Fekete-Szegö problem for certain classes of Ma-Minda bi-univalent functions

Lemma 1.1. (see [11]) If $p \in \mathcal{P}$ , then $|p_i| \leq 2$ for each i, where $\mathcal{P}$ is the family of all functions p, analytic in $\mathbb{U}$ , for which $$\Re\{p(z)\} > 0 \quad (z \in \mathbb{U}),$$ where $$p(z) = 1 + p_1 z + p_2 z^2 + \cdots \quad (z \in \mathbb{U}).$$ Motivated by the aforementioned works (especially [20] and [3, 10, 14]), we consider the following subclass of the function class $\Sigma$ (see also, [17]). A function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathcal{N}^{\mu,\lambda}_{\Sigma}(\varphi)$ if the following conditions are satisfied: $$(1 - \lambda) \left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z) \left(\frac{f(z)}{z}\right)^{\mu - 1} \prec \varphi(z) \qquad (\lambda \ge 1, \, \mu \ge 0, \, z \in \mathbb{U})$$ (1.8) and $$(1-\lambda)\left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w)\left(\frac{g(w)}{w}\right)^{\mu-1} \prec \varphi(w) \qquad (\lambda \ge 1, \ \mu \ge 0, \ w \in \mathbb{U}), \tag{1.9}$$ where $q(w) = f^{-1}(w)$ . <span id="page-2-2"></span>Remark 1.2. From among the many choices of $\mu$ , $\lambda$ and the function $\varphi$ which would provide the following known subclasses: - (1) $\mathcal{N}_{\Sigma}^{1,1}(\varphi) = \mathcal{H}_{\Sigma}^{\varphi}$ [1, p.345]. - (1) $\mathcal{N}_{\Sigma}$ $(\varphi) = \mathcal{H}_{\Sigma}$ (z) (z) $\mathcal{N}_{\Sigma}^{1,1}(\left(\frac{1+z}{1-z}\right)^{\beta}) = \mathcal{H}_{\Sigma}^{\beta} (0 < \beta \le 1)$ and $\mathcal{N}_{\Sigma}^{1,1}(\frac{1+(1-2\alpha)z}{1-z}) = \mathcal{H}_{\Sigma}^{\alpha} (0 \le \alpha < 1)$ [15, Definitions 1 and 2]. - (3) $\mathcal{N}_{\Sigma}^{1,\lambda}(\varphi) = \mathcal{R}_{\Sigma}(\lambda,\varphi) \quad (\lambda \geq 0) \quad [12, \text{ Definition 1.1}].$ - (4) $\mathcal{N}_{\Sigma}^{1,\lambda}((\frac{1+z}{1-z})^{\beta}) = \mathcal{B}_{\Sigma}(\beta,\lambda) \quad (\lambda \geq 1; 0 < \beta \leq 1) \text{ and } \mathcal{N}_{\Sigma}^{1,\lambda}((\frac{1+(1-2\alpha)z}{1-z})) = \mathcal{B}_{\Sigma}(\alpha,\lambda)$ $(\lambda \geq 1; 0 \leq \alpha < 1)$ [5, Definitions 2.1 and 3.1]. - (5) $\mathcal{N}_{\Sigma}^{\mu,1}(\varphi) = \mathcal{F}_{\Sigma}^{\mu}(\varphi) \quad (\mu \ge 0) \quad [12, \text{ Definition 2.1}].$ - (6) $\mathcal{N}_{\Sigma}^{0,1}(\left(\frac{1+z}{1-z}\right)^{\beta}) = \mathcal{S}_{\Sigma,\beta}^{}$ (0 < $\beta \le 1$ ) and $\mathcal{N}_{\Sigma}^{0,1}(\frac{1+(1-2\alpha)z}{1-z}) = \mathcal{S}_{\Sigma}^{}(\alpha)$ (0 $\le \alpha < 1$ ). (7) $\mathcal{N}_{\Sigma}^{\mu,\lambda}(\left(\frac{1+z}{1-z}\right)^{\beta}) = \mathcal{N}_{\Sigma}^{\mu,\lambda}(\beta)$ ( $\lambda \ge 1; \mu \ge 0; 0 < \beta \le 1$ ) [3, Definitions 2.1]. $\mathcal{N}_{\Sigma}^{\mu,\lambda}(\frac{1+(1-2\alpha)z}{1-z}) = \mathcal{N}_{\Sigma}^{\mu,\lambda}(\alpha) \quad (\lambda \geq 1; \mu \geq 0; 0 \leq \alpha < 1) \quad [3, \text{ Definitions 3.1}].$ In this paper we shall obtain the Fekete-Szegö inequalities for $\mathcal{N}^{\mu,\lambda}_{\Sigma}(\varphi)$ and its special classes. These inequalities will result in bounds of the third coefficient which are, in some cases, better than these obtained in [1, 3, 5, 14, 15, 17].

Theorem 2.1. Let f of the form (1.1) be in $\mathcal{N}^{\mu,\lambda}_{\Sigma}(\varphi)$ and $\delta \in \mathbb{R}$ . Then $$|a_3 - \delta a_2^2| \le \begin{cases} \frac{B_1}{2\lambda + \mu} & ; |\delta - 1| \le \frac{\mu + 1}{2} \left| 1 + \frac{2(B_1 - B_2)(\lambda + \mu)^2}{B_1^2(2\lambda + \mu)(1 + \mu)} \right| \\ \frac{2B_1^3 |\delta - 1|}{|(2\lambda + \mu)(1 + \mu)B_1^2 + 2(B_1 - B_2)(\lambda + \mu)^2|} & ; |\delta - 1| \ge \frac{\mu + 1}{2} \left| 1 + \frac{2(B_1 - B_2)(\lambda + \mu)^2}{B_1^2(2\lambda + \mu)(1 + \mu)} \right|. \end{cases}$$ (2.1)

