Third Hankel determinant for starlike and convex functions with respect to symmetric points

The objective of this paper is to obtain best possible upper bound to theH3(1)Hankel determinant for starlike and convex functions with respect to symmetric points, using Toeplitz determinants.

One can easily observe that the Fekete-Szegő functional is H 2 (1). Fekete-Szegő then further generalized the estimate |a 3 −µa 2 2 | with µ real and f ∈ S. Ali [1] found sharp bounds on the first four coefficients and sharp estimate for the Fekete-Szegő functional |γ 3 − tγ 2 2 |, where t is real, for the inverse function of f defined as f −1 (w) = w + ∞ n=2 γ n w n , when f ∈ ST (α), the class of strongly starlike functions of order α (0 < α ≤ 1). Further, sharp bounds for the functional when q = 2 and n = 2, known as the second Hankel determinant, were obtained for various subclasses of univalent and multivalent analytic functions. For our discussion, in this paper, we consider the Hankel determinant in the case of q = 3 and n = 1, denoted by H 3 (1), given by For f ∈ A, a 1 = 1, so that, we have ) and by applying triangle inequality, we obtain [2] obtained sharp upper bounds to the functional |a 2 a 3 − a 4 | and |H 3 (1)| for the familiar subclasses namely starlike and convex functions respectively denoted by ST and CV of S. The sharp upper bounds to the second Hankel determinant |a 2 a 4 − a 2 3 | for the classes ST and CV were obtained by Janteng et al. [6].
Motivated by the results obtained by Babalola [2] and recently by Raja and Malik [11] in finding the sharp upper bound to the Hankel determinant |H 3 (1)| for certain subclasses of S, in this paper, we obtain an upper bound to the functional |a 2 a 3 − a 4 | and hence for |H 3 (1)|, for the function f given in (1.1), belonging to the classes namely starlike with respect to symmetric points and convex with respect to symmetric points denoted by ST s and CV s respectively, defined as follows.
The class ST s was introduced and studied by Sakaguchi [15]. Further, he has shown that the functions in ST s are close-to-convex and hence are univalent. The concept of starlike functions with respect to symmetric points have been extended to starlike functions with respect to N -symmetric points by Ratanchand [14] and Prithvipalsingh [10]. RamReddy [12] studied the class of close-to-convex functions with respect to N -symmetric points and proved that this class is closed under convolution with convex univalent functions.
The class CV s was introduced and studied by Das and Singh [3]. From the Definitions 1.1 and 1.2, it is evident that f ∈ CV s if and only if zf ∈ ST s . Some preliminary lemmas required for proving our results are as follows: 2. Preliminary Results. Let P denote the class of functions consisting of p, such that which are regular in the open unit disc E and satisfy Re p(z) > 0, for any z ∈ E. Here p(z) is called the Carathéodory function [4]. 8,16]). If p ∈ P, then |c k | ≤ 2, for each k ≥ 1 and the inequality is sharp for the function 1+z 1−z . Lemma 2.2 ([5]). The power series for p(z) = 1 + ∞ n=1 c n z n given in

1) converges in the open unit disc E to a function in P if and only if the Toeplitz determinants
, n = 1, 2, 3 . . . . and c −k = c k , are all non-negative. They are strictly positive except for This necessary and sufficient condition found in [5] is due to Carathéodory and Toeplitz. We may assume without restriction that c 1 > 0. On using Lemma 2.2, for n = 2, we have for some x, |x| ≤ 1. For n = 3, and is equivalent to In obtaining our results, we refer to the classical method devised by Libera and Złotkiewicz [7] and used by several authors in the literature.
n=2 a n z n ∈ CV s , from the Definition 1.2, there exists an analytic function p ∈ P in the unit disc E with p(0) = 1 and Re p(z) > 0 such that Replacing f (z) , f (z), f (−z) and p(z) with their series equivalent expressions in (3.20) and applying the same procedure as described in Theorem 3.1, we get Substituting the values of a 2 , a 3 , and a 4 from (3.21) in |a 2 a 3 − a 4 | for the function f ∈ CV s , upon simplification, we obtain Applying the same procedure as described in Theorem 3.1, we arrive at Next, we maximize the function F (µ) on the closed region This completes the proof of our Theorem 3.5.
The following results are straightforward verification on applying the same procedure of Theorems 3.2 and 3.3 respectively. Theorem 3.6. If f (z) ∈ CV s , then |a 3 − a 2 2 | ≤ 1 3 and the inequality is sharp for the values c 1 = c = 0, c 2 = 2 and x = 1.