On sharp third Hankel determinant for certain starlike functions

Lemma 1.2. [10, 12] Let $p \in \mathcal{P}$ has the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then $$2p_2 = p_1^2 + \gamma(4 - p_1^2),$$ $$4p_3 = p_1^3 + 2p_1(4 - p_1^2)\gamma - p_1(4 - p_1^2)\gamma^2 + 2(4 - p_1^2)(1 - |\gamma|^2)\eta,$$ and $$8p_4 = p_1^4 + (4 - p_1^2)\gamma(p_1^2(\gamma^2 - 3\gamma + 3) + 4\gamma) - 4(4 - p_1^2)(1 - |\gamma|^2)(p_1(\gamma - 1)\eta + \bar{\gamma}\eta^2 - (1 - |\eta|^2)\rho),$$ for some $\gamma$ , $\eta$ and $\rho$ such that $|\gamma| \leq 1$ , $|\eta| \leq 1$ and $|\rho| \leq 1$ . 2. Sharp $$H_3(1)$$ for $\mathcal{S}^*(\alpha)$ Recently, Kowalczyk et al. [6] and Banga and Kumar [2] obtained the sharp bound of the thirdorder Hankel determinant for functions in the class $S^ := S^(0)$ , independently whereas Rath et al. [18] determined the sharp bound of $H_3(1)$ for functions in the class $S^(1/2)$ and corrected the proof provided in [11]. In this section, we extend our analysis to calculate the sharp bound of $H_3(1)$ for functions in the class $S^(\alpha)$ for some additional range of $\alpha$ . Below, is our main result.

Theorem 2.1. Let $f \in \mathcal{S}^*(\alpha)$ . Then <span id="page-2-4"></span> $$|H_3(1)| \le \frac{4(1-\alpha)^2}{9}, \quad \alpha \in [0, 1/6] \cup \{1/2\}.$$ (2.1) This result is sharp.

