Estimates for initial coefficients of certain bi-univalent functions

Theorem 1.3. Let $\varphi$ be an analytic function given by the series (1.2) such that $B_2 \in \mathbb{R}$ . For $\lambda \geq 0$ , let the function $f \in \mathcal{R}_{\sigma}(\lambda, \varphi)$ and $\tau := (1 + \lambda)^2/(1 + 2\lambda)$ . (a) If $\tau B_2 \leq B_1^2$ , then $$|a_2| \le \frac{B_1\sqrt{B_1}}{\sqrt{(1+2\lambda)(B_1^2-\tau B_2+\tau B_1)}} \quad and \quad |a_3| \le \frac{B_1}{1+2\lambda} \max\left\{\frac{B_1^2}{B_1^2-\tau B_2+\tau B_1}, 1\right\}.$$ (b) If $\tau B_2 \geq B_1^2$ , then $$|a_2| \le \frac{B_1\sqrt{B_1}}{\sqrt{(1+2\lambda)(\tau B_2 + \tau B_1 - B_1^2)}}$$ and $|a_3| \le \frac{B_1}{1+2\lambda} \max\left\{\frac{B_1^2}{\tau B_2 + \tau B_1 - B_1^2}, 1\right\}$ . <span id="page-3-0"></span>Remark 1.4. Theorem 1.3 is an improvement over the coefficient estimates obtained in [3, 9, 13, 23]. For an analytic function $\varphi$ of the form (1.2), Kumar et al. [13, Theorem 2.2] obtained a bound on the coefficient $a_2$ of the function $f \in \mathcal{R}_{\sigma}(\lambda, \varphi)$ for $\lambda \geq 0$ . In addition, if $B_2 \in \mathbb{R}$ , by the means of the following comparisons, it may be noted that Theorem 1.3 gives an estimate for $a_2$ which is smaller than the one given by [13, Theorem 2.2]. We can see that if $\tau B_2 \leq B_1^2$ and $B_2 \leq B_1$ , then $$\frac{2B_1 - B_2}{B_1} - \frac{B_1^2}{B_1^2 - \tau B_2 + \tau B_1} = \frac{(B_1 - B_2)(2\tau B_1 + B_1^2 - \tau B_2)}{B_1(B_1^2 - \tau B_2 + \tau B_1)} \ge 0.$$ Therefore, $\min\left\{\frac{2B_1-B_2}{B_1}, \frac{B_1^2}{B_1^2-\tau B_2+\tau B_1}\right\} = \frac{B_1^2}{B_1^2-\tau B_2+\tau B_1}$ which implies that the estimate obtained for $a_2$ using Theorem 1.3 for this case is less than $\sqrt{(2B_1-B_2)/(1+2\lambda)}$ . Similarly, if $\tau B_2 \leq B_1^2$ and $B_2 \geq B_1$ , then $$\frac{B_2}{B_1} - \frac{B_1^2}{B_1^2 - \tau B_2 + \tau B_1} = \frac{(B_1^2 - \tau B_2)(B_2 - B_1)}{B_1(B_1^2 - \tau B_2 + \tau B_1)} \ge 0.$$ Next, the case when the conditions $\tau B_2 \geq B_1^2$ and $B_2 \leq B_1$ hold, it follows that $$\frac{2B_1 - B_2}{B_1} - \frac{B_1^2}{\tau B_2 + \tau B_1 - B_1^2} \ge \frac{B_1(\tau - B_1)^2}{\tau (\tau B_2 + \tau B_1 - B_1^2)} \ge 0$$ and further, if $\tau B_2 \geq B_1^2$ and $B_2 \geq B_1$ , then the inequality $$\frac{B_2}{B_1} - \frac{B_1^2}{\tau B_2 + \tau B_1 - B_1^2} = \frac{(\tau B_2 - B_1^2)(B_2 + B_1)}{B_1(\tau B_2 + \tau B_1 - B_1^2)} \ge 0$$ holds. Now let us consider the class $\mathcal{R}_{\sigma,\alpha}(\lambda) := \mathcal{R}_{\sigma}(\lambda, ((1+z)/(1-z))^{\alpha})$ for $0 < \alpha \leq 1$ . Clearly, $B_1 = 2\alpha$ and $B_2 = 2\alpha^2$ . For a function f given by (1.1) in the class $\mathcal{R}_{\sigma,\alpha}(\lambda)$ , Theorem 1.3 yields $$|a_2| \le \begin{cases} \frac{2\alpha}{\sqrt{(1+\lambda)^2 + \alpha(1-\lambda^2 + 2\lambda)}} & \text{if } 1 \le \lambda \le 1 + \sqrt{2} \\ \frac{2\alpha}{\sqrt{(1+\lambda)^2 - \alpha(1-\lambda^2 + 2\lambda)}} & \text{if } \lambda \ge 1 + \sqrt{2}. \end{cases}$$ It can be verified that if $1 \le \lambda \le 1 + \sqrt{2}$ , then the bound derived for $a_2$ coincides with that obtained by Frasin and