Third Hankel Determinants for Subclasses of Univalent Functions

. The main aim of this paper is to discuss the third Hankel determinants for three classes: S ∗ of starlike functions, K of convex functions and R of functions whose derivative has a positive real part. Moreover, the sharp results for twofold and threefold symmetric functions from these classes are obtained.


Introduction
Let Δ be the unit disk {z ∈ C : |z| < 1} and A be the family of all functions f analytic in Δ, normalized by the condition f (0) = f (0) − 1 = 0. It means that f has the expansion f (z) = z + ∞ n=2 a n z n . Pommerenke (see, [11,12]) defined the q-th Hankel determinant for a function f as: H q (n) = a n a n+1 . . . a n+q−1 a n+1 a n+2 . . . a n+q · · · · · · · · · · · · a n+q−1 a n+q . . . a n+2q−2 , where n, q ∈ N. In recent years, the research on Hankel determinants has focused on the estimation of |H 2 (2)|. Many authors obtained results for various classes of univalent functions. It is worth citing a few of them. The exact estimates of |H 2 (2)| for the classes: S * of starlike functions, K of convex functions and R of functions whose derivative has a positive real part were proved by Janteng et al. [3,4]. They got the bounds: 1, 1/8 and 4/9, respectively. For the class S * (ϕ) of Ma-Minda starlike functions, the exact bound of the second Hankel determinant was obtained by Lee et al. [5]. The proof of the result |H 2 (2)| ≤ 1 for the class C of close-to-convex functions can be found in the paper [15] by Selvaraj and Kumar. Other results in this direction are presented in [2,7,10,16]. On the other hand, in [19] we obtained the sharp bounds: |H 2 (2)| ≤ 9 and |H 2 (3)| ≤ 15 for the class T of typically real functions.
The case q = 3 appears to be much more difficult than the case q = 2. Very few papers have been devoted to the third Hankel determinant. The first one was the paper by Babalola [1], who tried to estimate |H 3 (1)| for the classes S * , K and R. Following this paper, some other authors published their results concerning |H 3 (1)| (see, for example, [14,17,18]). In [1], it was proved that All results are sharp.
Moreover, Babalola claimed that the extremal functions for S * are the rotations of f (z) = z (1−z) 2 . The estimates given in Theorem 1.1 are true, but rather weak, and so, not sharp! We improve these estimates in the subsequent section. There we also discuss particular subclasses of S * , K and R consisting of functions with so-called n-fold symmetry. The results for these classes, which are presented in Theorem 3.1 and in Theorem 3.3, are sharp.
It appears interesting to discuss the third Hankel determinants for functions which in particular case reduce to f (z) = z sin(θ) , θ = arccos t are Chebyshev polynomials of the second kind. Then, applying the recurrence formula and the properties of determinants, we get It is obvious that for f t , t ∈ [−1, 1], this method yields that H q (n) = 0 for every positive integers q, n, such that q ≥ 3.
Moreover, Libera and Z lotkiewicz proved that Lemma 1.5 [6]. If p ∈ P, then for some x such that |x| ≤ 1.

Bounds of |H 3 (1)| for S * , K and R
At the beginning, observe that H 3 (1) can be written in the form or equivalently, where H 2 (k), k = 2, 3 are the second Hankel determinants defined by (1) and J 2 = a 3 a 4 − a 2 a 5 . The expression J 2 is a particular case of J n = a n+1 a n+2 − a n a n+3 .
It seems interesting to discuss this functional in a general case for n ∈ N.
As it can be seen in (1), H 3 (1) is a polynomial of four variables: a 2 , a 3 , a 4 , a 5 , where these numbers are successive coefficients of a function f in a given class. However, in many cases it is possible to connect the coefficients a 2 , a 3 , a 4 , a 5 with coefficients p 1 , p 2 , p 3 , p 4 of a function p ∈ P. To do this, we need to know the correspondence between f and p.
Let f , g, h be univalent. Then h ∈ R ⇔ h (z) ∈ P.
In the same way, we obtain the bound for f ∈ R taking into account that H(p 1 , p 2 , p 3 , p 4 ) = 1 20 An analogous calculation can be applied to obtain the result for J 2 defined by (7).
From the paper [9], we know that |J 1 | ≤ 2 for starlike functions. Hence, it is a natural question: whether |J n | ≤ 2 for all f ∈ S * and all positive integers n? Such a conjecture is supported by the fact that for the Koebe function f (z) = z (1−z) 2 we have |J n | = 2 for n = 1, 2, . . .

Bounds of |H 3 (1)| for Twofold and Threefold Symmetric Functions
Since the results in Theorem 2.1 are not sharp, it is interesting to pose a question about the magnitude of |H 3 (1)| for the discussed classes. We can give a partial answer considering functions satisfying an additional condition. For a given class A ⊂ A, a function f ∈ A is said to be n-fold symmetric if f (εz) = εf (z) holds for all z ∈ Δ, where ε = exp (2πi/n) means the principal n-th root of 1. The set of all n-fold symmetric functions belonging to A is denoted by A (n) . If f ∈ A (n) , then f has the Taylor series expansion f (z) = z + a n+1 z n+1 + a 2n+1 z 2n+1 + · · · . In case n = 2, the set A (2) consists of all functions in A which are odd.