Some $m$-Fold Symmetric Bi-Univalent Function Classes and Their Associated Taylor-Maclaurin Coefficient Bounds

Lemma 1. [2]. If $h \in \mathcal{P}$ with h(z) given by (1.2), then $|h_k| \leq 2$ for each $k \in \mathbb{N}$ .

Theorem 1. Let $f \in Q_{\Sigma_m}(\tau, \lambda, \gamma, \delta; \alpha)$ be given by (1.5). Then $$|a_{m+1}| \le \frac{2\sqrt{2}|\tau|\alpha}{\sqrt{(\delta+1)|\tau\alpha(\delta+2)(m+1)\Phi_1(\lambda,\gamma,m)+2(1-\alpha)(\delta+1)\Phi_2(\lambda,\gamma,m)|}},$$ (2.3) and $$|a_{2m+1}| \le \frac{2|\tau|\alpha}{(\delta+1)(\delta+2)\Phi_1(\lambda,\gamma,m)} + \frac{2|\tau|^2\alpha^2(m+1)}{(\delta+1)^2\Phi_2(\lambda,\gamma,m)},\tag{2.4}$$ where $$\Phi_1(\lambda, \gamma, m) = 1 + 2(\lambda + \gamma)m + \lambda\gamma\left((2m+1)^2 + 1\right),$$ and $$\Phi_2(\lambda, \gamma, m) = \left(1 + (\lambda + \gamma)m + \lambda\gamma\left((m+1)^2 + 1\right)\right)^2.$$

Theorem 2. Let f ∈ Θ^Σm(τ, λ, γ, δ; β) be given by [\(1.5\)](#page-2-0). Then $$|a_{m+1}| \leq \min \left\{ \frac{2|\tau|(1-\beta)}{(\delta+1)(1+m(\lambda+\gamma)+\lambda\gamma((m+1)^2+1))}, \frac{2\sqrt{\frac{2|\tau|(1-\beta)}{(\delta+1)(\delta+2)(m+1)\Phi_1(\lambda,\gamma,m)}}} \right\}, \quad (3.3)$$ and $$|a_{2m+1}| \le \frac{4|\tau|(1-\beta)}{(\delta+1)(\delta+2)(1+2(\lambda+\gamma)m+\lambda\gamma((2m+1)^2+1))}.$$ (3.4)

Corollary 1. Let $f \in \Theta_{\Sigma_m}(\tau, \lambda, \gamma; \beta)$ be given by (1.5). Then $$\left|a_{m+1}\right| \leq \min\left\{\frac{2|\tau|(1-\beta)}{1+m(\lambda+\gamma)+\lambda\gamma\left((m+1)^2+1\right)}, 2\sqrt{\frac{|\tau|(1-\beta)}{(m+1)\left(1+2m(\lambda+\gamma)+\lambda\gamma\left((2m+1)^2+1\right)\right)}}\right\}$$ and $$|a_{2m+1}| \le \frac{2|\tau|(1-\beta)}{1+2m(\lambda+\gamma)+\lambda\gamma((2m+1)^2+1)}$$ Remark 3. The bounds on $|a_{m+1}|$ and $|a_{2m+1}|$ given in Corollary 1 are better than those given in [28, Theorem 3.1]. By taking $\gamma = 0$ in Corollary 1, we conclude the following result.

Corollary 2. Let $f \in \Theta_{\Sigma_m}(\tau, \lambda; \beta)$ be given by (1.5). Then $$|a_{m+1}| \le \min \left\{ \frac{2|\tau|(1-\beta)}{1+m\lambda}, 2\sqrt{\frac{|\tau|(1-\beta)}{(m+1)(1+2m\lambda)}} \right\}$$ and $$|a_{2m+1}| \le \frac{2|\tau|(1-\beta)}{1+2m\lambda}.$$ Remark 4. The bounds on $|a_{m+1}|$ and $|a_{2m+1}|$ given in Corollary 2 are better than those given in [29, Theorem 3.1]. By setting $\gamma = 0$ and $\tau = 1$ in Corollary 1, we conclude the following result.

Corollary 3. Let $f \in \Theta_{\Sigma_m}(\lambda; \beta)$ be given by (1.5). Then $$|a_{m+1}| \le \min \left\{ \frac{2(1-\beta)}{1+m\lambda}, 2\sqrt{\frac{(1-\beta)}{(m+1)(1+2m\lambda)}} \right\}$$ and $$|a_{2m+1}| \leq \frac{2(1-\beta)}{1+2m\lambda}$$ . Remark 5. The bounds on $|a_{m+1}|$ and $|a_{2m+1}|$ given in Corollary 3 are better than those given in [19, Theorem 2]. By setting $\gamma = 0$ and $\lambda = \tau = 1$ in Corollary 1, we conclude the following result.