Aouf [9, Theorem 2.2], whereas the estimate obtained for $a_2$ for the part $\lambda \ge 1 + \sqrt{2}$ is smaller than that in [9, Theorem 2.2]. Likewise, using Theorem 1.3, we can see that $|a_3| \le 2\alpha/(1+2\lambda)$ which is less than the bound for $a_3$ derived in [9, Theorem 2.2]. Similarly, let $\varphi(z) = (1 + (1 - 2\beta)z)/(1 - z)$ for $0 \le \beta < 1$ . As a result of Theorem 1.3, the functions in the class $\mathcal{R}_{\sigma}(\lambda, \beta)$ satisfy $$|a_2| \le \begin{cases} \sqrt{\frac{2(1-\beta)}{1+2\lambda}} & \text{if } 0 \le \beta \le \frac{1-\lambda^2+2\lambda}{2(1+2\lambda)} \\ (1-\beta)\sqrt{\frac{2}{\lambda^2+\beta(1+2\lambda)}} & \text{if } \frac{1-\lambda^2+2\lambda}{2(1+2\lambda)} \le \beta < 1 \end{cases}$$ and $|a_3| \leq 2(1-\beta)/(1+2\lambda)$ . Again, the estimate for $a_2$ so determined for the part when $0 \leq \beta \leq (1-\lambda^2+2\lambda)/(2(1+2\lambda))$ is same as that obtained by Frasin and Aouf [9, Theorem 3.2]. For $(1-\lambda^2+2\lambda)/(2(1+2\lambda)) \leq \beta < 1$ , the estimate for $a_2$ , derived using Theorem 1.3, is refined in comparison with [9, Theorem 3.2]. The estimate for the coefficient $a_3$ obtained using Theorem 1.3 is smaller than the one in [9, Theorem 3.2]. Moreover, the coefficient estimates derived above for the functions in classes $\mathcal{R}_{\sigma,\alpha}(\lambda)$ and $\mathcal{R}_{\sigma}(\lambda,\beta)$ are valid for $\lambda \geq 0$ . Also, Ali et al. [3, Theorem 2.1] derived bound on the coefficients $a_2$ and $a_3$ of a function $f \in \mathcal{R}_{\sigma}(\varphi)$ of the form (1.1). It may be noted that the estimates for the coefficients $a_2$ and $a_3$ of the function $f \in \mathcal{R}_{\sigma}(\varphi)$ given using Theorem 1.3 improve the estimates given in [3, Theorem 2.1] provided $\varphi''(0) \in \mathbb{R}$ . Furthermore, the coefficient estimates for the functions in the classes $\mathcal{R}_{\sigma,\alpha}$ and $\mathcal{R}_{\sigma}(\beta)$ determined in [23, Theorem 1] and [23, Theorem 2], respectively are particular cases for the above-mentioned estimates. The next theorem determines the estimates for the initial coefficients for a function in the class $\mathcal{S}_{\sigma}^{*}(\varphi)$ . <span id="page-4-0"></span>Theorem 1.5. Let $f \in \mathcal{S}_{\sigma}^*(\varphi)$ , where $\varphi''(0) \in \mathbb{R}$ . (a) If $B_2 \leq B_1^2$ , then $$|a_2| \le \frac{B_1\sqrt{B_1}}{\sqrt{B_1^2 + B_1 - B_2}}$$ and $|a_3| \le \max\left\{\frac{B_1^3}{B_1^2 - B_2 + B_1}, \frac{B_1}{2}\right\}$ . (b) If $B_2 \leq B_1^2$ , then $$|a_2| \le \frac{B_1\sqrt{B_1}}{\sqrt{B_2 + B_1 - B_1^2}}$$ and $|a_3| \le \max\left\{\frac{B_1^3}{B_2 + B_1 - B_1^2}, \frac{B_1}{2}\right\}$ . Remark 1.6. Bohra et al. [5, Corollary 2.3] and Ali et al. [3, Corollary 2.1] gave estimates on the coefficients $a_2$ and $a_3$ of the functions in the class $\mathcal{S}^_{\sigma}(\varphi)$ . In addition, let us assume that $B_2 \in \mathbb{R}$ . By means of inequalities similar to those in Remark 1.4, we can see that the estimates for the coefficients $a_2$ and $a_3$ of a function in the class $\mathcal{S}^_{\sigma}(\varphi)$ , obtained using Theorem 1.5, improve those derived