Certain classes of bi-univalent functions related to Shell-like curves connected with Fibonacci numbers

Lemma 1.7. [14] If $p \in \mathcal{P}$ , then $|p_i| \leq 2$ for each i, where $\mathcal{P}$ is the family of all functions p, analytic in $\mathbb{D}$ , for which $$\Re\{p(z)\} > 0 \qquad (z \in \mathbb{D}),$$ where $$p(z) = 1 + p_1 z + p_2 z^2 + \cdots$$ $(z \in \mathbb{D}).$ In this investigation, we find the estimates for the coefficients $|a_2|$ and $|a_3|$ for functions in the subclass $\mathcal{WSL}_{\Sigma}(\gamma, \lambda, \alpha, \tilde{p})$ , $\mathcal{RSL}_{\Sigma}(\gamma, \lambda, \tilde{p})$ , $\mathcal{SLB}_{\Sigma}(\lambda; \tilde{p})$ and $\mathcal{PSL}_{\Sigma}(\lambda; \tilde{p})$ Also, we obtain the upper bounds using the results of $|a_2|$ and $|a_3|$ .

Theorem 2.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $WSL_{\Sigma}(\gamma, \lambda, \alpha, \tilde{p})$ . Then $$|a_2| \le \frac{|\gamma| |\tau|}{\sqrt{\gamma \tau (1 + 2\alpha + 2\lambda) + (1 - 3\tau)(1 + \alpha)^2}},$$ $$|a_3| \le \frac{|\gamma| |\tau| \{(1-3\tau)(1+\alpha)^2\}}{(1+2\alpha+2\lambda) [\gamma\tau(1+2\alpha+2\lambda)+(1-3\tau)(1+\alpha)^2]}$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{\gamma |\tau|}{(1 + 2\alpha + 2\lambda)} &; 0 \le |h(\mu)| \le \frac{\gamma |\tau|}{4(1 + 2\alpha + 2\lambda)} \\ 4|h(\mu)| &; |h(\mu)| \ge \frac{\gamma |\tau|}{4(1 + 2\alpha + 2\lambda)}, \end{cases}$$ where $$h(\mu) = \frac{(1-\mu)\gamma^2\tau^2}{4\left[\gamma\tau(1+2\alpha+2\lambda) + (1+\alpha)^2(1-3\tau)\right]}.$$

Theorem 2.2. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{RSL}_{\Sigma}(\gamma, \lambda, \tilde{p})$ . Then $$|a_2| \le \frac{\sqrt{2} |\gamma| |\tau|}{\sqrt{\gamma \tau (2+\lambda) (1+\lambda) + 2(1-3\tau)(1+\lambda)^2}},$$ $$|a_3| \le \frac{|\gamma| |\tau| \{\gamma \tau (2+\lambda) (1+\lambda) + 2(1-3\tau)(1+\lambda)^2 - 2(2+\lambda) \gamma \tau\}}{(2+\lambda) [\gamma \tau (2+\lambda) (1+\lambda) + 2(1-3\tau)(1+\lambda)^2]}$$ and $$\left| a_{3} - \mu a_{2}^{2} \right| \leq \begin{cases} \frac{\left| \gamma \right| \left| \tau \right|}{2 + \lambda} & ; 0 \leq \left| \mu - 1 \right| \leq \frac{M}{2 \left| \gamma \right| \left| \tau \right| (2 + \lambda)} \\ \frac{2 \left| 1 - \mu \right| \gamma^{2} \tau^{2}}{M} & ; \left| \mu - 1 \right| \geq \frac{M}{2 \left| \gamma \right| \left| \tau \right| (2 + \lambda)}, \end{cases}$$ where $$M = \gamma \tau (2 + \lambda) (1 + \lambda) + 2 (1 + \lambda)^{2} (1 - 3\tau).$$

Theorem 2.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{SLB}_{\Sigma}(\lambda; \tilde{p})$ . Then $$|a_2| \le \frac{|\tau|}{\sqrt{(2\lambda - 1)\left[\tau\left(3 - 5\lambda\right) + 2\lambda - 1\right]}},$$ $$|a_3| \le \frac{|\tau| \left[ (2\lambda - 1)^2 - 2 \left( 5\lambda^2 - 4\lambda + 1 \right) \tau \right]}{(2\lambda - 1)(3\lambda - 1)[(3 - 5\lambda)\tau + 2\lambda - 1]}$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\gamma| |\tau|}{3\lambda - 1} & ; 0 \le |\mu - 1| \le \frac{M}{|\tau| (3\lambda - 1)} \\ \frac{|1 - \mu| \tau^2}{M} & ; |\mu - 1| \ge \frac{M}{|\tau| (3\lambda - 1)}, \end{cases}$$ where $$M = (2\lambda - 1) \left[ \tau \left( 3 - 5\lambda \right) + 2\lambda - 1 \right].$$

