New Subclasses of Analytic and Bi-Univalent Functions Endowed with Coefficient Estimate Problems

Lemma 1.1. [13] If $p \in \mathcal{P}$ , then $|c_k| \le 2$ for each k, where $\mathcal{P}$ is the family of all functions p analytic in $\mathbb{U}$ for which Re(p(z)) > 0, $p(z) = 1 + c_1 z + c_2 z^2 + \cdots$ for $z \in \mathbb{U}$ .

Theorem 2.2. Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\alpha, \lambda, \delta)$ . Then $$|a_2| \le \frac{2\alpha}{\sqrt{(\lambda + \mu + 2\xi\delta)^2 + \alpha \left[2\lambda + \mu - (\lambda + 2\xi\delta)^2 + (12 - 4\mu)\xi\delta\right]}}$$ (2.3) and $$|a_3| \le \frac{4\alpha^2}{(\lambda + \mu + 2\xi\delta)^2} + \frac{2\alpha}{2\lambda + \mu + 6\xi\delta}.\tag{2.4}$$

Corollary 2.3. [16] Let the function f(z) given by (1.1) be in the class $\mathscr{B}_{\Sigma}(\alpha)$ . Then $$|a_2| \le \alpha \sqrt{\frac{2}{\alpha + 2}}$$ and $$|a_3| \le \frac{\alpha(3\alpha+2)}{3}.$$ If we choose $\mu = 1$ and $\delta = 0$ in Theorem 2.2, we get the following consequence.

Corollary 2.4. [9] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(\alpha, \lambda)$ . Then $$|a_2| \le \frac{2\alpha}{\sqrt{(\lambda+1)^2 + \alpha(1+2\lambda-\lambda^2)}}$$ and $$|a_3| \le \frac{4\alpha^2}{(\lambda+1)^2} + \frac{2\alpha}{(2\lambda+1)}.$$ If we choose $\delta = 0$ in Theorem 2.2, we get the following consequence.

Corollary 2.5. [7] Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\alpha,\lambda)$ . Then $$|a_2| \le \frac{2\alpha}{\sqrt{(\lambda + \mu)^2 + \alpha(2\lambda + \mu - \lambda^2)}}$$ and $$|a_3| \le \frac{4\alpha^2}{(\lambda + \mu)^2} + \frac{2\alpha}{(2\lambda + \mu)}.$$ If we choose $\lambda = 1$ , $\mu = 0$ and $\delta = 0$ in Theorem 2.2, we get the following consequence.

Corollary 2.6. [7] Let the function f(z) given by (1.1) be in the class $S_{\Sigma}^*[\alpha]$ . Then $$|a_2| \le \frac{2\alpha}{\sqrt{1+\alpha}}$$ and $$|a_3| \leq \alpha(4\alpha + 1).$$

Theorem 3.2. Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\beta,\lambda,\delta)$ . Then $$|a_2| \le \min\left\{\sqrt{\frac{4(1-\beta)}{(2\lambda+\mu)(1+\mu+\frac{12\delta}{2\lambda+1})}}, \frac{2(1-\beta)}{\lambda+\mu+2\xi\delta}\right\}$$ (3.3) and $$|a_{3}| \leq \begin{cases} \min\left\{\frac{(1-\beta)\left(4+\frac{24\delta}{2\lambda+1}\right)}{(2\lambda+\mu+6\xi\delta)(1+\mu+\frac{12\delta}{2\lambda+1})}, \frac{4(1-\beta)^{2}}{(\lambda+\mu+2\xi\delta)^{2}} + \frac{2(1-\beta)}{2\lambda+\mu+6\xi\delta}\right\}; & 0 \leq \mu < 1\\ \frac{2(1-\beta)}{2\lambda+\mu+6\xi\delta}; & \mu \geq 1 \end{cases}$$ (3.4)

Corollary 3.3. [7] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(\beta)$ . Then $$|a_2| \le \begin{cases} \sqrt{\frac{2(1-\beta)}{3}}; & 0 \le \beta < \frac{1}{3} \\ 1-\beta; & \frac{1}{3} \le \beta < 1 \end{cases}$$ and $$|a_3| \le \frac{2(1-\beta)}{3}.$$ If we choose $\mu = 1$ and $\delta = 0$ in Theorem 3.2, we get the following consequence.

Corollary 3.4. [7] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(\beta, \lambda)$ . Then $$|a_2| \le \min\left\{\sqrt{\frac{2(1-\beta)}{2\lambda+1}}, \frac{2(1-\beta)}{\lambda+1}\right\}$$ and $$|a_3| \le \frac{2(1-\beta)}{2\lambda + 1}.$$ If we choose $\delta = 0$ in Theorem 3.2, we get the following consequence.