in the above references. Particularly if $\varphi(z) = ((1+z)/(1-z))^{\alpha}$ , $(0 < \alpha \le 1)$ , the Theorem 1.5 readily yields that for a function $f \in \mathcal{S}_{\sigma}^*[\alpha]$ of the form (1.1), we have $|a_2| \le 2\alpha/(\sqrt{\alpha+1})$ , while $$|a_3| \le \alpha$$ if $0 < \alpha \le 1/3$ , and $|a_3| \le 4\alpha^2/(\alpha+1)$ if $1/3 \le \alpha \le 1$ which coincides with the estimates for $a_3$ as mentioned in [18, Theorem 2.1]. For a function $f \in \mathcal{S}_{\sigma}^*(\beta)$ (0 $\leq \beta <$ 1), a bi–starlike function of order $\beta$ , using Theorem 1.5, we may solve to get $|a_2| \le \sqrt{2(1-\beta)}$ if $0 \le \beta \le 1/2$ , whereas $|a_2| \le (1-\beta)\sqrt{2/\beta}$ if $1/2 \le \beta < 1$ . Further, $$|a_3| \le \begin{cases} 2(1-\beta) & \text{if } 0 \le \beta \le 1/2\\ 2(1-\beta)^2/\beta & \text{if } 1/2 \le \beta \le 2/3\\ 1-\beta & \text{if } 2/3 \le \beta < 1. \end{cases}$$ The bounds for $a_2$ and $a_3$ obtained above are smaller than those given by [17]. Also it can be seen that the bounds obtained as a result of Theorem 1.5 are an improvement over the ones given by Brannan and Taha [7].

Lemma 2.1. Let $\xi \in \mathbb{R}$ , $\eta > 0$ . Let the function $G: \mathbb{R} \to [0, \infty)$ be defined by $$G(x) := \max\{1, |\eta x - \xi|\}.$$ Then $$\inf_{x,y\in\mathbb{R}}\frac{G(x)+G(y)}{|2-x-y|}=\begin{cases} \frac{1}{1-\gamma} & \text{if } \xi\leq\eta\\ \frac{1}{\rho-1} & \text{if } \xi\geq\eta, \end{cases}$$ where $\gamma := (\xi - 1)/\eta \text{ and } \rho := (\xi + 1)/\eta.$

Lemma 2.2. Let $\xi \in \mathbb{R}$ and $\eta > 0$ . Let the function $G: \mathbb{R} \to [0, \infty)$ be defined as in Lemma 2.1. Then $$\inf_{x,y\in\mathbb{R}}\frac{|2-y|G(x)+|x|G(y)}{|2-x-y|}=\begin{cases} \frac{1}{1-\gamma} & \text{if} \quad 1\leq \xi\leq \eta\\ \frac{1}{\rho-1} & \text{if} \quad \eta\leq \xi\leq 2\eta-1\\ 1 & \text{otherwise,} \end{cases}$$ where $\gamma := (\xi - 1)/\eta$ and $\rho := (\xi + 1)/\eta$ .

Lemma 2.4. Let $\xi \in \mathbb{R}$ and $\eta > 0$ . Let the function $G: \mathbb{R} \to [0, \infty)$ be defined as in Lemma 2.1. Then $$\inf_{x,y\in\mathbb{R}}\frac{|3-y|G(x)+|x+1|G(y)}{|2-x-y|}=\begin{cases} \frac{2}{1-\gamma} & \text{if} \quad 1-\eta\leq\xi\leq\eta\\ \frac{2}{\rho-1} & \text{if} \quad \eta\leq\xi\leq3\eta-1\\ 1 & \text{otherwise}, \end{cases}$$ where $\gamma := (\xi - 1)/\eta$ and $\rho := (\xi + 1)/\eta$ . Proof of Theorem 1.5. Since the function $f \in \mathcal{S}_{\sigma}^*(\varphi)$ , the Definition 1.2 states that there exist two Schwarz functions r and s such that <span id="page-15-1"></span> $$\frac{zf'(z)}{f(z)} = \varphi(r(z)) \quad \text{and} \quad \frac{wg'(w)}{g(w)} = \varphi(s(w)). \tag{2.12}$$ Let the functions p and q be defined by equation (2.2). Clearly, the functions p and q are analytic functions in $\mathbb{D}$ with positive real part and p(0) = 1 = q(0). Therefore, equation (2.12) and (2.2) yield $$\frac{zf'(z)}{f(z)} = \varphi\left(\frac{p(z) - 1}{p(z) + 1}\right) \quad \text{and} \quad \frac{wg'(w)}{q(w)} = \varphi\left(\frac{q(w) - 1}{q(w) + 1}\right). \tag{2.13}$$ Comparing the coefficients on each side of the above two relations, we get $$a_2 = \frac{B_1 