We conclude with two more applictions of Remark 2.