On $λ-$ Pseudo bi-starlike functions related with Fibonacci numbers

Theorem 1.2. ([5]). The function $\tilde{p}(z) = \frac{1+\tau^2z^2}{1-\tau z-\tau^2z^2}$ belongs to the class $\mathcal{P}(\sqrt{5}/10)$ , where $\sqrt{5}/10 \approx 0.2236$ . Now we recall the following lemma which will be relevant for our study.

Lemma 1.3. ([12]). Let $$p \in \mathcal{P}$$ with $p(z) = 1 + c_1 z + c_2 z^2 + \cdots$ , then $|c_n| < 2$ for $n > 1$ . (6) In this present work, we introduce a new subclass of $\Sigma$ associated with shell-like functions connected with Fibonacci numbers and obtain the initial Taylor coefficients $|a_2|$ and $|a_3|$ for this function class. Also, we give bounds for the Fekete-Szegö functional $|a_3 - \mu a_2^2|$ for this class.

Theorem 2.4. Let f given by (1) be in the class $\mathcal{PSL}^{\lambda}_{\Sigma}(\tilde{p}(z))$ , then $$|a_2| \le \frac{|\tau|}{\sqrt{(2\lambda - 1)^2 - (10\lambda^2 - 11\lambda + 3)\tau}}$$ (19) and $$|a_3| \le \frac{|\tau| \left[ (2\lambda - 1)^2 - 2(5\lambda^2 - 4\lambda + 1)\tau \right]}{(3\lambda - 1) \left[ (2\lambda - 1)^2 - (10\lambda^2 - 11\lambda + 3)\tau \right]},\tag{20}$$ where $\lambda \geq 1$ .

Corollary 2.6. Let f given by (1) be in the class $\mathcal{GSL}_{\Sigma}(\tilde{p}(z))$ , then $$|a_2| \le \frac{|\tau|}{\sqrt{9 - 21\tau}}\tag{40}$$ $$|a_3| \le \frac{|\tau|(9 - 26\tau)}{5(9 - 21\tau)}. (41)$$

Theorem 3.1. Let f given by (1) be in the class $\mathcal{PSL}^{\lambda}_{\Sigma}(\tilde{p}(z))$ and $\mu \in \mathbb{R}$ , then $$|a_3 - \mu a_2^2| \leq \left\{ \begin{array}{ll} \frac{|\tau|}{4(3\lambda - 1)}, & 0 \leq |h(\mu)| \leq \frac{|\tau|}{4(3\lambda - 1)}, \\ 4|h(\mu)|, & |h(\mu)| \geq \frac{|\tau|}{4(3\lambda - 1)}, \end{array} \right.$$ where $$h(\mu) = \frac{(1-\mu)\tau^2}{4\left[(2\lambda - 1)^2 - (10\lambda^2 - 11\lambda + 3)\tau\right]}.$$ (42)

Corollary 3.3. ([7]). Let f given by (1) be in the class $\mathcal{SL}_{\Sigma}(\tilde{p}(z))$ and $\mu \in \mathbb{R}$ , then we have $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\tau|}{8}, & 0 \le |h(\mu)| \le \frac{|\tau|}{8}, \\ 4|h(\mu)|, & |h(\mu)| \ge \frac{|\tau|}{8}, \end{cases}$$ where $$h(\mu) = \frac{(1-\mu)\tau^2}{4[1-2\tau]}. (47)$$ Further if $\mu = 1$ we get $$|a_3 - a_2^2| \le \frac{|\tau|}{8}.$$

Corollary 3.4. Let f given by (1) be in the class $\mathcal{GSL}_{\Sigma}(\tilde{p}(z))$ and $\mu \in \mathbb{R}$ , then we have $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\tau|}{20}, & 0 \le |h(\mu)| \le \frac{|\tau|}{20}, \\ 4|h(\mu)|, & |h(\mu)| \ge \frac{|\tau|}{20}, \end{cases}$$ where $$h(\mu) = \frac{(1-\mu)\tau^2}{4[9-21\tau]}. (48)$$ Further if $\mu = 1$ we get $$|a_3 - a_2^2| \le \frac{|\tau|}{20}.$$