Corollary 4. Let $f \in \Theta_{\Sigma_m}(\beta)$ be given by (1.5). Then $$|a_{m+1}| \le \min \left\{ \frac{2(1-\beta)}{1+m}, 2\sqrt{\frac{(1-\beta)}{(m+1)(1+2m)}} \right\}$$ and $$|a_{2m+1}| \le \frac{2(1-\beta)}{1+2m}.$$ Remark 6. The bounds on $|a_{m+1}|$ and $|a_{2m+1}|$ given in Corollary 4 are better than those given in [18, Theorem 3]. Remark 7. For 1-fold symmetric bi-univalent functions, the classes $Q_{\Sigma_1}(\tau, \lambda, \gamma, \delta; \alpha) \equiv Q_{\Sigma}(\tau, \lambda, \gamma, \delta; \alpha)$ and $\Theta_{\Sigma_1}(\tau, \lambda, \gamma, \delta; \beta) \equiv \Theta_{\Sigma}(\tau, \lambda, \gamma, \delta; \beta)$ are special cases of these classes illustrated below: - 1. For $\delta = 0$ , the classes $\mathcal{Q}_{\Sigma}(\tau, \lambda, \gamma, \delta; \alpha)$ and $\Theta_{\Sigma}(\tau, \lambda, \gamma, \delta; \beta)$ reduces to the classes $\mathcal{WS}_{\Sigma}(\lambda, \gamma, \tau; \alpha)$ and $\mathcal{WS}_{\Sigma}^{*}(\lambda, \gamma, \tau; \beta)$ , respectively, which were introduced recently by Srivastava and Wanas [28]. - 2. For $\delta = \gamma = 0$ and $\tau = 1$ , the classes $Q_{\Sigma}(\tau, \lambda, \gamma, \delta; \alpha)$ and $\Theta_{\Sigma}(\tau, \lambda, \gamma, \delta; \beta)$ reduces to the classes $\mathcal{B}_{\Sigma}(\alpha, \lambda)$ and $\mathcal{B}_{\Sigma}(\beta, \lambda)$ , respectively, which were investigated recently by Frasin and Aouf [8]. - 3. For $\delta = \gamma = 0$ and $\tau = \lambda = 1$ , the classes $Q_{\Sigma}(\tau, \lambda, \gamma, \delta; \alpha)$ and $\Theta_{\Sigma}(\tau, \lambda, \gamma, \delta; \beta)$ reduces to the classes $\mathcal{H}_{\Sigma}(\alpha)$ and $\mathcal{H}_{\Sigma}(\beta)$ , respectively, which were given by Srivastava et al. [6]. For 1-fold symmetric bi-univalent functions, Theorem 1 reduce to the following corollary:

Corollary 9. Let f ∈ ΘΣ(β)(0 ≦ β < 1) be given by [\(1.1\)](#page-1-0). Then $$|a_2| \le \min\left\{1 - \beta, \sqrt{\frac{2(1-\beta)}{3}}\right\}$$ and $$|a_3| \leqq \frac{2(1-\beta)}{3}.$$ Remark 11. The bounds on |a²| and |a³| given in Corollary [9](#page-12-4) are better than those given in [\[6,](#page-13-1) Theorem 2].

Definition 1. A function $f \in \Sigma_m$ given by (1.5) is called in the class $Q_{\Sigma_m}(\tau, \lambda, \gamma, \delta; \alpha)$ if it satisfies the conditions: $$\left| \arg \left( 1 + \frac{1}{\tau} \left[ (1 - \lambda)(1 - \gamma) \frac{\mathcal{R}^{\delta} f(z)}{z} \right] + (\lambda(\gamma + 1) + \gamma) \left( \mathcal{R}^{\delta} f(z) \right)' + \lambda \gamma \left( z \left( \mathcal{R}^{\delta} f(z) \right)'' - 2 \right) - 1 \right] \right) \right| < \frac{\alpha \pi}{2}, \quad (2.1)$$ and $$\left| \arg \left( 1 + \frac{1}{\tau} \left[ (1 - \lambda)(1 - \gamma) \frac{\mathcal{R}^{\delta} g(w)}{z} \right] + (\lambda(\gamma + 1) + \gamma) \left( \mathcal{R}^{\delta} g(w) \right)' + \lambda \gamma \left( w \left( \mathcal{R}^{\delta} g(w) \right)'' - 2 \right) - 1 \right] \right) \right| < \frac{\alpha \pi}{2}, \quad (2.2)$$ where $z, w \in \mathbb{U}$ and the function $g = f^{-1}$ is given by (1.7).

Definition 2. A function f ∈ Σ^m given by [\(1.5\)](#page-2-0) is called in the class Θ^Σm(τ, λ, γ, δ; β) if it satisfies the conditions: $$\operatorname{Re}\left\{1 + \frac{1}{\tau}\left[(1-\lambda)(1-\gamma)\frac{\mathcal{R}^{\delta}f(z)}{z} + (\lambda(\gamma+1)+\gamma)\left(\mathcal{R}^{\delta}f(z)\right)' + \lambda\gamma\left(z\left(\mathcal{R}^{\delta}f(z)\right)'' - 2\right) - 1\right]\right\} > \beta, \quad (3.1)$$ and $$\operatorname{Re}\left\{1 + \frac{1}{\tau}\left[(1-\lambda)(1-\gamma)\frac{\mathcal{R}^{\delta}g(w)}{z} + (\lambda(\gamma+1)+\gamma)\left(\mathcal{R}^{\delta}g(w)\right)' + \lambda\gamma\left(w\left(\mathcal{R}^{\delta}g(w)\right)'' - 2\right) - 1\right]\right\} > \beta, \quad (3.2)$$ where z, w ^∈ ^U and the function ^g ⁼ ^f −1 is given by [\(1.7\)](#page-2-1).