Coefficient Bounds for a New Families of m-Fold Symmetric Bi-Univalent Functions Defined by Bazilevic Convex Functions

Lemma 1.1. [3]. If $h \in \mathcal{P}$ , then $|c_k| \leq 2$ for each $k \in \mathbb{N}$ , where $\mathcal{P}$ is the family of all f all functions h holomorphic in U for which $$Re(h(z)) > 0, \quad (z \in U),$$ where $$h(z) = 1 + c_1 z + c_2 z^2 + \cdots$$ , $(z \in U)$ .

Theorem 2.1. Let $f \in L_{\Sigma_m}(\delta, \gamma; \alpha)$ $(0 < \alpha \le 1, 0 \le \delta \le 1, 0 \le \gamma \le 1, m \in \mathbb{N})$ be given by (1.3). Then $$|a_{m+1}| \le \frac{2\alpha}{\sqrt{\frac{\left|2\alpha\left[(1-\delta)(2m+\gamma)+2m\delta(2m+1)+\frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)\right]\right|}{+(1-\alpha)[(1-\delta)(m+\gamma)+\delta m(m+1)]^2}}}$$ (2.3) and $$|a_{2m+1}| \le \frac{4\alpha^2}{[(1-\delta)(m+\gamma) + \delta m(m+1)]^2} + \frac{2\alpha}{(1-\delta)(2m+\gamma) + 2m\delta(2m+1)}. \quad (2.4)$$ Proof. It follows from conditions (2.1) and (2.2) that $$\left( (1 - \delta) \frac{z^{1 - \gamma} f'(z)}{\left( f(z) \right)^{1 - \gamma}} + \delta (1 + \frac{z^{2 - \gamma} f''(z)}{(z f'(z))^{1 - \gamma}} \right) = [p(z)]^{\alpha}$$ (2.5) $$\left( (1 - \delta) \frac{w^{1-\gamma} g'(w)}{\left( g(w) \right)^{1-\gamma}} + \delta \left( 1 + \frac{w^{2-\gamma} g''(w)}{(wg'(w))^{1-\gamma}} \right) = [q(w)]^{\alpha}, \tag{2.6}$$ where $g = f^{-1}$ and p, q in $\mathcal{P}$ have the following series representations: $$p(z) = 1 + p_m z^m + p_{2m} z^{2m} + p_{3m} z^{3m} + \cdots$$ (2.7) and $$q(w) = 1 + q_m w^m + q_{2m} w^{2m} + q_{3m} w^{3m} + \cdots$$ (2.8) Comparing the corresponding coefficients of (2.5) and (2.6) yields $$[(1 - \delta)(m + \gamma) + \delta m(m + 1)] a_{m+1} = \alpha p_m$$ (2.9) $$[(1-\delta)(2m+\gamma) + 2m\delta(2m+1)]a_{2m+1} + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)a_{m+1}^2$$ $$= \alpha p_{2m} + \frac{\alpha(\alpha - 1)}{2} p_m^2, \tag{2.10}$$ $$-[(1-\delta)(m+\gamma) + \delta m(m+1)]a_{m+1} = \alpha q_m$$ (2.11) and $$[(1-\delta)(2m+\gamma) + 2m\delta(2m+1)](2a_{m+1}^2 - a_{2m+1})$$ $$+ \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)a_{m+1}^2 = \alpha q_{2m} + \frac{\alpha(\alpha-1)}{2}q_m^2.$$ (2.12) Making use of (2.9) and (2.11), we obtain $$p_m = -q_m \tag{2.13}$$ and $$2[(1-\delta)(m+\gamma) + \delta m(m+1)]^2 a_{m+1}^2 = \alpha^2 (p_m^2 + q_m^2). \tag{2.14}$$ Also, from (2.10), (2.12) and (2.14), we find that $$2\left[(1-\delta)(2m+\gamma) + 2m\delta(2m+1) + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)\right]a_{m+1}^2$$ $$= \alpha(p_{2m} + q_{2m}) + \frac{\alpha(\alpha - 1)}{2} (p_m^2 + q_m^2)$$ $$= \alpha(p_{2m} + q_{2m}) + \frac{(\alpha - 1)[(1 - \delta)(m + \gamma) + \delta m(m + 1)]^2}{\alpha} a_{m+1}^2.$$ Therefore, we have $$a_{m+1}^{2} = \frac{\alpha^{2}(p_{2m} + q_{2m})}{2\alpha \left[ (1 - \delta)(2m + \gamma) + 2m\delta(2m + 1) + \frac{1}{2}(1 - \delta)(\gamma - 1)(2(m + \gamma) + \gamma) \right]}.$$ (2.15) $+ (1 - \alpha)[(1 - \delta)(m + \gamma) + \delta m(m + 1)]^{2}$ Now, taking the absolute value of (2.15) and applying Lemma 1.1 for the coefficients $p_{2m}$ and $q_{2m}$ , we obtain $$|a_{m+1}| \le \frac{2\alpha}{\sqrt{\left|2\alpha\left[(1-\delta)(2m+\gamma) + 2m\delta(2m+1) + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)\right]\right|} + (1-\alpha)[(1-\delta)(m+\gamma) + \delta m(m+1)]^2}}$$ This gives the desired estimate for $|a_{m+1}|$ as asserted in (2.3). In order to find the bound on $|a_{2m+1}|$ , by subtracting (2.12) from (2.10), we get $$2[(1-\delta)(2m+\gamma) + 2m\delta(2m+1)](a_{2m+1} - a_{m+1}^2)$$ $$= \alpha(p_{2m} - q_{2m}) + \frac{\alpha(\alpha-1)}{2}(p_m^2 - q_m^2). \quad (2.16)$$ It follows from (2.13), (2.14) and (2.16) that $$a_{2m+1} = \frac{\alpha^2(p_m^2 + q_m^2)}{2[(1-\delta)(m+\gamma) + \delta m(m+1)]^2} + \frac{\alpha(p_{2m} - q_{2m})}{2[(1-\delta)(2m+\gamma) + 2m\delta(2m+1)]}.$$ (2.17) Taking the absolute value of (2.17) and applying Lemma 1.1 once again for the coefficients $p_m$ , $p_{2m}$ , $q_m$ and $q_{2m}$ , we obtain $$|a_{2m+1}| \leq \frac{4\alpha^2}{[(1-\delta)(m+\gamma)+\delta m(m+1)]^2} + \frac{2\alpha}{(1-\delta)(2m+\gamma)+2m\delta(2m+1)}.$$ For one-fold symmetric bi-univalent functions, Theorem 2.1 reduces to the following corollary:

Corollary 2.1. Let $f \in L_{\Sigma}(\delta, \gamma; \alpha)$ $(0 < \alpha \le 1, 0 \le \delta \le 1, 0 \le \gamma \le 1)$ be given by (1.1). Then $$|a_2| \leq \frac{2\alpha}{\sqrt{\left|2\alpha[(1-\delta)(2+\gamma) + 6\delta + \frac{1}{2}(1-\delta)(\gamma-1)(2(1+\gamma) + \gamma) + (1-\alpha)[(1-\delta)(1+\gamma) + 2\delta]^2\right|}}$$ $$|a_3| \le \frac{4\alpha^2}{[(1-\delta)(1+\gamma)+2\delta]^2} + \frac{2\alpha}{(1-\delta)(2+\gamma)+6\delta}.$$

Theorem 3.1. Let $f \in L^*_{\Sigma_m}(\delta, \gamma; \beta)$ $(0 \le \beta < 1, 0 \le \delta \le 1, 0 \le \gamma \le 1, m \in \mathbb{N})$ be given by (1.3). Then $$|a_{m+1}| \le \sqrt{\frac{2(1-\beta)}{\left|(1-\delta)(2m+\gamma) + 2m\delta(2m+1) + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma) + \gamma)\right|}}$$ (3.3) and $$|a_{2m+1}| \le \frac{4(1-\beta)^2}{[(1-\delta)(m+\gamma) + \delta m(m+1)]^2} + \frac{4(1-\beta)}{(1-\delta)(2m+\gamma) + 2m\delta(2m+1)}.$$ (3.4) Proof. It follows from conditions (3.1) and (3.2) that there exist $p, q \in \mathcal{P}$ such that $$\left( (1 - \delta) \frac{z^{1 - \gamma} f'(z)}{\left( f(z) \right)^{1 - \gamma}} + \delta (1 + \frac{z^{2 - \gamma} f''(z)}{\left( z f'(z) \right)^{1 - \gamma}}) \right) = \beta + (1 - \beta) p(z) \tag{3.5}$$ and $$\left( (1 - \delta) \frac{w^{1-\gamma} g'(w)}{(g(w))^{1-\gamma}} + \delta (1 + \frac{w^{2-\gamma} g''(w)}{(wg'(w))^{1-\gamma}}) \right) = \beta + (1 - \beta) q(w), \tag{3.6}$$ where p(z) and q(w) have the forms (2.7) and (2.8), respectively. Equating coefficients (3.5) and (3.6) yields $$[(1 - \delta)(m + \gamma) + \delta m(m + 1)]a_{m+1} = (1 - \beta)p_m, \qquad (3.7)$$ $$[(1-\delta)(2m+\gamma) + 2m\delta(2m+1)]a_{2m+1} + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)a_{m+1}^{2}$$ $$= (1-\beta)p_{2m}, \qquad (3.8)$$ $$-[(1-\delta)(m+\gamma) + \delta m(m+1)]a_{m+1} = (1-\beta)q_m$$ (3.9) $$[(1-\delta)(2m+\gamma) + 2m\delta(2m+1)](2a_{m+1}^2 - a_{2m+1})$$ + $\frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)a_{m+1}^2 = (1-\beta)q_{2m}.