p_1}{2}, \quad 2a_3 - a_2^2 = \frac{B_2 p_1^2}{4} + \frac{B_1}{2} \left( p_2 - \frac{p_1^2}{2} \right),$$ $$a_2 = -\frac{B_1 q_1}{2}$$ , and $3a_2^2 - 2a_3 = \frac{B_2 q_1^2}{4} + \frac{B_1}{2} \left( q_2 - \frac{q_1^2}{2} \right)$ . A similar computations as that in proof of Theorem 1.3 leads to the following inequalities: <span id="page-16-0"></span> $$|2a_3 - (x+1)a_2^2| \le B_1G(x)$$ and $|2a_3 - (3-y)a_2^2| \le B_1G(y)$ , (2.14) where $G(x) := \max\{1, |xB_1 - B_2/B_1|\}$ . On computing using triangle's inequality, it is easy to see that $$|(2-x-y)a_2^2| \le |a_3-xa_2^2| + |a_3-(2-y)a_2^2| \le B_1(G(x)+G(y))$$ which implies $$|a_2|^2 \le B_1 \inf_{x,y \in \mathbb{R}} \frac{G(x) + G(y)}{|2 - x - y|}.$$ Since $B_2 \in \mathbb{R}$ , upon taking $\xi = B_2/B_1$ and $\eta = B_1$ , Lemma 2.1 gives $$|a_2| \le \sqrt{\frac{B_1}{1-\gamma}} \quad \left(\text{if} \quad \frac{B_2}{B_1} \le B_1\right) \quad \text{and} \quad \sqrt{\frac{B_1}{\rho-1}} \quad \left(\text{if} \quad \frac{B_2}{B_1} \ge B_1,\right)$$ where $\gamma = \frac{1}{B_1} \left( \frac{B_2}{B_1} - 1 \right)$ and $\rho = \frac{1}{B_1} \left( \frac{B_2}{B_1} + 1 \right)$ . Besides, keeping in view the relation (2.14), we may solve to get $$|a_3| \le \frac{B_1}{2} \inf_{x,y \in \mathbb{R}} \frac{|3-y|G(x)+|x+1|G(y)}{|2-x-y|}.$$ By means of Lemma 2.4 with $\xi = B_2/B_1$ and $\eta = B_1$ again, on simplifying the above relations, we get the desired estimates for the second and third coefficient of a function in class $\mathcal{S}_{\sigma}^*(\varphi)$ . Illustration 2.5. Let $\varphi(z) = (1+z)/(1-z)$ . For $\nu \ge 1$ , the function $f_{\nu}(z) := \nu z/(\nu-z) \in \mathcal{S}_{\sigma}^((1+z)/(1-z))$ . The function $f_{\nu}$ and its inverse, denoted by $g_{\nu}$ , are univalent in $\mathbb{D}$ for $\nu \ge 1$ . For $f_{\nu} \in \mathcal{S}_{\sigma}^((1+z)/(1-z))$ , the following subordinations must hold: $$\frac{zf_{\nu}'(z)}{f_{\nu}(z)} = \frac{\nu}{\nu - z} \prec \frac{1 + z}{1 - z} \quad \text{and} \quad \frac{wg_{\nu}'(w)}{g_{\nu}(w)} = \frac{\nu}{\nu + w} \prec \frac{1 + w}{1 - w}.$$ As in Illustration 2.3, the functions $zf'_{\nu}(z)/f_{\nu}(z)$ and $wg'_{\nu}(w)/g_{\nu}(w)$ map the unit disk onto the region contained in the right-half plane if and only if $\nu \geq 1$ . Hence, $f_{\nu} \in \mathcal{S}^*_{\sigma}((1+z)/(1-z))$ for $\nu \geq 1$ . Further, according to Theorem 1.5, it is required that $1/|\nu| < \sqrt{2}$ which is true as $\nu \geq 1$ . Assuming $\nu \geq \sqrt{2}(\sqrt{2}+1)$ , using [2, Lemma 2.2], we can see that the mappings $zf'_{\nu}(z)/f_{\nu}(z)$ and $wg'_{\nu}(w)/g_{\nu}(w)$ map the unit disk onto the disks that are contained in the region $\{w: |w^2-1|<1\}$ . Hence, the function $f_{\nu} \in \mathcal{S}^*_{\sigma}(\sqrt{1+z})$ . In this case, Theorem 1.5 implies that $1/|\nu| \leq 1/\sqrt{7}$ which is true as $\nu \geq \sqrt{2}(\sqrt{2}+1)$ . Remark 2.6. It may be noted that with $\varphi(z) = (1+z)/(1-z)$ , whenever $\nu \geq 1$ , the function $f_{\nu} := \nu z/(\nu - z) \in \mathcal{S}_{\sigma}^*(\varphi)$ and $f_{\nu} \in \mathcal{R}_{\sigma}(0,\varphi)$ but for $1 \leq \nu < \sqrt{2}$ , $f_{\nu} \notin \mathcal{R}_{\sigma}(1,\varphi)$ .