Corollary 3.5. [7] Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\beta,\lambda)$ . Then $$|a_2| \le \min \left\{ \sqrt{\frac{4(1-\beta)}{(2\lambda+\mu)(\mu+1)}}, \frac{2(1-\beta)}{(\lambda+\mu)} \right\}$$ and $$|a_3| \le \left\{ \begin{array}{l} \min\left\{ \frac{4(1-\beta)}{(2\lambda+\mu)(\mu+1)}, \frac{4(1-\beta)^2}{(\lambda+\mu)^2} + \frac{2(1-\beta)}{2\lambda+\mu} \right\}; \ 0 \le \mu < 1 \\ \frac{2(1-\beta)}{2\lambda+\mu}; \qquad \mu \ge 1 \end{array} \right..$$ If we choose $\lambda = 1$ , $\mu = 0$ and $\delta = 0$ in Theorem 3.2, we get the following consequence.

Corollary 3.6. [7] Let the function f(z) given by (1.1) be in the class $S_{\Sigma}^*(\beta)$ . Then $$|a_2| \le \sqrt{2(1-\beta)}$$ and $$|a_3| \le \left\{ \begin{array}{l} 2(1-\beta); & 0 \le \beta < \frac{3}{4} \\ (1-\beta)(5-4\beta); & \frac{3}{4} \le \beta < 1 \end{array} \right..$$

Definition 2.1. For $\lambda \ge 1$ , $\mu \ge 0$ , $\delta \ge 0$ and $0 < \alpha \le 1$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathscr{B}^{\mu}_{\Sigma}(\alpha, \lambda, \delta)$ if the following conditions hold for all $z, w \in \mathbb{U}$ : $$\left| \arg \left( (1 - \lambda) \left( \frac{f(z)}{z} \right)^{\mu} + \lambda f'(z) \left( \frac{f(z)}{z} \right)^{\mu - 1} + \xi \delta z f''(z) \right) \right| < \frac{\alpha \pi}{2}$$ (2.1) and $$\left| \arg \left( (1 - \lambda) \left( \frac{g(w)}{w} \right)^{\mu} + \lambda g'(w) \left( \frac{g(w)}{w} \right)^{\mu - 1} + \xi \delta w g''(w) \right) \right| < \frac{\alpha \pi}{2}, \tag{2.2}$$ where the function $g(w) = f^{-1}(w)$ is defined by (1.6) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . Remark 1. Note that for $\lambda=1, \mu=1$ and $\delta=0$ , the class of functions $\mathscr{B}^1_{\Sigma}(\alpha,1,0):=\mathscr{B}_{\Sigma}(\alpha)$ have been introduced and studied by Srivastava et al. [16], for $\mu=1$ and $\delta=0$ , the class of functions $\mathscr{B}^1_{\Sigma}(\alpha,\lambda,0):=\mathscr{B}_{\Sigma}(\alpha,\lambda)$ have been introduced and studied by Frasin and Aouf [9], for $\delta=0$ , the class of functions $\mathscr{B}^\mu_{\Sigma}(\alpha,\lambda,0):=\mathscr{B}^\mu_{\Sigma}(\alpha,\lambda)$ have been introduced and studied by Çağlar et al. [7], and for $\lambda=1,\mu=0$ and $\delta=0$ , we obtain the well-known class $\mathscr{B}^0_{\Sigma}(\alpha,1,0):=\mathscr{S}^*_{\Sigma}[\alpha]$ of strongly bi-starlike functions of order $\alpha$ . We first state and prove the following result.

Definition 3.1. For $\lambda \ge 1$ , $\mu \ge 0$ , $\delta \ge 0$ and $0 \le \beta < 1$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathscr{B}^{\mu}_{\Sigma}(\beta,\lambda,\delta)$ if the following conditions hold for all $z,w \in \mathbb{U}$ : $$\operatorname{Re}\left((1-\lambda)\left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z)\left(\frac{f(z)}{z}\right)^{\mu-1} + \xi \delta z f''(z)\right) > \beta \tag{3.1}$$ and $$\operatorname{Re}\left((1-\lambda)\left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w)\left(\frac{g(w)}{w}\right)^{\mu-1} + \xi \delta w g''(w)\right) > \beta,\tag{3.2}$$ where the function $g(w) = f^{-1}(w)$ is defined by (1.6) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . Remark 2. Note that for $\lambda = 1, \mu = 1$ and $\delta = 0$ , the class of functions $\mathcal{B}^1_{\Sigma}(\beta, 1, 0) := \mathcal{B}_{\Sigma}(\beta)$ have been introduced and studied by Srivastava et al. [16], for $\mu = 1$ and $\delta = 0$ , the class of functions $\mathcal{B}^1_{\Sigma}(\beta, \lambda, 0) := \mathcal{B}_{\Sigma}(\beta, \lambda)$ have been introduced and studied by Frasin and Aouf [9], for $\delta = 0$ , the class of functions $\mathcal{B}^{\mu}_{\Sigma}(\beta, \lambda, 0) := \mathcal{B}^{\mu}_{\Sigma}(\beta, \lambda)$ have been introduced and studied by Çağlar et al. [7], and for $\lambda = 1, \mu = 0$ and $\delta = 0$ , we obtain the well-known class $\mathcal{B}^0_{\Sigma}(\beta, 1, 0) := \mathcal{S}^*_{\Sigma}(\beta)$ of bi-starlike functions of order $\beta$ .