$ (3.10) From (3.7) and (3.9), we get $$p_m = -q_m \tag{3.11}$$ and $$2[(1-\delta)(m+\gamma) + m\delta(m+1)]^2 a_{m+1}^2 = (1-\beta)^2 (p_m^2 + q_m^2).$$ (3.12) Adding (3.8) and (3.10), we obtain $$2\left[(1-\delta)(2m+\gamma) + 2m\delta(2m+1) + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)\right]a_{m+1}^{2}$$ $$= (1-\beta)(p_{2m}+q_{2m}). \tag{3.13}$$ Therefore, we have $$a_{m+1}^2 = \frac{(1-\beta)(p_{2m} + q_{2m})}{2\left[(1-\delta)(2m+\gamma) + 2m\delta(2m+1) + \frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)\right]}.$$ Applying Lemma 1.1 for the coefficients $p_{2m}$ and $q_{2m}$ , we obtain $$|a_{m+1}| \leq \sqrt{\frac{2(1-\beta)}{\left|(1-\delta)(2m+\gamma)(2m\delta(2m+1)+\frac{1}{2}(1-\delta)(\gamma-1)(2(m+\gamma)+\gamma)\right|}}.$$ This gives the desired estimate for $|a_{m+1}|$ as asserted in (3.3). In order to find the bound on $|a_{2m+1}|$ , by subtracting (3.10) from (3.8), we get $$[(1-\delta)(2m+\gamma)+2m\delta(2m+1)](a_{2m+1}-a_{m+1}^2)=(1-\beta)(p_{2m}-q_{2m}),$$ or equivalently $$a_{2m+1} = a_{m+1}^2 + \frac{(1-\beta)(p_{2m} - q_{2m})}{(1-\delta)(2m+\gamma) + 2m\delta(2m+1)}.$$ Upon substituting the value of $a_{m+1}^2$ from (3.12), it follows that $$a_{2m+1} = \frac{(1-\beta)^2(p_m^2 + q_m^2)}{2[(1-\delta)(m+\gamma) + \delta m(m+1)]^2} + \frac{(1-\beta)(p_{2m} - q_{2m})}{(1-\delta)(2m+\gamma) + 2m\delta(2m+1)}.$$ Applying Lemma 1.1 once again for the coefficients $p_m$ , $p_{2m}$ , $q_m$ and $q_{2m}$ , we obtain $$|a_{2m+1}| \le \frac{4(1-\beta)^2}{[(1-\delta)(m+\gamma)+\delta m(m+1)]^2} + \frac{4(1-\beta)}{(1-\delta)(2m+\gamma)+2m\delta(2m+1)},$$ which completes the proof of Theorem 3.1. For one-fold symmetric bi-univalent functions, Theorem 3.1 reduces to the following corollary:

Corollary 3.1. Let $f \in L_{\Sigma}^*(\delta, \gamma; \beta)$ $(0 \le \beta < 1, 0 \le \delta \le 1, 0 \le \gamma \le 1)$ be given by (1.1). Then $$|a_2| \leq \sqrt{\frac{2(1-\beta)}{\left|(1-\delta)(2+\gamma) + 6\delta + \frac{1}{2}(1-\delta)(\gamma-1)(2(1+\gamma) + \gamma)\right|}}$$ and $$|a_3| \le \frac{4(1-\beta)^2}{[(1-\delta)(1+\gamma)+2\delta]^2} + \frac{4(1-\beta)}{(1-\delta)(2+\gamma)+6\delta}.$$

