Coefficient estimates of some classes of rational functions

Lemma 1.1. (see [37]) If $$p(z) \in P$$ , then $|c_n| \le 2$ $(n \in \mathbb{N} = \{1, 2, ...\})$ . (6)

Theorem 2.4. Let $\psi(z) \in \mathcal{H}_{\sigma}(\phi)$ be of the form (5). Then <span id="page-3-6"></span> $$|a_1| \le \frac{B_1\sqrt{B_1}}{\sqrt{|3B_1^2 - 4B_2 + 4B_1|}} \quad and \ |a_2| \le \frac{1}{3}B_1.$$ (7)

Theorem 2.10. Let $\psi(z) \in S_{\sigma}(\alpha, \phi)$ be of the form (5). Then <span id="page-6-5"></span> $$|a_1| \le \frac{B_1 \sqrt{B_1}}{\sqrt{|B_1^2 (1 + 4\alpha) + (B_1 - B_2)(1 + 2\alpha)^2|}},\tag{20}$$ and <span id="page-6-6"></span> $$|a_2| \le \frac{B_1}{1+3\alpha}.\tag{21}$$

Corollary 2.12. For $0 \le \alpha \le 1$ and $0 < \gamma \le 1$ , let the function $\psi \in S_{\sigma}(\alpha, \gamma)$ be of the form (5). Then $$|a_1| \le \frac{2\gamma}{\sqrt{(1+2\alpha)^2 + \gamma [1+4\alpha-4\alpha^2]}}$$ and $|a_2| \le \frac{2\gamma}{1+3\alpha}$ . Using the parameter setting of Definition 2.9 in Theorem 2.10 we get the following corollary. Corollary 2.13. For $0 \le \alpha \le 1$ and $0 < \nu \le 1$ , let the function $\psi \in S_{\sigma}(\alpha, \nu)$ be of the form (5). Then $$|a_1| \le \sqrt{\frac{2(1-\nu)}{1+4\alpha}}$$ and $|a_2| \le \frac{2(1-\nu)}{1+3\alpha}$ .

Theorem 2.18. Let $\psi(z) \in M_{\sigma}(\alpha, \phi)$ be of the form (5). Then <span id="page-8-4"></span> $$|a_1| \le \frac{B_1 \sqrt{B_1}}{\sqrt{(1+\alpha)|B_1^2 + (1+\alpha)(B_1 - B_2)|}},\tag{28}$$ and <span id="page-8-5"></span> $$|a_2| \le \frac{B_1}{2(1+2\alpha)}. (29)$$

Theorem 2.23. Let $\psi(z) \in \Im_{\alpha}(\alpha, \phi)$ be of the form (5). Then <span id="page-10-4"></span> $$|a_1| \le \frac{2B_1\sqrt{B_1}}{\sqrt{|2(\alpha^2 - 3\alpha + 4)B_1^2 + 4(\alpha - 2)^2(B_1 - B_2)|}},\tag{37}$$ and <span id="page-10-5"></span> $$|a_2| \le \frac{B_1}{2|3 - 2\alpha|}. (38)$$

Theorem 2.25. Let $\psi(z) \in \beta_{\alpha}(\lambda, \phi)$ , $\lambda \geq 0$ be of the form (5). Then <span id="page-11-5"></span> $$|a_1| \le \frac{B_1 \sqrt{B_1}}{\sqrt{|(1+2\lambda)B_1^2 + (1+\lambda)^2 (B_1 - B_2)|}},\tag{46}$$ and <span id="page-11-6"></span> $$|a_2| \le \frac{B_1}{1+2\lambda}.\tag{47}$$

Definition 2.1. A function $\psi \in \sigma$ given by (5) is said to be in the class $\mathcal{H}_{\sigma}(\phi)$ if the following conditions are satisfied: $$\psi'(z) \prec \phi(z) (z \in \Delta)$$ and $g'(w) \prec \phi(w) (w \in \Delta)$ , where $q(w) := \psi^{-1}(w)$ . If we set $$\phi(z) = \left(\frac{1+z}{1-z}\right)^{\gamma} = 1 + 2\gamma z + 2\gamma^2 z^2 + \dots (0 < \gamma \le 1, \ z \in \Delta)$$ in Definition 2.1 of the bi-univalent function class $\mathcal{H}_{\sigma}(\phi)$ we obtain a new class $\mathcal{H}_{\sigma}(\gamma)$ given by Definition 2.2 below.

Definition 2.2. For $0 < \gamma \le 1$ , a function $\psi \in \sigma$ given by (5) is said to be in the class $\mathcal{H}_{\sigma}(\gamma)$ if the following conditions are satisfied: $$\psi'(z) \prec \left(\frac{1+z}{1-z}\right)^{\gamma} (z \in \Delta) \text{ and } g'(w) \prec \left(\frac{1+w}{1-w}\right)^{\gamma} (w \in \Delta),$$ where $g(w) := \psi^{-1}(w)$ . If we set $$\phi(z) = \frac{1 + (1 - 2\nu)z}{1 - z} = 1 + 2(1 - \nu)z + 2(1 - \nu)z^2 + \dots (0 < \nu \le 1, \ z \in \Delta)$$ in Definition 2.1 of the bi-univalent function class $\mathcal{H}_{\sigma}(\phi)$ we obtain, a new class $\mathcal{H}_{\sigma}(\nu)$ given by Definition 2.3 below.

