A Comprehensive subclass of analytic and bi-univalent functions associated with subordination

Theorem 3.1. For $\lambda \geq 1$ , $\mu \geq 0$ , and $\delta \geq 0$ , let the function $f \in \mathfrak{B}_{\Sigma}(\mu, \lambda, \Phi, \xi)$ be given by (1.1). Then $$|a_2| \le \min \left\{ \frac{2}{\mu + \lambda + 2\xi\delta}, \sqrt{\frac{8}{(\mu + 2\lambda + 6\xi\delta)(\mu + 1)}} \right\}$$ and $$|a_3| \le \min \left\{ \begin{array}{c} \frac{1}{\left(1 + \frac{6\delta}{2\lambda + 1}\right)} \left[ \frac{4}{(\mu + \lambda + 2\xi\delta)^2} + \frac{2}{(\mu + 2\lambda + 6\xi\delta)} \right], \\ \frac{1}{\left(1 + \frac{6\delta}{2\lambda + 1}\right)} \left[ \frac{8}{(\mu + 2\lambda + 6\xi\delta)(\mu + 1)} + \frac{2}{(\mu + 2\lambda + 6\xi\delta)} \right] \end{array} \right\}.$$

Theorem 3.2. Let $\mathfrak{B}_{\Sigma}(\mu, \lambda, \Phi, \xi)$ . If $a_m = 0$ with $2 \leq m \leq n-1$ , then $$|a_n| \le \frac{2}{\mu + (n-1)\lambda + n(n-1)\xi\delta} \quad (n \ge 4).$$ (3.11)

Definition 2.1. For $\lambda \geq 1$ , $\mu \geq 0$ , $\delta \geq 0$ and $0 \leq \alpha < 1$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathfrak{B}^{\mu}_{\Sigma}(\alpha, \lambda, \delta)$ if the following conditions hold for all $z, w \in \mathbb{U}$ : $$\operatorname{Re}\left((1-\lambda)\left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z)\left(\frac{f(z)}{z}\right)^{\mu-1} + \xi \delta z f''(z)\right) > \alpha \tag{2.1}$$ and $$\operatorname{Re}\left((1-\lambda)\left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w)\left(\frac{g(w)}{w}\right)^{\mu-1} + \xi \delta w g''(w)\right) > \alpha, \tag{2.2}$$ where the function $g(w) = f^{-1}(w)$ is defined by (1.2) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . Using the Faber polynomial expansion of functions $f \in \mathcal{A}$ of the form (1.1), the coefficients of its inverse map $g = f^{-1}$ may be expressed as in [1]: $$g(w) = f^{-1}(w) = w + \sum_{n=2}^{\infty} \frac{1}{n} K_{n-1}^{-n} (a_2, a_3, ...) w^n,$$ (2.3) where $$K_{n-1}^{-n} = \frac{(-n)!}{(-2n+1)! (n-1)!} a_2^{n-1} + \frac{(-n)!}{(2(-n+1))! (n-3)!} a_2^{n-3} a_3 + \frac{(-n)!}{(-2n+3)! (n-4)!} a_2^{n-4} a_4$$ $$+ \frac{(-n)!}{(2(-n+2))! (n-5)!} a_2^{n-5} \left[ a_5 + (-n+2) a_3^2 \right] + \frac{(-n)!}{(-2n+5)! (n-6)!} a_2^{n-6}$$ $$\left[ a_6 + (-2n+5) a_3 a_4 \right] + \sum_{i \ge 7} a_2^{n-j} V_j,$$ $$(2.4)$$ such that $V_j$ with $7 \le j \le n$ is a homogeneous polynomial in the variables $a_2, a_3, ..., a_n$ [2]. In particular, the first three terms of $K_{n-1}^{-n}$ are $$K_1^{-2} = -2a_2, \ K_2^{-3} = 3\left(2a_2^2 - a_3\right), \ K_3^{-4} = -4\left(5a_2^3 - 5a_2a_3 + a_4\right).$$ (2.5) In general, for any $p \in \mathbb{N} := \{1, 2, 3, ...\}$ , an expansion of $K_n^p$ is as in [1], $$K_n^p = pa_n + \frac{p(p-1)}{2}D_n^2 + \frac{p!}{(p-3)!3!}D_n^3 + \dots + \frac{p!}{(p-n)!n!}D_n^n,$$ (2.6) where $D_n^p = D_n^p(a_2, a_3, ...)$ , and by [22], $D_n^m(a_1, a_2, ..., a_n) = \sum_{n=1}^{\infty} \frac{m!}{i_1!...i_n!} a_1^{i_1} ... a_n^{i_n}$ while $a_1 = 1$ , and the sum is taken over all non-negative integers $i_1, ..., i_n$ satisfying $i_1 + i_2 + \cdots + i_n = m$ , $i_1 + 2i_2 + \cdots + ni_n = n$ , it is clear that $D_n^m(a_1, a_2, ..., a_n) = a_1^n$ . Now, we are ready to establish a new subclass of analytic and bi-univalent functions based on subordination.

Definition 2.2. For $\lambda \geq 1$ , $\mu \geq 0$ , and $\delta \geq 0$ , A function $f \in \Sigma$ is said to be in the class $\mathfrak{B}_{\Sigma}(\mu, \lambda, \Phi, \xi)$ , if the following subordinations are satisfied: $$(1 - \lambda) \left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z) \left(\frac{f(z)}{z}\right)^{\mu - 1} + \xi \delta z f''(z) \prec \Phi(z) \tag{2.7}$$ and $$(1 - \lambda) \left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w) \left(\frac{g(w)}{w}\right)^{\mu - 1} + \xi \delta w g''(w) \prec \Phi(w)$$ (2.8) where the function $g(w) = f^{-1}(w)$ is defined by (1.2) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . 3. Coefficient bounds for the function class $\mathfrak{B}_{\Sigma}(\mu,\lambda,\Phi,\xi)$