Second Hankel determinant for a certain subclass of bi-close to convex functions defined by Kaplan

Lemma 1.1. [24] If the function $p \in \mathcal{P}$ is given by the series (1.11), then $|p_k| \leq 2, k = 1, 2, ...$

Lemma 1.2. [10] If the function $p \in \mathcal{P}$ is given by the series (1.11), then $$\begin{array}{rcl} 2p_2 & = & p_1^2 + x(4 - p_1^2), \\ 4p_3 & = & p_1^3 + 2p_1(4 - p_1^2)x - p_1(4 - p_1^2)x^2 + 2(4 - p_1^2)(1 - |x|^2)z, \end{array}$$ for some x, z with $|x| \le 1$ and $|z| \le 1$ .

Lemma 1.3. [13] If the function $\phi \in \mathcal{CV}$ , then for $\lambda \in \mathbb{R}$ , $$|c_3 - \lambda c_2^2| \le \begin{cases} 1 - \lambda & for \quad \lambda < 2/3, \\ 1 & for \quad 2/3 \le \lambda \le 4/3, \\ \lambda - 1 & for \quad \lambda > 4/3. \end{cases}$$ (1.12)

Lemma 1.4. [12] If the function $\phi \in \mathcal{CV}$ , then $|c_2c_4 - c_3^2| \leq \frac{1}{8}$ .

Lemma 1.5. [2] If the function $\phi \in \mathcal{CV}$ , then $|c_2c_3-c_4| \leq \frac{1}{6}$ . 2. SECOND HANKEL DETERMINANT IN CLASS $\mathcal{K}_{\Sigma}[\alpha]$ AND $\mathcal{K}_{\Sigma}(\beta)$ The first aim of this section is to find the best bound of the second Hankel determinant in the class $\mathcal{K}_{\Sigma}[\alpha]$ . A successful method of finding such bound has been exploited in [28] and other related publications. 2.1. The class $\mathcal{K}_{\Sigma}(\beta)$ . In the family of strongly bi-close-to-convex of order $\alpha$ we have the following non-sharp estimates of $H_2(2)$ however, this bound, for a particular selection of $\alpha$ , improves the earlier results in [6].

Theorem 2.1. Let $0 \le \alpha \le 1$ , and let the function f, given by (1.1), be in the class $\mathcal{K}_{\Sigma}[\alpha]$ . Then $$|a_2 a_4 - a_3^2| \le \frac{1}{8} + \frac{3}{2}\alpha + \frac{43}{9}\alpha^2 + \frac{1}{3}\alpha^3 + \frac{4}{3}\alpha^4. \tag{2.1}$$

Corollary 2.1. For $0 \le \alpha < 1$ , and $f \in \mathcal{K}_{\Sigma} \equiv \mathcal{K}_{\Sigma}[0]$ we have $$|a_2 a_4 - a_3^2| \le \frac{1}{8}. (2.17)$$ 2.2. The class $\mathcal{K}_{\Sigma}(\beta)$ . In order to estimate the second Hankel determinat in $\mathcal{K}_{\Sigma}(\beta)$ we apply consideration similar to that used in the proof of 2.1.

Theorem 2.2. Let $0 \le \beta < 1$ , and let the function f given by (1.1) be in the class $\mathcal{K}_{\Sigma}(\beta)$ . Then $$|a_2a_4 - a_3^2| \le \frac{1}{8} + \frac{19}{6}(1 - \beta) + 6(1 - \beta)^2 + 4(1 - \beta)^3 + (1 - \beta)^4.$$ (2.18)

Definition 1.1. [28] A function $f \in \Sigma$ of the form (1.1) belongs to the class of bi-close to convex functions $\mathcal{K}_{\Sigma}$ , if there exist a function $\phi$ , convex and univalent for $z \in \mathbb{D}$ , such that $$\Re\left\{\frac{f'(z)}{\phi'(z)}\right\} \ge 0$$ , and $\Re\left\{\frac{g'(w)}{\phi'(w)}\right\} \ge 0$ $(z, w \in \mathbb{D})$ , where g is the analytic continuation of $f^{-1}$ to $\mathbb{D}$ with a series expansion (1.2).

Definition 1.2. [28] Let $0 \le \alpha \le 1$ . A function $f \in \Sigma$ , given by (1.1), is said to be strongly bi-close-to-convex of order $\alpha$ if there exist bi-convex functions $\phi$ and $\psi$ such that <span id="page-1-0"></span> $$\left| \arg \left( \frac{f'(z)}{\phi'(z)} \right) \right| < \alpha \pi/2 \quad and \quad \left| \arg \left( \frac{g'(w)}{\psi'(w)} \right) \right| < \alpha \pi/2 \quad (z, w \in \mathbb{D}). \tag{1.7}$$ Here, g is the analytic continuation of $f^{-1}$ to $\mathbb{D}$ . We denote the class of strongly bi-close-to-convex functions of order $\alpha$ by $K_{\Sigma}[\alpha]$ . Remark 1.1. We note that $\mathcal{K}_{\Sigma}[1] \equiv \mathcal{K}_{\Sigma}$ , and $\mathcal{K}_{\Sigma}[0] \equiv \mathcal{CV}_{\Sigma}[4]$ .