Corollary 3.8. If $f \in \mathcal{S}^*_{\Sigma}(\varphi)$ then $$|a_3 - a_2^2| \le \frac{B_1}{2}.$$ <span id="page-5-2"></span>Corollary 3.9. If $f \in \mathcal{S}^*_{\Sigma}(\varphi)$ then $$|a_3| \leq \left\{ \begin{array}{cc} \frac{B_1}{2} & ; \frac{(B_1 - B_2)}{B_1^2} \in (-\infty, -3] \bigcup [0, \infty) \\ \frac{B_1^3}{|B_1^2 + (B_1 - B_2)|} & ; \frac{(B_1 - B_2)}{B_1^2} \in [-2, -1) \bigcup (-1, 1] \, . \end{array} \right.$$

Corollary 3.10. If $f \in \mathcal{S}_{\Sigma,\beta}^*$ then $$|a_3| \le \beta$$ and $|a_3 - a_2^2| \le \beta$ . <span id="page-5-3"></span>Corollary 3.11. If $f \in \mathcal{S}^*_{\Sigma}(\alpha)$ then $$|a_3| \le 1 - \alpha$$ and $|a_3 - a_2^2| \le 1 - \alpha$ . Remark 3.12. The inequalities estimated in Corollaries 3.9 to 3.11 are improvement of the inequalities obtained by Zaprawa [20, Corollaries 11 and 12, p.174].

Corollary 3.13. If $f \in \mathcal{R}_{\Sigma}(\lambda; \varphi)$ then $$|a_3 - a_2^2| \le \frac{B_1}{2\lambda + 1}.$$ <span id="page-5-4"></span>Corollary 3.14. If $f \in \mathcal{R}_{\Sigma}(\lambda; \varphi)$ then $$|a_3| \leq \left\{ \begin{array}{cc} \frac{B_1}{2\lambda + 1} & ; \frac{(B_1 - B_2)}{B_1^2} \in \left( -\infty, \frac{2(2\lambda + 1)}{(\lambda + 1)^2} \right] \bigcup [0, \infty) \\ \frac{B_1^3}{|(2\lambda + 1)B_1^2 + (B_1 - B_2)(\lambda + 1)^2|} & ; \frac{(B_1 - B_2)}{B_1^2} \in \left[ \frac{2(2\lambda + 1)}{(\lambda + 1)^2}, \frac{-(2\lambda + 1)}{(\lambda + 1)^2} \right) \bigcup \left( \frac{-(2\lambda + 1)}{(\lambda + 1)^2}, 0 \right]. \end{array} \right.$$ Remark 3.15. Corollary 3.14 provides an improvement of $|a_3|$ obtained by Sivaprasad Kumar et al. [12, Theorem 2.1, p.3]. <span id="page-5-5"></span>Corollary 3.16. If $f \in \mathcal{B}_{\Sigma}(\beta, \lambda)$ then $$|a_3| \le \frac{2\beta}{2\lambda + 1}$$ and $|a_3 - a_2^2| \le \frac{2\beta}{2\lambda + 1}$ . <span id="page-6-10"></span>Corollary 3.17. If $f \in \mathcal{B}_{\Sigma}(\alpha, \lambda)$ then $$|a_3| \le \frac{2(1-\alpha)}{2\lambda+1}$$ and $|a_3 - a_2^2| \le \frac{2(1-\alpha)}{2\lambda+1}$ . Remark 3.18. The bounds $|a_3|$ obtained in Corollaries 3.16 and 3.17 are improvement of the bounds $|a_3|$ estimated by Frasin and Aouf [5, Theorems 2.2 and 3.2, p.1570 and 1572], respectively. <span id="page-6-11"></span>Remark 3.19. If we take $$\varphi = \varphi_0 = \frac{1+z}{1-z} = 1 + 2z + 2z^2 + \dots$$ (3.1) in the class $\mathcal{N}_{\Sigma}^{\mu,\lambda}(\varphi)$ , we are led to the class which we denote, for convenience, by $\mathcal{N}_{\Sigma}^{\mu,\lambda}(\varphi_0)$ . In particular, $\mathcal{N}_{\Sigma}^{1,1}(\varphi_0) =: \mathcal{H}_{\Sigma}^{\varphi_0}, \, \mathcal{N}_{\Sigma}^{0,\mu}(\varphi_0) =: \mathcal{S}_{\Sigma}^(\varphi_0)$ and $\mathcal{N}_{\Sigma}^{1,\lambda}(\varphi) =: \mathcal{B}_{\Sigma}^(\lambda,\varphi_0)$ . In view of Remark 3.19, the Corollaries 3.1 and 3.2 yield the following corollaries. <span id="page-6-12"></span>Corollary 3.20. If $f \in \mathcal{N}^{\mu,\lambda}_{\Sigma}(\varphi_0)$ then $$|a_3| \le \frac{2}{2\lambda + \mu}$$ and $|a_3 - a_2^2| \le \frac{2}{2\lambda + \mu}$ . Remark 3.21. For $\mu = \lambda = 1$ the estimates in Corollary 3.20 would reduce to a known result in [20, Corollary 5, p.173]

Corollary 3.22. If $f \in \mathcal{S}^*_{\Sigma}(\varphi_0)$ then $$|a_3| \le 1$$ and $|a_3 - a_2^2| \le 1$ .

Corollary 3.23. If $f \in \mathcal{B}_{\Sigma}(\lambda, \varphi_0)$ then $$|a_3| \le \frac{2}{2\lambda + 1}$$ and $|a_3 - a_2^2| \le \frac{2}{2\lambda + 1}$ .