Definition 1.1. The function $f \in \mathcal{A}$ belongs to the class $\mathcal{SL}$ if it satisfies the condition that $$\frac{zf'(z)}{f(z)} \prec \tilde{p}(z)$$ with $$\tilde{p}(z) = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2},$$ where $\tau = (1 - \sqrt{5})/2 \approx -0.618$ . It should be observed $\mathcal{SL}$ is a subclass of the starlike functions $\mathcal{S}^*$ . The function $\tilde{p}$ is not univalent in $\mathbb{U}$ , but it is univalent in the disc of radius $(3-\sqrt{5})/2$ . For example, $\tilde{p}(0) = \tilde{p}(-1/2\tau) = 1$ and $\tilde{p}(e^{\mp i \arccos(1/4)}) = \sqrt{5}/5$ , and it may also be noticed that $$\frac{1}{|\tau|} = \frac{|\tau|}{1 - |\tau|},$$ which shows that the number $|\tau|$ divides [0,1] such that it fulfils the golden section. The image of the unit circle |z|=1 under $\tilde{p}$ is a curve described by the equation given by $$(10x - \sqrt{5})y^2 = (\sqrt{5} - 2x)(\sqrt{5}x - 1)^2$$ which is translated and revolved trisectrix of Maclaurin. The curve $\tilde{p}(re^{it})$ is a closed curve without any loops for $0 < r \le r_0 = (3 - \sqrt{5})/2 \approx 0.38$ . For $r_0 < r < 1$ , it has a loop, and for r = 1, it has a vertical asymptote. Since $\tau$ satisfies the equation $\tau^2 = 1 + \tau$ , this expression can be used to obtain higher powers $\tau^n$ as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of $\tau$ and 1. The resulting recurrence relationships yield Fibonacci numbers $u_n$ : $$\tau^n = u_n \tau + u_{n-1}.$$ In [13] Raina and Sokół showed that $$\tilde{p}(z) = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}$$ $$= \left(t + \frac{1}{t}\right) \frac{t}{1 - t - t^2}$$ $$= \frac{1}{\sqrt{5}} \left(t + \frac{1}{t}\right) \left(\frac{1}{1 - (1 - \tau)t} - \frac{1}{1 - \tau t}\right)$$ $$= \left(t + \frac{1}{t}\right) \sum_{n=1}^{\infty} u_n t^n$$ $$= 1 + \sum_{n=1}^{\infty} (u_{n-1} + u_{n+1}) \tau^n z^n,$$ (3) where $$u_n = \frac{(1-\tau)^n - \tau^n}{\sqrt{5}}, \tau = \frac{1-\sqrt{5}}{2} \quad (n=1,2,\ldots).$$ (4) This shows that the relevant connection of $\tilde{p}$ with the sequence of Fibonacci numbers $u_n$ , such that $u_0 = 0$ , $u_1 = 1$ , $u_{n+2} = u_n + u_{n+1}$ for $n = 0, 1, 2, \cdots$ . And they got $$\tilde{p}(z) = 1 + \sum_{n=1}^{\infty} \tilde{p}_n z^n$$ $$= 1 + (u_0 + u_2)\tau z + (u_1 + u_3)\tau^2 z^2 + \sum_{n=3}^{\infty} (u_{n-3} + u_{n-2} + u_{n-1} + u_n)\tau^n z^n$$ $$= 1 + \tau z + 3\tau^2 z^2 + 4\tau^3 z^3 + 7\tau^4 z^4 + 11\tau^5 z^5 + \cdots$$ (5) Let $\mathcal{P}(\beta)$ , $0 \leq \beta < 1$ , denote the class of analytic functions p in $\mathbb{U}$ with p(0) = 1 and $Re\{p(z)\} > \beta$ . Especially, we will use $\mathcal{P}$ instead of $\mathcal{P}(0)$ .

Definition 2.1. For $\lambda \geq 1$ and $\lambda \in \mathbb{R}$ , a function $f \in \Sigma$ of the form (1) is said to be in the class $\mathcal{PSL}^{\lambda}_{\Sigma}(\tilde{p}(z))$ if the following subordination hold: $$\frac{z(f'(z))^{\lambda}}{f(z)} \prec \tilde{p}(z) = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}$$ (13) and $$\frac{w(g'(w))^{\lambda}}{g(w)} \prec \tilde{p}(w) = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2}$$ (14) where $\tau = (1 - \sqrt{5})/2 \approx -0.618$ where $z, w \in \mathbb{U}$ and g is given by (2). Specialising the parameter $\lambda=1$ and $\lambda=2$ , we have the following remarks, respectively: Remark 2.2 ([7]). For $\lambda = 1$ a function $f \in \Sigma$ is in the class $\mathcal{PSL}^1_{\Sigma}(\tilde{p}(z)) \equiv \mathcal{SL}_{\Sigma}(\tilde{p}(z))$ if the following conditions are satisfied: $$\frac{zf'(z)}{f(z)} \prec \tilde{p}(z) = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}$$ (15) and $$\frac{wg'(w)}{g(w)} \prec \tilde{p}(w) = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2},\tag{16}$$ where $\tau = (1 - \sqrt{5})/2 \approx -0.618$ where $z, w \in \mathbb{U}$ and g is given by (2). Remark 2.3. For $\lambda = 2$ a function $f \in \Sigma$ is in the class $\mathcal{PSL}^2_{\Sigma}(\tilde{p}(z)) \equiv \mathcal{GSL}_{\Sigma}(\tilde{p}(z))$ if the following conditions are satisfied: $$\left(f'(z)\frac{zf'(z)}{f(z)}\right) \prec \tilde{p}(z) = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2} \tag{17}$$ and $$\left(g'(w)\frac{wg'(w)}{g(w)}\right) \prec \tilde{p}(w) = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2},$$ (18) where $\tau = (1 - \sqrt{5})/2 \approx -0.618$ where $z, w \in \mathbb{U}$ and g is given by (2). In the following theorem we determine the initial Taylor coefficients $|a_2|$ and $|a_3|$ for the function class $\mathcal{PSL}^{\lambda}_{\Sigma}(\tilde{p}(z))$ . Later we will reduce these bounds to other classes for special cases.