Definition 1.3. [28] Let $0 \le \beta < 1$ . A function $f \in \Sigma$ , given by (1.1) is said to be bi-close-to-convex of order $\beta$ if there exist the bi-convex functions $\phi$ and $\psi \in \mathcal{CV}_{\Sigma}$ such that: <span id="page-1-1"></span> $$\Re\left(\frac{f'(z)}{\phi'(z)}\right) > \beta \quad and \quad \Re\left(\frac{g'(w)}{\psi'(w)}\right) > \beta \quad (z, w \in \mathbb{D}),$$ (1.8) where g is the analytic continuation of $f^{-1}$ to $\mathbb{D}$ . We denote the class of bi-close-to-convex functions of order $\beta$ by $\mathcal{K}_{\Sigma}(\beta)$ . Remark 1.2. We note that $\mathcal{K}_{\Sigma}(0) \equiv \mathcal{K}_{\Sigma}$ . Also, for $\phi(z) = z$ , the class $N_{\Sigma}(\alpha)$ $(0 \le \alpha \le 1)$ reduces to the family of functions $f \in \Sigma$ , satisfying the condition, $$\left|\arg f'(z)\right| < \alpha\pi/2 \quad and \quad \left|\arg g'(w)\right| < \alpha\pi/2 \quad (z, w \in \mathbb{D}),$$ and $\mathcal{K}_{\Sigma}(\beta)$ reduces to $N_{\Sigma}(\beta)$ defined by the conditions $$\Re\left(f'(z)\right) > \beta$$ and $\Re\left(g'(w)\right) > \beta$ $(z, w \in \mathbb{D}),$ where the function g is defined by (1.2). These classes were studied by Çağlar et al. [5] Observe that if f is given by (1.1), then $g = f^{-1}$ is given by (1.2), and if <span id="page-2-0"></span> $$\phi(z) = z + c_2 z^2 + c_3 z^3 + c_4 z^4 + \cdots, \tag{1.9}$$ then <span id="page-2-1"></span> $$\psi(w) = \phi^{-1}(w) = w - c_2 w^2 + (2c_2^2 - c_3)w^3 - (5c_2^3 - 5c_2 c_3 + c_4)w^4 + \cdots$$ (1.10) In the sequel we assume that $q, \phi, \psi$ have Taylor expansions as in (1.2), (1.9) and (1.10). 1.3. Hankel determinant. Towards the full understanding of a behavior of bi-univalence, it is necessary to extend our attention to the Hankel determinants, that is one of the most important tool in Geometric Function Theory, defined by Pommerenke [25, 26]. Noonan and Thomas [17] defined the $q^{th}$ Hankel determinant of f given by (1.1) for integral $n \ge 1$ and $q \ge 1$ by $$H_q(n) = \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \cdots & a_{n+q} \\ \vdots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots \\ a_{n+q-1} & a_{n+q} & \cdots & a_{n+2q-2} \end{vmatrix}$$ The importance of the Hankel determinants was recognized over half a century ago and it has been studied in great details, see for example [25, 26]. The significance of the Hankel determinants follows from the study of singularities of analytic functions [7, p. 329], see also [8], and from the fact that it contains the Fekete-Szegö functional with its generalization [9]. Moreover, $H_2(2) = a_2 a_4 - a_3^2$ is well known second Hankel determinant. The Hankel determinant is useful for estimating the modulus of coefficients and the rate of growth of the coefficients. Both estimates determine the behavior of the studied function when the function itself and its properties are unknown. Extensive studies of the Hankel determinant in the theory of meromorphic functions are due to Wilson [29]; numerous applications in mathematical physics are given by Vein and Dale [30]. Recently, many authors have discussed upper bounds for the Hankel determinant and Fekete-Szegö functional for numerous subclasses of univalent functions [7, 14, 16, 17, 19, 21] and references therein. Very recently, the upper bounds of $H_2(2)$ for the classes $\mathcal{S}_{\Sigma}^*(\alpha)$ and $K_{\Sigma}(\alpha)$ were investigated by Deniz et al. [6], and extended by Orhan et al. [18, 23]. Sivasubramanian et al. [28] found the estimates for $|a_2|$ and $|a_3|$ for the classes $\mathcal{K}_{\Sigma}$ , $\mathcal{K}_{\Sigma}[\alpha]$ and $\mathcal{K}_{\Sigma}(\beta)$ . Further they verified Brannan and Clunie's conjecture $|a_2| \leq \sqrt{2}$ for some of their subclasses. Therefore, a naturally arising problem addressed in this paper is to investigate the behavior of the Hankel determinants in the newly defined families. 1.4. Some useful bounds. Let $\mathcal{P}$ denote the class of functions p(z) of the form <span id="page-2-2"></span> $$p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \dots, (1.11)$$ which are analytic in the open unit disk $\mathbb{D}$ and such that $\Re p(z) > 0, z \in \mathbb{D}$ .