Definition 2.3. For $0 < \nu \le 1$ , a function $\psi \in \sigma$ given by (5) is said to be in the class $\mathcal{H}_{\sigma}(\nu)$ if the following conditions hold true: $$\psi'(z) \prec \frac{1 + (1 - 2\nu)z}{1 - z} (z \in \Delta)$$ and $g'(w) \prec \frac{1 + (1 - 2\nu)w}{1 - w} (w \in \Delta)$ , where $g(w) := \psi^{-1}(w)$ .

Definition 2.14. A function $\psi \in \sigma$ given by (5) belongs to the class $M_{\sigma}(\alpha, \phi)$ $(0 \le \alpha \le 1)$ , if the following subordinations hold: $$(1 - \alpha) \frac{z\psi'(z)}{\psi(z)} + \alpha (1 + \frac{z\psi''(z)}{\psi'(z)}) \prec \phi(z) (z \in \Delta),$$ and $$(1-\alpha)\frac{wg'(w)}{g(w)} + \alpha(1 + \frac{wg''(w)}{g'(w)}) \prec \phi(w), (w \in \Delta),$$ where $g(w) := \psi^{-1}(w)$ . If we set $$\phi(z) = \left(\frac{1+z}{1-z}\right)^{\gamma} = 1 + 2\gamma z + 2\gamma^2 z^2 + \dots (0 < \gamma \le 1, \ z \in \Delta)$$ in Definition 2.14 of the bi-univalent function class $M_{\sigma}(\alpha, \phi)$ , we obtain a new class $M_{\sigma}(\alpha, \gamma)$ given by Definition 2.15 below.

Definition 2.15. For $0 \le \alpha \le 1$ and $0 < \gamma \le 1$ , a function $\psi \in \sigma$ given by (5) is said to be in the class $M_{\sigma}(\alpha, \gamma)$ if the following subordinations hold: $$(1-\alpha)\frac{z\psi'(z)}{\psi(z)} + \alpha(1 + \frac{z\psi''(z)}{\psi'(z)}) \prec \left(\frac{1+z}{1-z}\right)^{\gamma} (z \in \Delta),$$ and $$(1-\alpha)\frac{wg'(w)}{g(w)} + \alpha(1 + \frac{wg''(w)}{g'(w)}) \prec \left(\frac{1+w}{1-w}\right)^{\gamma} (w \in \Delta),$$ $q(w) := \psi^{-1}(w).$ Corollary 2.16. If we set $$\phi(z) = \frac{1 + (1 - 2\nu)z}{1 - z} = 1 + 2(1 - \nu)z + 2(1 - \nu)z^2 + \dots (0 < \nu \le 1, \ z \in \Delta)$$ in Definition 2.14 of the bi-univalent function class $M_{\sigma}(\alpha, \phi)$ we obtain a new class $M_{\sigma}(\alpha, \nu)$ given by Definition 2.17 below.

Definition 2.17. For $0 \le \alpha \le 1$ and $0 < \nu \le 1$ , a function $\psi \in \sigma$ given by (5) is said to be in the class $M_{\sigma}(\alpha, \nu)$ if the following subordinations hold: $$(1-\alpha)\frac{z\psi'(z)}{\psi(z)} + \alpha(1+\frac{z\psi''(z)}{\psi'(z)}) \prec \frac{1+(1-2\nu)z}{1-z} \left(z \in \Delta\right),$$ and $$(1-\alpha)\frac{w\psi'(w)}{\psi(w)} + \alpha(1+\frac{w\psi''(w)}{\psi'(w)}) \prec \frac{1+(1-2\nu)w}{1-w} (w \in \Delta),$$ where $g(w) := \psi^{-1}(w)$ . A function in the class $M_{\sigma}(\alpha, \phi)$ is called bi-Mocanu-convex function of Ma-Minda type. This class unifies the classes $S(\alpha)$ and $C(\alpha)$ . For functions in the class $M_{\sigma}(\alpha, \phi)$ , the following coefficients estimates hold.

Definition 2.22. A function $\psi \in \sigma$ given by (5) is said to be in the class $\Im_{\alpha}(\alpha,\phi)$ ( $0 \le \alpha \le 1$ ), if the following subordinations hold: $$\left(\frac{z\psi'(z)}{\psi(z)}\right)^{\alpha}\left(1+\frac{z\psi''(z)}{\psi'(z)}\right)^{1-\alpha} \prec \phi(z) \, (z \in \Delta) \, ,$$ and $$\left(\frac{wg'(w)}{g(w)}\right)^{\alpha} \left(1 + \frac{wg''(w)}{g'(w)}\right)^{1-\alpha} \prec \phi(w) \left(w \in \Delta\right),$$ $g(w) := \psi^{-1}(w)$ . This class also reduces to classes of Ma-Minda bi-starlike and bi-convex functions. For functions in this class, the following coefficient estimates are obtained.

Definition 2.24. A function $\psi \in \sigma$ given by (5) is said to be in the class $\beta_{\alpha}(\lambda, \phi)$ , $\lambda \geq 0$ , if the following subordinations hold: $$(1-\lambda)\frac{\psi(z)}{z} + \lambda \psi'(z) \prec \phi(z) (z \in \Delta),$$ and $$(1 - \lambda) \frac{g(w)}{w} + \lambda g'(w) \prec \phi(w) (w \in \Delta),$$ where $g(w) := \psi^{-1}(w)$ .