Theorem 2.4. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{PSL}_{\Sigma}(\lambda; \tilde{p})$ . Then $$|a_2| \le \frac{|\tau|}{\sqrt{(1+\lambda)^2 - 2\tau (2\lambda^2 + 2\lambda + 1)}},$$ $$|a_3| \le \frac{|\tau| (1 - 4\tau) (1 + \lambda)^2}{2(1 + 2\lambda) \left[ (1 + \lambda)^2 - 2\tau (2\lambda^2 + 2\lambda + 1) \right]}$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\tau|}{2+4\lambda} & ; 0 \le |\mu - 1| \le \frac{M}{2|\tau|(1+2\lambda)} \\ \frac{|1-\mu|\tau^2}{M} & ; |\mu - 1| \ge \frac{M}{2|\tau|(1+2\lambda)}, \end{cases}$$ where $$M = (1 + \lambda)^2 - 2\tau \left(2\lambda^2 + 2\lambda + 1\right).$$

Corollary 3.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{FSL}_{\Sigma}(\gamma, \lambda, \tilde{p})$ . Then $$|a_2| \le \frac{|\gamma| |\tau|}{\sqrt{3\gamma\tau (1+2\lambda) + 4(1-3\tau)(1+\lambda)^2}}, \qquad |a_3| \le \frac{4 |\gamma| |\tau| (1-3\tau)(1+\alpha)^2}{3(1+2\lambda) [3\gamma\tau (1+2\lambda) + 4(1-3\tau)(1+\lambda)^2]}$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\gamma| |\tau|}{3+6\lambda} &; 0 \le |h(\mu)| \le \frac{|\gamma| |\tau|}{12+24\lambda} \\ 4|h(\mu)| &; |h(\mu)| \ge \frac{|\gamma| |\tau|}{12+24\lambda}, \end{cases}$$ where $$h(\mu) = \frac{(1-\mu)\,\gamma^2\tau^2}{4\,[3\gamma\tau(1+2\lambda)+4(1-3\tau)(1+\lambda)^2]}.$$

Corollary 3.2. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{BSL}_{\Sigma}(\gamma, \alpha, \tilde{p})$ . Then $$|a_2| \le \frac{|\gamma| |\tau|}{\sqrt{\gamma \tau (1 + 2\alpha) + (1 - 3\tau)(1 + \alpha)^2}}, \qquad |a_3| \le \frac{|\gamma| |\tau| \{(1 - 3\tau)(1 + \alpha)^2\}}{(1 + 2\alpha) [\gamma \tau (1 + 2\alpha) + (1 - 3\tau)(1 + \alpha)^2]}$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\gamma| |\tau|}{1 + 2\alpha} & ; 0 \le |h(\mu)| \le \frac{|\gamma| |\tau|}{4 + 8\alpha} \\ 4 |h(\mu)| & ; |h(\mu)| \ge \frac{|\gamma| |\tau|}{4 + 8\alpha}, \end{cases}$$ where $$h(\mu) = \frac{(1-\mu)\gamma^2\tau^2}{4\left[\gamma\tau(1+2\alpha) + (1+\alpha)^2(1-3\tau)\right]}.$$

Corollary 3.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{HSL}_{\Sigma}(\gamma, \tilde{p})$ . Then $$|a_2| \le \frac{|\gamma| |\tau|}{\sqrt{3\gamma\tau + 4(1 - 3\tau)}}, \qquad |a_3| \le \frac{4 |\gamma| |\tau| (1 - 3\tau)}{3 [3\gamma\tau + 4(1 - 3\tau)]},$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\gamma| |\tau|}{3} & ; 0 \le |h(\mu)| \le \frac{|\gamma| |\tau|}{12} \\ 4 |h(\mu)| & ; |h(\mu)| \ge \frac{|\gamma| |\tau|}{12}, \end{cases}$$ where $$h(\mu) = \frac{(1-\mu)\,\gamma^2\tau^2}{4\,[3\gamma\tau + 4\,(1-3\tau)]}.$$ <span id="page-16-11"></span>Corollary 3.4. [10] Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{SL}_{\Sigma}(\tilde{p})$ . Then $$|a_2| \le \frac{|\tau|}{\sqrt{1-2\tau}}, \quad |a_3| \le \frac{|\tau|(1-4\tau)}{2-4\tau}$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\tau|}{2} & ; 0 \le |\mu - 1| \le \frac{1 - 2\tau}{2|\tau|} \\ \frac{|1 - \mu|\tau^2}{1 - 2\tau} & ; |\mu - 1| \ge \frac{1 - 2\tau}{2|\tau|}. \end{cases}$$ <span id="page-16-12"></span>Corollary 3.5. [10] Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $KSL_{\Sigma}(\tilde{p})$ . Then $$|a_2| \le \frac{|\tau|}{\sqrt{4 - 10\tau}}, \quad |a_3| \le \frac{|\tau|(1 - 4\tau)}{6 - 15\tau}.$$ and $$\left|a_{3}-\mu a_{2}^{2}\right| \leq \begin{cases} \frac{|\tau|}{6} & ; 0 \leq |\mu-1| \leq \frac{2-5\tau}{3|\tau|} \\ \frac{|1-\mu|\tau^{2}}{4-10\tau} & ; |\mu-1| \geq \frac{2-5\tau}{6|\tau|}. \end{cases}$$ Remark 3.6. Results discussed in Corollaries 3.4 and 3.5 are coincide with bounds obtained in [10, Corollary 1, Corollary 2, Corollary 4 and Corollary 5].