Definition 1.1. [15] For $0 \le \alpha < 1$ , we say that a function $f \in \mathcal{A}$ is starlike of order $\alpha$ if and only if $$\operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) > \alpha, \quad z \in \mathbb{D}.$$ The class of all such functions is represented by $\mathcal{S}^*(\alpha)$ . In 1992, Ma and Minda [13] introduced a more general class of starlike functions through subordination, defined as follows: $$\mathcal{S}^*(\varphi) = \left\{ f \in \mathcal{A} : \frac{zf'(z)}{f(z)} \prec \varphi(z) \right\},$$ where $\varphi$ is an analytic univalent function such that $\operatorname{Re} \varphi(z) > 0$ , $\varphi(\mathbb{D})$ is symmetric about the real axis and starlike with respect to $\varphi(0) = 1$ with $\varphi'(0) > 0$ . Through this concept, we can re-define the class $\mathcal{S}^*(\alpha)$ as: $$\mathcal{S}^*(\alpha) = \left\{ f \in \mathcal{A} : \frac{zf'(z)}{f(z)} \prec \frac{1 + (1 - 2\alpha)z}{1 - z}, \quad \alpha \in [0, 1) \right\}.$$ Note that $S^(0) = S$ and $S^(\varphi) \subset S^(\alpha)$ for some $\alpha$ depending upon the choice of $\varphi$ . ²⁰¹⁰ Mathematics Subject Classification. 30C45, 30C50. $Key\ words\ and\ phrases.$ Starlike, Sharp, Hankel determinant, Order alpha . | Class | Sharp bound | Reference | |------------------------------------------------------------|-------------|-----------| | $\mathcal{S}^ := \mathcal{S}^(0)$ | 4/9 | [2, 6] | | $\mathcal{S}^*(1/2)$ | 1/9 | [11, 18] | | $\mathcal{S}_{\rho}^ := \mathcal{S}^(1 + ze^z)$ | 1/9 | [19] | | $\mathcal{SL}^ := \mathcal{S}^(\sqrt{1+z})$ | 1/36 | [1] | | $\mathcal{S}_e^ := \mathcal{S}^(e^z)$ | 1/9 | [16] | | $\mathcal{S}_{\rho}^ := \mathcal{S}^(1 + \sinh^{-1}(z))$ | 1/9 | [17] | | $S_{Ne}^ := S^(1+z-z^3/3)$ | | _ | <span id="page-1-2"></span>Table 1. List of sharp third order Hankel determinants The Bieberbach conjecture, as documented on [4, Page no. 17], has been a significant source of inspiration in the development of univalent function theory and in the formulation of coefficient problems. Building on this foundation, in 1966, Pommerenke [14] introduced the concept of qth Hankel determinants, denoted as $H_q(n)$ , where n and q are both natural numbers, associated with analytic functions as in (1.1), defined as follows: <span id="page-1-0"></span> $$H_q(n) = \begin{vmatrix} a_n & a_{n+1} & \dots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \dots & a_{n+q} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n+q-1} & a_{n+q} & \dots & a_{n+2q-2} \end{vmatrix}.$$ (1.2) By choosing specific values for both n and q, we can examine particular cases of this concept. For instance, when we set q=2, we obtain the expression for the second-order Hankel determinant. Numerous studies have investigated and established sharp bounds for second-order Hankel determinants and other determinants within various subclasses of $\mathcal{S}$ , see [5,7,8] for more details. Now, if we choose q=3 and n=1 in (1.2), assuming $a_1:=1$ , we arrive at the expression for the Hankel determinant of order three, given by <span id="page-1-1"></span> $$H_3(1) := \begin{vmatrix} 1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} = a_3(a_2a_4 - a_3^2) - a_4(a_4 - a_2a_3) + a_5(a_3 - a_2^2).$$ (1.3) Determining the third-order Hankel determinant poses a greater challenge compared to the second-order, as evidenced in [9,22]. We also list some of the sharp estimates for the third-order Hankel determinant concerning functions within the class $\mathcal{S}^(\varphi)$ , considering various selections of $\varphi(z)$ in Table 1. However, the sharp estimate of $H_3(1)$ for $\mathcal{S}^_{Ne}$ is yet to be estimated. For the class $S^(\alpha)$ , Krishna and Ramreddy [7] computed the bound of the second order Hankel determinant, $|a_2a_4 - a_3^2| \leq (1 - \alpha)^2$ , $\alpha \in [0, 1/2]$ while Xu and Fang [21] calculated the sharp bounds of the Fekete and Szegö functional $|a_3 - \lambda a_2^2| \leq (1 - \alpha) \max\{1, |3 - 2\alpha - 4\lambda(1 - \alpha)|\}$ , $\lambda \in \mathbb{C}$ and $\alpha \in [0, 1)$ . We refer [3] for further information on Hankel determinants associated with the class $S^(\alpha)$ . The purpose of this study is to establish the sharp bound of third order Hankel determinant for functions belonging to the class, $S^(\alpha)$ . At the end of this paper, we demonstrate the validation of our main result by considering the class $S^(\alpha)$ specifically for the case when $\alpha = 0$ , and we also present some relevant applications. 1.1. Preliminary. In this part of the section, we mention the initial coefficient bounds $a_i$ (i = 2, 3, 4, 5) in terms of the Carathéodory coefficients and a lemma which will be used in our forthcoming results. Let $f \in \mathcal{S}^*(\alpha)$ , then a Schwarz function w(z) exists such that <span id="page-2-0"></span> $$\frac{zf'(z)}{f(z)} = \frac{1 + (1 - 2\alpha)w(z)}{1 - w(z)}. (1.4)$$ Let $p(z) = 1 + \sum_{n=2}^{\infty} p_n z^n \in \mathcal{P}$ and w(z) = (p(z) - 1)/(p(z) + 1). The expressions of $a_i (i = 2, 3, 4, 5)$ are obtained in terms of $p_j (j = 1, 2, 3, 4)$ by substituting w(z), p(z), and f(z) in equation (1.4) with suitable comparison of coefficients so that <span id="page-2-2"></span> $$a_2 = p_1(1 - \alpha), \tag{1.5}$$ $$a_3 = \frac{(1-\alpha)}{2} \left( p_2 + p_1^2 (1-\alpha) \right), \tag{1.6}$$ $$a_4 = \frac{(1-\alpha)}{6} \left( 2p_3 + 3p_1p_2(1-\alpha) + p_1^3(1-\alpha)^2 \right)$$ (1.7) and <span id="page-2-3"></span> $$a_5 = \frac{(1-\alpha)}{24} \left( 6p_4 + (1-\alpha) \left( 3p_2^2 + 8p_1p_3 \right) + (1-\alpha)^2 \left( 6p_1^2p_2 + p_1^4(1-\alpha) \right) \right). \tag{1.8}$$ The formula for $p_j$ (j = 2, 3, 4), which plays a significant role in finding the sharp bound of the Hankel determinant and has been prominently exploited in the main theorem, is contained in the Lemma 1.2 below.