Coefficient and Fekete-Szegö problem estimates for certain subclass of analytic and bi-univalent functions

Lemma 1.1. Let $f(z) \in A$ and $\pi/2 \le \alpha < \pi$ . Then $f \in \mathcal{M}(\alpha)$ if, and only if, $$\left(\frac{zf'(z)}{f(z)}-1\right) \prec \mathcal{B}_{\alpha}(z) \quad (z \in \Delta),$$ where (1.3) $$\mathcal{B}_{\alpha}(z) := \frac{1}{2i\sin\alpha} \log\left(\frac{1+ze^{i\alpha}}{1+ze^{-i\alpha}}\right) \quad (z \in \Delta).$$ The function $\mathcal{B}_{\alpha}(z)$ is convex univalent and has the form (1.4) $$\mathcal{B}_{\alpha}(z) = \sum_{n=1}^{\infty} A_n z^n \quad (z \in \Delta),$$ where $$A_n := \frac{(-1)^{(n-1)} \sin n\alpha}{n \sin \alpha} \quad (n = 1, 2, \ldots).$$ Also we have $\mathcal{B}_{\alpha}(\Delta) = \Omega_{\alpha}$ where $$\Omega_{\alpha} := \left\{ \zeta \in \mathbb{C} : \frac{\alpha - \pi}{2 \sin \alpha} < \operatorname{Re} \left\{ \zeta \right\} < \frac{\alpha}{2 \sin \alpha}, \quad \frac{\pi}{2} \le \alpha < \pi \right\}.$$ Very recently Sun et al. (see [22]) and Kwon and Sim (see [11]) have studied the class $\mathcal{M}(\alpha)$ . Sun et al. showed if the function f of the form (1.1) belongs to the class $\mathcal{M}(\alpha)$ , then $|a_n| \leq 1$ while the estimate is not sharp. Subsequently, Kwon and Sim obtained sharp estimates on the initial coefficients $a_2$ , $a_3$ , $a_4$ and $a_5$ of the functions f belonging to the class $\mathcal{M}(\alpha)$ . The coefficient estimate problem for each of the Taylor-Maclaurin coefficients $|a_n|$ (n = 6, 7, ...) is still an open question. Also, the logarithmic coefficients of the function $f \in \mathcal{M}(\alpha)$ were estimated by Kargar, see [7]. It is interesting to mention this subject that Brannan and Taha [4] introduced certain subclass of the bi-univalent function class $\Sigma$ , denoted by $\mathcal{S}_{\Sigma}^(\gamma)$ similar to the class of the starlike functions of order $\gamma$ (0 $\leq \gamma < 1$ ). For each function $f \in \mathcal{S}_{\Sigma}^(\gamma)$ they found non-sharp estimates for the initial Taylor-Maclaurin coefficients. Recently, motivated by the Brannan and Taha's work, many authors investigated the coefficient bounds for various subclasses of the bi-univalent function class $\Sigma$ , see for instance [5, 15, 16, 17, 18, 19, 20, 21]. In this paper, motivated by the aforementioned works, we introduce and investigate a certain subclass of $\Sigma$ similar to the class $\mathcal{M}(\alpha)$ as follows.

Lemma 1.2. (see [13]) Let the function p be of the form belongs to the class $\mathcal{P}$ . Then for any complex number $\mu$ we have $$|p_2 - \mu p_1^2| \le \begin{cases} -4\mu + 2, & \text{if } \mu \le 0; \\ 2, & \text{if } 0 \le \mu \le 1; \\ 4\mu - 2, & \text{if } \mu \ge 1. \end{cases}$$ The result is sharp for the cases $\mu < 0$ or $\mu > 1$ if and only if $p(z) = \frac{1+z}{1-z}$ or one of its rotations. If $0 < \mu < 1$ , then the equality holds if and only if $p(z) = \frac{1+z^2}{1-z^2}$ or one of its rotations. For the case $\mu = 0$ , the equality holds if and only if $$p(z) = \frac{1}{2}(1+\nu)\frac{1+z}{1-z} + \frac{1}{2}(1-\nu)\frac{1-z}{1+z} \quad (0 \le \nu \le 1),$$ or one of its rotations. If $\mu = 1$ , the equality holds if and only if $$\frac{1}{p(z)} = \frac{1}{2}(1+\nu)\frac{1+z}{1-z} + \frac{1}{2}(1-\nu)\frac{1-z}{1+z} \quad (0 \le \nu \le 1),$$ or one of its rotations. 4 This paper is organized as follows. In Section 2 we derive the Fekete-Szegö coefficient functional associated with the k-th root transform for functions in the class $\mathcal{M}(\alpha)$ . In Section 3 we propose to find the estimates on the Taylor-Maclaurin coefficients $|a_2|$ , $|a_3|$ and Fekete-Szegö problem for functions in the class $\mathcal{M}_{\Sigma}(\alpha)$ which we introduced in Definition 1.1.

Theorem 2.1. Let $\pi/2 \le \alpha < \pi$ and $f \in \mathcal{M}(\alpha)$ . If F is the k-th $(k \ge 1)$ root transform of the function f defined by (2.1), then for any complex number $\mu$ we have (2.4) $$|b_{2k+1} - \mu b_{k+1}^2| \le \begin{cases} \frac{1}{2k} \left( 1 - \cos \alpha - \frac{2\mu + k - 1}{k} \right), & \text{if } \mu \le \delta_1; \\ \frac{1}{2k}, & \text{if } \delta_1 \le \mu \le \delta_2; \\ \frac{1}{2k} \left( \cos \alpha + \frac{2\mu + k - 1}{k} - 1 \right), & \text{if } \mu \ge \delta_2, \end{cases}$$ where $\delta_1 := (1 - k(1 + \cos \alpha))/2$ , $\delta_2 := (1 + k(1 - \cos \alpha))/2$ and $b_{2k+1}$ and $b_{k+1}$ are defined by (2.3). The result is sharp.

