Fekete-Szegö problem and Second Hankel Determinant for a class of bi-univalent functions

Theorem 2.1. If f given by (1.1) is in the class $\mathcal{G}_{\sigma}^{\lambda}(\varphi)$ , then $$|a_2| \le \frac{B_1 \sqrt{B_1}}{\sqrt{4B_1 + |(3-\lambda)B_1^2 - 4B_2|}} \tag{2.1}$$ and <span id="page-2-4"></span> $$|a_3| \le \begin{cases} \left(1 - \frac{4}{3(1+\lambda)B_1}\right) \frac{B_1^3}{4B_1 + |(3-\lambda)B_1^2 - 4B_2|} + \frac{B_1}{3(1+\lambda)}, & if \quad B_1 \ge \frac{4}{3(1+\lambda)}; \\ \frac{B_1}{3(1+\lambda)}, & if \quad B_1 < \frac{4}{3(1+\lambda)}. \end{cases}$$ $$(2.2)$$

Corollary 2.1. Let $f \in \mathcal{K}_{\sigma}(\varphi)$ . Then $$|a_2| \le \frac{B_1 \sqrt{B_1}}{\sqrt{4B_1 + |2B_1^2 - 4B_2|}} \tag{2.19}$$ and $$|a_3| \le \begin{cases} \left(1 - \frac{2}{3B_1}\right) \frac{B_1^3}{4B_1 + |2B_1^2 - 4B_2|} + \frac{B_1}{6} ; B_1 \ge \frac{2}{3}; \\ \frac{B_1}{3(1+\lambda)} ; B_1 < \frac{2}{3}. \end{cases}$$ $$(2.20)$$

Lemma 3.1. (see [12] or [16]) Let $p(z) = 1 + p_1 z + p_2 z^2 + \cdots \in \mathcal{P}$ , where $\mathcal{P}$ is the family of all functions p, analytic in $\mathbb{U}$ , for which $\Re\{p(z)\} > 0$ , $z \in \mathbb{U}$ . Then $$|p_n| \le 2;$$ $n = 1, 2, 3, ...,$ and $$\left| p_2 - \frac{1}{2}p_1^2 \right| \le 2 - \frac{1}{2}|p_1|^2.$$

Theorem 3.1. Let f of the form (1.1) be in $\mathcal{G}^{\lambda}_{\sigma}(\varphi)$ . Then <span id="page-4-4"></span> $$|a_2| \le \begin{cases} \sqrt{\frac{B_1}{3-\lambda}}, & if \quad |B_2| \le B_1; \\ \sqrt{\frac{|B_2|}{3-\lambda}}, & if \quad |B_2| \ge B_1 \end{cases}$$ (3.1) and <span id="page-4-5"></span> $$\left| a_3 - \frac{4\lambda}{3+3\lambda} a_2^2 \right| \le \begin{cases} \frac{B_1}{3+3\lambda}, & \text{if } |B_2| \le B_1; \\ \frac{|B_2|}{3+3\lambda}, & \text{if } |B_2| \ge B_1. \end{cases}$$ (3.2)

Lemma 4.1. [25] If the function $p \in \mathcal{P}$ is given by the series <span id="page-6-0"></span> $$p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \cdots, (4.1)$$ then the following sharp estimate holds: $$|p_n| \le 2, \qquad n = 1, 2, \cdots.$$ (4.2)

Lemma 4.2. [15] If the function $p \in \mathcal{P}$ is given by the series (4.1), then $$2c_2 = c_1^2 + x(4 - c_1^2)$$ $$4c_3 = c_1^3 + 2c_1(4 - c_1^2)x - c_1(4 - c_1^2)x^2 + 2(4 - c_1^2)(1 - |x|^2)z$$ for some x, z with $|x| \le 1$ and $|z| \le 1$ . The following theorem provides a bound for the second Hankel determinant of the functions in the class $\mathcal{G}_{\sigma}^{\lambda}(\beta)$ .

