Upper bound of second Hankel determinant of subclass of bi-univalent functions defined by subordination

Lemma 1.2. If $u(z) = \sum_{n=1}^{\infty} c_n z^n$ , $z \in \mathbb{U}$ , is a Schawrz function. Then, $$c_2 = (1 - c_1^2)x,$$ $c_3 = (1 - c_1^2)(1 - |x|^2)\xi - c_1(1 - c_1^2)x^2,$ for some complex values $x, \xi$ with $|x| \le 1$ and $|\xi| \le 1$ .

Lemma 1.3. Let u(z) be analytic function in the unit disc $\mathbb{U}$ with u(0) = 0 and |u(z)| < 1 for all $z \in U$ with the power series expansion $$u(z) = \sum_{n=1}^{\infty} c_n z^n \quad , z \in U$$ then $|c_n| \le 1$ for all n = 1, 2, 3, .... Furthermore, $|c_n| = 1$ for some n(n = 1, 2, 3, ...) if and only if $$u(z) = e^{i\theta} z^n, \quad \theta \in \mathbb{R}.$$ In this present work, we determine the upper bound of the second order Hankel determinant $|H_2(2)| = |a_2a_4 - a_3^2|$ of subclass of the class of analytic bi-univalant functions $\Sigma$ which defined by subordination. Furthermore, we introduce some results as a special case of our main results and correct some previous results.

Theorem 2.1. For $\lambda \geq 1$ , $0 \leq \delta \leq 1$ and $\tau \in \mathbb{C} - \{0\}$ , let f(z) defined by (1.1) belonging to the class $\mathcal{H}_{\Sigma}(\tau, \lambda, \delta; \varphi)$ , then $$|a_2a_4 - a_3^2| \le B_1|\tau|^2(P + Q + R) \tag{2.1}$$ where, <span id="page-3-1"></span> $$P = \left| \frac{B_3}{(1+\lambda+2\delta)(1+3\lambda+12\delta)} - \frac{B_1^3\tau^2}{(1+\lambda+2\delta)^4} \right| + \frac{B_1}{(1+2\lambda+6\delta)^2} + \frac{B_1^2|\tau|}{2(1+\lambda+2\delta)^2(1+2\lambda+6\delta)} + \frac{B_1+2|B_2|}{(1+\lambda+2\delta)(1+3\lambda+12\delta)},$$ $$Q = \frac{B_1+2|B_2|}{(1+\lambda+2\delta)(1+3\lambda+12\delta)} + \frac{2B_1}{(1+2\lambda+6\delta)^2} + \frac{B_1^2|\tau|}{2(1+\lambda+2\delta)^2(1+2\lambda+6\delta)},$$ $$R = \frac{B_1}{(1+2\lambda+6\delta)^2}.$$ (2.2)

Corollary 3.1. Let $f \in \mathcal{R}_{\sigma}(\lambda, \varphi)$ , then $$|a_{2}a_{4} - a_{3}^{2}| \leq B_{1} \left[ \left| \frac{B_{3}}{(1+\lambda)(1+3\lambda)} - \frac{B_{1}^{3}}{(1+\lambda)^{4}} \right| + \frac{4B_{1}}{(1+2\lambda)^{2}} + \frac{B_{1}^{2}}{(1+\lambda)^{2}(1+2\lambda)} + \frac{2B_{1}+4|B_{2}|}{(1+\lambda)(1+3\lambda)} \right].$$

Corollary 3.2. Let $f \in \mathcal{B}_{\Sigma}(\alpha, \lambda)$ , then $$|a_{2}a_{4} - a_{3}^{2}| \leq 2\alpha \left[ \left| \frac{4\alpha^{3} + 2\alpha}{3(1+\lambda)(1+3\lambda)} - \frac{8\alpha^{3}}{(1+\lambda)^{4}} \right| + \frac{8\alpha}{(1+2\lambda)^{2}} + \frac{4\alpha^{2}}{(1+\lambda)^{2}(1+2\lambda)} + \frac{4\alpha + 8\alpha^{2}}{(1+\lambda)(1+3\lambda)} \right].$$

Corollary 3.3. Let $f \in \mathcal{H}_{\Sigma}(\alpha, \beta)$ , then $$|a_{2}a_{4} - a_{3}^{2}| \leq 2\alpha \left[ \left| \frac{2\alpha^{3} + \alpha}{12(1+\beta)(1+3\beta)} - \frac{\alpha^{3}}{4(1+\beta)^{4}} \right| + \frac{8\alpha}{9(1+2\beta)^{2}} + \frac{\alpha^{2}}{3(1+\beta)^{2}(1+2\beta)} + \frac{\alpha+2\alpha^{2}}{2(1+\beta)(1+3\beta)} \right].$$

Corollary 3.4. Let $f \in \mathcal{N}_{\sigma}^{\alpha}$ , then $$|a_2a_4 - a_3^2| \le 2\alpha \left[ \left| \frac{4\alpha^3 - \alpha}{12} \right| + \frac{25\alpha}{18} + \frac{4\alpha^2}{3} \right].$$