Definition 1.2. A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ belongs to the class $\mathcal{RSL}_{\Sigma}(\gamma, \lambda, \tilde{p})$ , $\gamma \in \mathbb{C} \setminus \{0\}$ and $\lambda \geq 0$ , if the following conditions are satisfied: $$1 + \frac{1}{\gamma} \left( \frac{z^{1-\lambda} f'(z)}{(f(z))^{1-\lambda}} - 1 \right) \prec \widetilde{p(z)} = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}, \ z \in \mathbb{D}$$ (1.7) and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} \left( \frac{w^{1-\lambda} g'(w)}{(g(w))^{1-\lambda}} - 1 \right) \prec \widetilde{p(w)} = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2}, \ w \in \mathbb{D}, \tag{1.8}$$ where $$\tau = \frac{1 - \sqrt{5}}{2} \approx -0.618$$ . (1) For λ = 0, we have the class RSLΣ(γ, 0, p˜) ≡ SLΣ(γ, p˜). A function f ∈ Σ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in SLΣ(γ, p˜), if the following conditions $$1 + \frac{1}{\gamma} \left( \frac{zf'(z)}{f(z)} - 1 \right) \prec \widetilde{p(z)} = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}, \ z \in \mathbb{D}$$ and for g(w) = f −1 (w) $$1 + \frac{1}{\gamma} \left( \frac{wg'(w)}{g(w)} - 1 \right) \prec \widetilde{p(w)} = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2}, \ w \in \mathbb{D}$$ hold, where τ = 1 − √ 5 2 ≈ −0.618. Remark 1.3. For γ = 1 the class SLΣ(1, p˜) ≡ SLΣ(˜p) was introduced and studied G¨uney et al. [\[10\]](#page-16-1). (2) For λ = 1, we have the class RSLΣ(γ, 1, p˜) ≡ HSLΣ(γ, p˜).

Definition 1.5. A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ belongs to the class $\mathcal{PSL}_{\Sigma}(\lambda; \tilde{p})$ , $0 \leq \lambda \leq 1$ , if the following conditions are satisfied: $$\frac{zf'(z) + \lambda z^2 f''(z)}{(1 - \lambda)f(z) + \lambda z f'(z)} \prec \widetilde{p(z)} = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}, \ z \in \mathbb{D}$$ (1.11) and for $g(w) = f^{-1}(w)$ $$\frac{wf'(w) + \lambda w^2 g''(w)}{(1 - \lambda)g(w) + \lambda w g'(w)} \prec \widetilde{p(w)} = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2}, \ w \in \mathbb{D},\tag{1.12}$$ where $\tau = \frac{1 - \sqrt{5}}{2} \approx -0.618$ . (1) For $\lambda = 0$ , we have the class $\mathcal{PSL}_{\Sigma}(0; \tilde{p}) \equiv \mathcal{SL}_{\Sigma}(\tilde{p})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $\mathcal{SL}_{\Sigma}(\tilde{p})$ , if the following conditions $$\frac{zf'(z)}{f(z)} \prec \widetilde{p(z)} = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$\frac{wg'(w)}{g(w)} \prec \widetilde{p(w)} = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2}, \ w \in \mathbb{D},$$ hold, where $\tau = \frac{1 - \sqrt{5}}{2} \approx -0.618$ . (2) For $\lambda = 1$ , we have the class $\mathcal{PSL}_{\Sigma}(1; \tilde{p}) \equiv \mathcal{KSL}_{\Sigma}(\tilde{p})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $\mathcal{KSL}_{\Sigma}(\tilde{p})$ , if the following conditions $$1 + \frac{z^2 f''(z)}{f'(z)} \prec \widetilde{p(z)} = \frac{1 + \tau^2 z^2}{1 - \tau z - \tau^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{w^2 g''(w)}{g'(w)} \prec \widetilde{p(w)} = \frac{1 + \tau^2 w^2}{1 - \tau w - \tau^2 w^2}, \ w \in \mathbb{D},$$ hold, where $\tau = \frac{1 - \sqrt{5}}{2} \approx -0.618$ . Remark 1.6. For $\gamma = 0$ , $\mathcal{PSL}_{\Sigma}(0, \tilde{p}) \equiv \mathcal{SL}_{\Sigma}(\tilde{p})$ and $\gamma = 1$ , $\mathcal{PSL}_{\Sigma}(1, \tilde{p}) \equiv \mathcal{KSL}_{\Sigma}(\tilde{p})$ the classes were introduced and studied Güney et al. [10]. In order to prove our results for the function in the classes $WSL_{\Sigma}(\gamma, \lambda, \alpha, \tilde{p})$ , $RSL_{\Sigma}(\gamma, \lambda, \tilde{p})$ , $SLB_{\Sigma}(\lambda; \tilde{p})$ and $PSL_{\Sigma}(\lambda; \tilde{p})$ , we need the following lemma.