Theorem 4.1. Let f of the form (1.1) be in $\mathcal{G}^{\lambda}_{\sigma}(\beta)$ . Then $$|a_{2}a_{4}-a_{3}^{2}| \leq \begin{cases} \frac{(1-\beta)^{2}}{2(1+2\lambda)} \left[ (2-\lambda)(1-\beta)^{2}+1 \right] ; \\ \beta \in \left[ 0, 1 - \frac{(1+2\lambda)+\sqrt{(1+2\lambda)^{2}+18(1+\lambda)^{2}(2-\lambda)}}{6(1+\lambda)(2-\lambda)} \right] \\ \frac{36[8(1+2\lambda)(2-\lambda) - (1+2\lambda)^{2}](1-\beta)^{2}}{-324(1+\lambda)(1+2\lambda)(1-\beta) + 288(1+2\lambda) - 729(1+\lambda)^{2}} \\ \frac{-324(1+\lambda)(1+2\lambda)(1-\beta) + 288(1+2\lambda) - 729(1+\lambda)^{2}}{9(1+\lambda)^{2}(2-\lambda)(1-\beta)^{2} - 6(1+\lambda)(1+2\lambda)(1-\beta)} \end{cases} ; \\ +8(1+2\lambda) - 18(1+\lambda)^{2} \\ \beta \in \left( 1 - \frac{(1+2\lambda)+\sqrt{(1+2\lambda)^{2}+18(1+\lambda)^{2}(2-\lambda)}}{6(1+\lambda)(2-\lambda)}, 1 \right) . \end{cases}$$

Corollary 4.1. Let f of the form (1.1) be in $\mathcal{H}^{\beta}_{\sigma}$ . Then $$|a_2 a_4 - a_3^2| \le \begin{cases} \frac{(1-\beta)^2 [1+2(1-\beta)^2]}{2} ; & \beta \in \left[0, \frac{11-\sqrt{37}}{12}\right] \\ \frac{(1-\beta)^2 [60\beta^2 - 84\beta - 25]}{16(9\beta^2 - 15\beta + 1)} ; & \beta \in \left(\frac{11-\sqrt{37}}{12}, 1\right). \end{cases}$$

Corollary 4.2. Let f of the form (1.1) be in $\mathcal{K}_{\sigma}(\beta)$ . Then $$|a_2 a_4 - a_3^2| \le \frac{(1-\beta)^2}{24} \left[ \frac{5\beta^2 + 8\beta - 32}{3\beta^2 - 3\beta - 4} \right].$$

Corollary 4.3. Let f of the form (1.1) be in $\mathcal{H}_{\sigma}$ . Then $$|a_2a_4 - a_3^2| \le \frac{3}{2} .$$

Corollary 4.4. Let f of the form (1.1) be in $\mathcal{K}_{\sigma}$ . Then $$|a_2a_4 - a_3^2| \le \frac{1}{3} .$$

Definition 1.1. For $0 \le \lambda \le 1$ and $0 \le \beta < 1$ , a function $f \in \sigma$ given by (1.1) is said to be in the class $\mathcal{G}^{\lambda}_{\sigma}(\varphi)$ if the following conditions are satisfied: $$(1-\lambda)f'(z) + \lambda\left(1 + \frac{zf''(z)}{f'(z)}\right) \prec \varphi(z), \qquad 0 \le \lambda \le 1, z \in \mathbb{U}$$ and for $g = f^{-1}$ given by (1.3) $$(1 - \lambda)g'(w) + \lambda \left(1 + \frac{wg''(w)}{g'(w)}\right) \prec \varphi(w), \qquad 0 \le \lambda \le 1, \ w \in \mathbb{U}.$$ From among the many choices of $\varphi$ and $\lambda$ which would provide the following known subclasses: - (1) $\mathcal{G}_{\sigma}^{0}(\varphi) := \mathcal{H}_{\sigma}(\varphi)$ [3], (2) $\mathcal{G}_{\sigma}^{1}(\varphi) := \mathcal{K}_{\sigma}(\varphi)$ [3], (3) $\mathcal{G}_{\sigma}^{\lambda}(\frac{1+(1-2\beta)z}{1-z}) := \mathcal{G}_{\sigma}^{\lambda}(\beta)$ $(0 \le \beta < 1)$ [4]. (4) $\mathcal{G}_{\sigma}^{0}(\frac{1+(1-2\beta)z}{1-z}) := \mathcal{H}_{\sigma}^{\beta}$ $(0 \le \beta < 1)$ [28] - (5) $\mathcal{G}_{\sigma}^{1}(\frac{1+(1-2\beta)z}{1-z}) := \mathcal{K}_{\sigma}(\beta) \quad (0 \le \beta < 1)$ [5]. In this paper we shall obtain the Fekete-Szegö inequalities for $\mathcal{G}_{\sigma}^{\lambda}(\varphi)$ as well as its special classes. Further, the second Hankel determinant obtained for the class $\mathcal{G}^{\lambda}_{\sigma}(\beta)$ .