Definition 2.1. A function $f \in \Sigma_m$ given by (1.3) is said to be in the family $L_{\Sigma_m}(\delta, \gamma; \alpha)$ ( $0 < \alpha \le 1, 0 \le \delta \le 1, 0 \le \gamma \le 1$ ) if it satisfies the following conditions: $$\left| \arg \left( (1 - \delta) \frac{z^{1 - \gamma} f'(z)}{\left( f(z) \right)^{1 - \gamma}} + \delta (1 + \frac{z^{2 - \gamma} f''(z)}{\left( z f'(z) \right)^{1 - \gamma}}) \right) \right| < \frac{\alpha \pi}{2} , \qquad (z \in U)$$ (2.1) $$\left| \arg \left( (1 - \delta) \frac{w^{1 - \gamma} g'(w)}{\left( g(w) \right)^{1 - \gamma}} + \delta (1 + \frac{w^{2 - \gamma} g''(w)}{\left( w g'(w) \right)^{1 - \gamma}}) \right) \right| < \frac{\alpha \pi}{2}, \quad (w \in U), \quad (2.2)$$ where the function $g = f^{-1}$ is given by (1.4). In particular, for one-fold symmetric bi-univalent functions, we denote the family $L_{\Sigma_1}(\delta, \gamma; \alpha) = L_{\Sigma}(\delta, \gamma; \alpha)$ . Remark 2.1. It should be remarked that the family $L_{\Sigma}(\delta, \gamma; \alpha)$ is a generalization of well-known families consider earlier. These families are: - (1) For $\delta = 0$ , the family $L_{\Sigma}(\delta, \gamma; \alpha)$ reduce to the family $P_{\Sigma}(\alpha, \gamma)$ which was introduced by Prema and Keerthi [17]; - (2) For $\delta = 1$ , the family $L_{\Sigma}(\delta, \gamma; \alpha)$ reduce to the family $B_{\Sigma}(\gamma, \alpha)$ which was introduced by Sakar and Wanas [22]; - (3) For $\delta = \gamma = 0$ , the family $L_{\Sigma}(\delta, \gamma; \alpha)$ reduce to the family $S_{\Sigma}^{*}(\alpha)$ which was given by Brannan and Taha [6]; - (4) $\delta = 0$ and $\gamma = 1$ , the family $L_{\Sigma}(\delta, \gamma; \alpha)$ reduce to the family $H_{\Sigma}^{\alpha}$ which was investigated by Srivastava et al. [25].

Definition 3.1. A function $f \in \Sigma_m$ given by (1.3) is said to be in the family $L_{\Sigma_m}^*(\delta, \gamma; \beta)$ ( $0 \le \beta < 1, 0 \le \delta \le 1, 0 \le \gamma \le 1$ ) if it satisfies the following conditions: $$Re\left\{ (1-\delta) \frac{z^{1-\gamma} f'(z)}{\left(f(z)\right)^{1-\gamma}} + \delta(1 + \frac{z^{2-\gamma} f''(z)}{\left(z f'(z)\right)^{1-\gamma}}) \right\} > \beta, \tag{3.1}$$ and $$Re\left\{ (1 - \delta) \frac{w^{1 - \gamma} g'(w)}{(g(w))^{1 - \gamma}} + \delta (1 + \frac{w^{2 - \gamma} g''(w)}{(w g'(w))^{1 - \gamma}}) \right\} > \beta, \tag{3.2}$$ where the function $g = f^{-1}$ is given by (1.4). In particular, for one-fold symmetric bi-univalent functions, we denote the family $L_{\Sigma_1}^(\delta, \gamma; \beta) = L_{\Sigma}^(\delta, \gamma; \beta)$ . Remark 3.1. It should be remarked that the family $L_{\Sigma}^*(\delta, \gamma; \beta)$ is a generalization of well-known families consider earlier. These families are: - (1) For $\delta = 0$ , the family $L_{\Sigma}^{*}(\delta, \gamma; \beta)$ reduce to the family $P_{\Sigma}(\beta, \gamma)$ which was introduced by Prema and Keerthi [17]; - (2) For $\delta = 1$ , the family $L_{\Sigma}^(\delta, \gamma; \beta)$ reduce to the family $\beta_{\Sigma(\gamma;\beta)}$ which was introduced by Sakar and Wanas [22]; - (3) For $\delta = \gamma = 0$ , the family $L_{\Sigma}^(\delta, \gamma; \beta)$ reduce to the family $S_{\Sigma}^(\beta)$ which was given by Brannan and Taha [6]; - (4) For $\delta = 0$ and $\gamma = 1$ , the family $L_{\Sigma}^*(\delta, \gamma; \beta)$ reduce to the family $H_{\Sigma}(\beta)$ which was investigated by Srivastava et al. [25].