Definition 1.1. Let $\lambda \geq 0$ . A bi–univalent function f given by (1.1) is in class $\mathcal{R}_{\sigma}(\lambda, \varphi)$ , if it satisfies $$(1-\lambda)\frac{f(z)}{z} + \lambda f'(z) \prec \varphi(z)$$ and $(1-\lambda)\frac{g(w)}{w} + \lambda g'(w) \prec \varphi(w)$ , where g denotes the univalent extension of $f^{-1}$ to the unit disk. With the particular values of $\lambda$ and $\varphi$ , the class $\mathcal{R}_{\sigma}(\lambda,\varphi)$ reduces to many earlier classes as mentioned below: - (i) $\mathcal{R}_{\sigma}(\lambda, (1+(1-2\beta)z)/(1-z)) = \mathcal{R}_{\sigma}(\lambda, \beta)$ $(\lambda \ge 1; 0 \le \beta < 1)$ [9, Definition 3.1] - (ii) $\mathcal{R}_{\sigma}(\lambda, ((1+z)/(1-z))^{\alpha}) = \mathcal{R}_{\sigma,\alpha}(\lambda) \quad (\lambda \ge 1; 0 < \alpha \le 1)$ [9, Definition 2.1] - (iii) $\mathcal{R}_{\sigma}(1,\varphi) = \mathcal{R}_{\sigma}(\varphi)$ [3, p. 345]. - (iv) $\mathcal{R}_{\sigma}(1, (1+(1-2\beta)z)/(1-z)) = \mathcal{R}_{\sigma}(\beta)$ $(0 \le \beta < 1)$ [23, Definition 2]. (v) $\mathcal{R}_{\sigma}(1, ((1+z)/(1-z))^{\alpha}) = \mathcal{R}_{\sigma,\alpha}$ $(0 < \alpha \le 1)$ [23, Definition 1] The class of bi-starlike functions of Ma-Minda type was given by Ali et al. [3].

Definition 1.2. A function $f \in \sigma$ of the form (1.1), is said to be in the class of Ma-Minda bi-starlike functions, denoted by $\mathcal{S}_{\sigma}^{*}(\varphi)$ , if the following subordinations hold: $$\frac{zf'(z)}{f(z)} \prec \varphi(z)$$ and $\frac{wg'(w)}{g(w)} \prec \varphi(w)$ , where g denotes the univalent extension of $f^{-1}$ to $\mathbb{D}$ and $\varphi$ is the function of the form (1.2) satisfying the conditions as in the definition of the class $S^*(\varphi)$ as mentioned earlier. The class $\mathcal{S}_{\sigma}^*(\varphi)$ includes some well-known classes of the bi–univalent functions. For example: - (i) $S_{\sigma}^((1+(1-2\beta)z)/(1-z)) =: S_{\sigma}^(\beta), 0 \le \beta < 1.$ - (ii) $S_{\sigma}^{}(((1+z)/(1-z))^{\alpha}) =: S_{\sigma}^{}[\alpha], \ 0 < \alpha \leq 1.$ Using the Fekete-Szegö inequalities and principles of subordination, in this paper, the estimates for the coefficients $a_2$ and $a_3$ of the functions of the form (1.1) in the classes $\mathcal{R}_{\sigma}(\lambda,\varphi)$ and $\mathcal{S}_{\sigma}^{*}(\varphi)$ have been obtained. Moreover, the estimates so obtained are observed to be an improvement over the ones derived in [3, 5, 13]. For some particular choices of $\lambda$ and $\varphi$ , the bounds determined are smaller than those mentioned in [7, 9, 17, 18, 23] for the coefficients of the functions in the respective classes. More precisely, the following theorem derives the estimates for the coefficients $a_2$ and $a_3$ for the functions given by (1.1) that belong to the class $\mathcal{R}_{\sigma}(\lambda,\varphi)$ .