Corollary 3.5. Let $f \in \mathcal{N}_{\Sigma}(\alpha, \lambda, \delta)$ , then $$|a_{2}a_{4} - a_{3}^{2}| \leq 2(1 - \alpha)^{2} \left[ \frac{8}{(1 + 2\lambda + 6\delta)^{2}} + \left| \frac{2}{(1 + \lambda + 2\delta)(1 + 3\lambda + 12\delta)} - \frac{8(1 - \alpha)^{2}}{(1 + \lambda + 2\delta)^{4}} \right| + \frac{4(1 - \alpha)}{(1 + \lambda + 2\delta)^{2}(1 + 2\lambda + 6\delta)} + \frac{12}{(1 + \lambda + 2\delta)(1 + 3\lambda + 12\delta)} \right].$$

Corollary 3.6. Let $f \in \mathcal{H}_{\Sigma}(\alpha, \delta)$ , then $$|a_2a_4 - a_3^2| \le 2(1 - \alpha)^2 \left[ \frac{8}{9(1 + 2\delta)^2} + \left| \frac{1}{4(1 + \delta)(1 + 3\delta)} - \frac{(1 - \alpha)^2}{2(1 + \delta)^2} \right| + \frac{(1 - \alpha)}{3(1 + \delta)^2(1 + 2\delta)} + \frac{3}{2(1 + \delta)(1 + 3\delta)} \right].$$

Corollary 3.7. Let $f \in \mathcal{N}_{\sigma}(\beta)$ , then $$|a_2a_4 - a_3^2| \le 2(1 - \alpha)^2 \left[ \frac{49}{18} - \frac{1}{3}\alpha + \left| \frac{1}{4} - \frac{(1 - \alpha)^2}{2} \right| \right].$$ Remark 2. Previous researchers got wrong results by miscalculation. We corrected their mistakes and obtained the correct result. - 1. Corollary (3.4) is a correction of the obtained estimates given in [7, Theorem 1]. - 2. Corollary (3.7) is a correction of the obtained estimates given in [7, Theorem 2].

Definition 1.1. Let $\lambda \ge 1$ , $\tau \in \mathbb{C}^* = \mathbb{C} - \{0\}$ , $0 \le \delta \le 1$ and let $f \in \Sigma$ given by (1.1), then f is said to be in the class $\mathcal{H}_{\Sigma}(\tau, \lambda, \delta; \varphi)$ if it satisfy the following condition $$1 + \frac{1}{\tau} \left( (1 - \lambda) \frac{f(z)}{z} + \lambda f'(z) + \delta z f''(z) - 1 \right) < \varphi(z)$$ $$\tag{1.4}$$ and $$1 + \frac{1}{\tau} \left( (1 - \lambda) \frac{g(w)}{w} + \lambda g'(w) + \delta w g''(w) - 1 \right) < \varphi(w). \tag{1.5}$$ where $z, w \in \mathbb{U}$ and $g = f^{-1} \in \Sigma$ given by (1.2). REMARK 1. For special choices of the parameters $\lambda$ , $\tau$ , $\delta$ and the function $\varphi(z)$ , we can obtain the following subclasses as a special case of our class $\mathcal{H}_{\Sigma}(\tau, \lambda, \delta; \varphi)$ : - 1. $\mathcal{H}_{\Sigma}(\tau, 1, \gamma; \varphi) = \Sigma(\tau, \gamma, \varphi)$ which introduced by Tudor [21] and Srivastava and Bansel [5]. - 2. $\mathcal{H}_{\Sigma}(1,\lambda,0;\varphi) = \mathcal{R}_{\sigma}(\lambda,\varphi)$ which defined and studied by Kumar et al. [15]. - 3. $\mathcal{H}_{\Sigma}\left(1,\lambda,0;\left(\frac{1+z}{1-z}\right)^{\alpha}\right) = \mathcal{B}_{\Sigma}(\alpha,\lambda)$ and $\mathcal{H}_{\Sigma}\left(1-\beta,\lambda,0;\frac{1+z}{1-z}\right) = \mathcal{B}_{\Sigma}(\beta,\lambda)$ which is introduced by Frasin and Aouf [11]. - 4. $\mathcal{H}_{\Sigma}\left(1,1,\beta;\left(\frac{1+z}{1-z}\right)^{\alpha}\right) = \mathcal{H}_{\Sigma}(\alpha,\beta)$ and $\mathcal{H}_{\Sigma}\left(1-\gamma,1,\beta;\frac{1+z}{1-z}\right) = \mathcal{H}_{\Sigma}(\gamma,\beta)$ which introduced by Frasin [12]. - 5. $\mathcal{H}_{\Sigma}\left(1,1,0;\left(\frac{1+z}{1-z}\right)^{\alpha}\right) = \mathcal{H}_{\Sigma}^{\alpha} = \mathcal{N}_{\sigma}^{\alpha}$ and $\mathcal{H}_{\Sigma}\left(1-\beta,1,0;\frac{1+z}{1-z}\right) = \mathcal{H}_{\Sigma}(\beta) = \mathcal{N}_{\sigma}(\beta)$ which introduced by Srivastava et al. [20] and recently studied by Çagler et al. [7]. - 6. $\mathcal{H}_{\Sigma}\left(1-\alpha,\lambda,\delta;\frac{1+z}{1-z}\right) = \mathcal{N}_{\Sigma}(\alpha,\lambda,\delta)$ which introduced by Bulut [6]. - 7. $\mathcal{H}_{\Sigma}\left(\tau, 1, \gamma; \frac{1+Az}{1+Bz}\right) = \mathcal{R}_{\gamma, \sigma}^{\tau}(A, B)$ which introduced by Tudor [21]. For proof of the main results we shall need the following lemma which proved by Kanas et al. [14],