Corollary 2.1. Let $\alpha \in [\pi/2, \pi)$ and $f \in \mathcal{M}(\alpha)$ . Then for any complex number $\mu$ we have $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{1}{2}(1 - \cos \alpha) - \mu, & \text{if } \mu \le -\frac{1}{2}\cos \alpha; \\ \frac{1}{2}, & \text{if } -\frac{1}{2}\cos \alpha \le \mu \le 1 - \frac{1}{2}\cos \alpha; \\ \frac{1}{2}(\cos \alpha - 1) + \mu, & \text{if } \mu \ge 1 - \frac{1}{2}\cos \alpha. \end{cases}$$ The result is sharp. Putting $\alpha = \pi/2$ in the Corollary 2.1 we get the following.

Corollary 2.2. Let the function f be given by (1.1) satisfies the inequality $$\left| \operatorname{Re} \left\{ \frac{zf'(z)}{f(z)} \right\} - 1 \right| < \frac{\pi}{4} \quad (z \in \Delta).$$ 6 Then for any complex number $\mu \in \mathbb{C}$ we have the following sharp inequalities $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{1}{2} - \mu, & \text{if } \mu \le 0; \\ \frac{1}{2}, & \text{if } 0 \le \mu \le 1; \\ \mu - \frac{1}{2}, & \text{if } \mu \ge 1. \end{cases}$$ If we let $\alpha \to \pi^-$ in the Corollary 2.1, then we have:

Corollary 2.3. If the function f of the form (1.1) is starlike of order 1/2, then for any complex number $\mu \in \mathbb{C}$ the following sharp inequalities hold true. $$|a_3 - \mu a_2^2| \le \begin{cases} 1 - \mu, & \text{if } \mu \le \frac{1}{2}; \\ \frac{1}{2}, & \frac{1}{2} \le \mu \le \frac{3}{2}; \\ \mu - 1, & \text{if } \mu \ge \frac{3}{2}. \end{cases}$$ From (2.7) and (2.8) and the first case of the Lemma 1.2 we get

Corollary 2.4. If a function $f \in A$ of the form (1.1) belongs to the class $\mathcal{M}(\alpha)$ ( $\pi/2 \leq \alpha < \pi$ ), then the following sharp inequalities hold. $$|a_2| \le 1$$ and $|a_3| \le \frac{1}{2}(1 - \cos \alpha)$ .

Theorem 3.1. Let the function f given by (1.1) be in the class $\mathcal{M}_{\Sigma}(\alpha)$ and $\pi/2 \leq \alpha < \pi$ . Then $$|a_2| \le \sqrt{\frac{2}{2 + \cos \alpha}}$$ and for any real number $\mu$ we have $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{1}{2}, & \text{if } |1 - \mu| \le \frac{1}{2} \left( 1 + \frac{1}{2} \cos \alpha \right); \\ \frac{|1 - \mu|}{1 + \frac{1}{2} \cos \alpha}, & \text{if } |1 - \mu| \ge \frac{1}{2} \left( 1 + \frac{1}{2} \cos \alpha \right). \end{cases}$$

Corollary 3.1. Let f of the form (1.1) be in the class $\mathcal{M}_{\Sigma}(\alpha)$ . Then $$|a_3| \le \frac{1}{1 + \frac{1}{2}\cos\alpha} \quad (\pi/2 \le \alpha < \pi).$$ If we let $\alpha \to \pi^-$ in the Theorem 3.1, we get the following.

Corollary 3.2. If the function f of the form (1.1) belongs to the class $\mathcal{M}_{\Sigma}(1/2)$ , then $|a_2| \leq 1$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{1}{2}, & \text{if } |1 - \mu| \le \frac{1}{4}; \\ 2|1 - \mu|, & \text{if } |1 - \mu| \ge \frac{1}{4}, \end{cases}$$ where $\mu$ is real.

Definition 1.1. Let $\pi/2 \leq \alpha < \pi$ . A function $f \in \Sigma$ is said to be in the class $\mathcal{M}_{\Sigma}(\alpha)$ , if the following inequalities hold: $$1 + \frac{\alpha - \pi}{2\sin\alpha} < \operatorname{Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1 + \frac{\alpha}{2\sin\alpha} \quad (z \in \Delta)$$ and $$1 + \frac{\alpha - \pi}{2 \sin \alpha} < \operatorname{Re} \left\{ \frac{w g'(w)}{g(w)} \right\} < 1 + \frac{\alpha}{2 \sin \alpha} \quad (w \in \Delta),$$ where g is defined by (1.2). Remark 1.1. Upon letting $\alpha \to \pi^-$ it is readily seen that a function $f \in \Sigma$ is in the class $\mathcal{M}_{\Sigma}(1/2)$ if the following inequalities are satisfied: $$\operatorname{Re}\left\{\frac{zf'(z)}{f(z)}\right\} > \frac{1}{2} \quad (z \in \Delta)$$ and $$\operatorname{Re}\left\{\frac{wg'(w)}{g(w)}\right\} > \frac{1}{2} \quad (w \in \Delta),$$ where g is defined by (1.2). The following lemma will be useful.