Initial estimates for certain subclasses of bi-univalent functions with $κ-$Fibonacci numbers

Lemma 1.1. [26] 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 $$p(z) = 1 + p_1 z + p_2 z^2 + \cdots \qquad z \in \mathbb{D}.$$ In this investigation, we find the estimates for the coefficients $|a_2|$ and $|a_3|$ for functions in the subclasses $\mathcal{WSL}^{\kappa}_{\Sigma}(\gamma, \lambda, \alpha, \tilde{p}_{\kappa})$ , $\mathcal{RSL}^{\kappa}_{\Sigma}(\gamma, \lambda, \tilde{p}_{\kappa})$ , $\mathcal{SLB}^{\kappa}_{\Sigma}(\lambda; \tilde{p}_{\kappa})$ and $\mathcal{PSL}^{\kappa}_{\Sigma}(\lambda; \tilde{p}_{\kappa})$ . Further, Fekete-Szegö problems of the aforementioned classes. 2. Initial Coefficient Estimates and Fekete-Szegő Inequalities In the following theorem, we obtain coefficient estimates for functions in the class $f \in \mathcal{WSL}^{\kappa}_{\Sigma}(\gamma, \lambda, \alpha, \tilde{p}_{\kappa})$ .

Theorem 2.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $WSL_{\Sigma}^{\kappa}(\gamma, \lambda, \alpha, \tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\gamma| |\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{\gamma \kappa^2 \tau_{\kappa} (1 + 2\alpha + 2\lambda) + (\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \alpha)^2}},$$ $$|a_3| \le \frac{|\gamma| |\kappa \tau_{\kappa}| (\kappa - (\kappa^2 + 2)\tau_{\kappa}) (1 + \alpha)^2}{(1 + 2\alpha + 2\lambda) [\gamma \kappa^2 \tau_{\kappa} (1 + 2\alpha + 2\lambda) + (\kappa - (\kappa^2 + 2)\tau_{\kappa}) (1 + \alpha)^2]}$$ $$|a_{3} - \mu a_{2}^{2}| \leq \begin{cases} \frac{\gamma |\kappa \tau_{\kappa}|}{(1 + 2\alpha + 2\lambda)}; \\ 0 \leq |\mu - 1| \leq \frac{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha + 2\lambda) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \alpha)^{2}}{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha + 2\lambda)} \\ \frac{|1 - \mu| \gamma^{2} \kappa^{3} \tau_{\kappa}^{2}}{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha + 2\lambda) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \alpha)^{2}}; \\ |\mu - 1| \geq \frac{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha + 2\lambda) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \alpha)^{2}}{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha + 2\lambda)}. \end{cases}$$

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

Theorem 2.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{SLB}_{\Sigma}^{\kappa}(\lambda; \tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\kappa \tau_{\kappa}|}{\sqrt{\left[\lambda (2\lambda - 1) \kappa^2 \tau_{\kappa} + (\kappa - (\kappa^2 + 2)\tau_{\kappa}) (2\lambda - 1)^2\right]}},$$ $$|a_{3}| \leq \frac{|\kappa \tau_{\kappa}| \left[ (\kappa - (\kappa^{2} + 2)\tau_{\kappa}) (2\lambda - 1)^{2} + (2\lambda^{2} - 4\lambda + 1) \kappa^{2} \tau_{\kappa} \right]}{(3\lambda - 1) \left[ \lambda (2\lambda - 1) \kappa^{2} \tau_{\kappa} + (\kappa - (\kappa^{2} + 2)\tau_{\kappa}) (2\lambda - 1)^{2} \right]}$$ $$|a_{3} - \mu a_{2}^{2}| \leq \begin{cases} \frac{|\kappa \tau_{\kappa}|}{3\lambda - 1}; \\ 0 \leq |\mu - 1| \leq \frac{\left[\lambda (2\lambda - 1) \kappa^{2} \tau_{\kappa} + (\kappa - (\kappa^{2} + 2)\tau_{\kappa}) (2\lambda - 1)^{2}\right]}{\kappa^{2} \tau_{\kappa} (3\lambda - 1)} \\ \frac{|1 - \mu| \kappa^{3} \tau_{\kappa}^{2}}{\left[\lambda (2\lambda - 1) \kappa^{2} \tau_{\kappa} + (\kappa - (\kappa^{2} + 2)\tau_{\kappa}) (2\lambda - 1)^{2}\right]}; \\ |\mu - 1| \geq \frac{\left[\lambda (2\lambda - 1) \kappa^{2} \tau_{\kappa} + (\kappa - (\kappa^{2} + 2)\tau_{\kappa}) (2\lambda - 1)^{2}\right]}{|\kappa^{3} \tau_{\kappa}| (3\lambda - 1)}. \end{cases}$$

Theorem 2.4. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{PSL}_{\Sigma}^{\kappa}(\lambda; \tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{[\kappa^2 \tau_{\kappa} (1 + 2\lambda - \lambda^2) + (\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \lambda)^2]}},$$ $$|a_3| \le \frac{|\kappa \tau_{\kappa}| \left(\kappa - 2(\kappa^2 + 1)\tau_{\kappa}\right) (1 + \lambda)^2}{2(1 + 2\lambda) \left[\kappa^2 \tau_{\kappa} (1 + 2\lambda - \lambda^2) + (\kappa - (\kappa^2 + 2)\tau_{\kappa}) (1 + \lambda)^2\right]}$$ $$|a_{3} - \mu a_{2}^{2}| \leq \begin{cases} \frac{|\kappa \tau_{\kappa}|}{2 + 4\lambda}; \\ 0 \leq |\mu - 1| \leq \frac{\left[\kappa^{2} \tau_{\kappa} (1 + 2\lambda - \lambda^{2}) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \lambda)^{2}\right]}{2\kappa^{2} |\tau_{\kappa}| (1 + 2\lambda)} \\ \frac{|1 - \mu| \kappa^{3} \tau_{\kappa}^{2}}{\left[\kappa^{2} \tau_{\kappa} (1 + 2\lambda - \lambda^{2}) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \lambda)^{2}\right]}; \\ |\mu - 1| \geq \frac{\left[\kappa^{2} \tau_{\kappa} (1 + 2\lambda - \lambda^{2}) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \lambda)^{2}\right]}{2\kappa^{2} |\tau_{\kappa}| (1 + 2\lambda)}. \end{cases}$$

Corollary 3.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{FSL}_{\Sigma}^{\kappa}(\gamma, \lambda, \tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\gamma| |\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{3\gamma \kappa^2 \tau_{\kappa} (1 + 2\lambda) + 4(\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \lambda)^2}}$$ $$|a_3| \le \frac{4|\gamma| |\kappa \tau_{\kappa}| (\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \lambda)^2}{3(1 + 2\lambda) [3\gamma \kappa^2 \tau_{\kappa} (1 + 2\lambda) + 4(\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \lambda)^2]}$$ $$|a_{3} - \mu a_{2}^{2}| \leq \begin{cases} \frac{|\gamma| |\kappa \tau_{\kappa}|}{3 + 6\lambda}; \\ 0 \leq |\mu - 1| \leq \frac{3\gamma \kappa^{2} \tau_{\kappa} (1 + 2\lambda) + 4(\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \lambda)^{2}}{(3 + 6\lambda) |\gamma| \kappa^{2} |\tau_{\kappa}|} \\ \frac{|1 - \mu| \gamma^{2} \kappa^{3} \tau_{\kappa}^{2}}{3\gamma \kappa^{2} \tau_{\kappa} (1 + 2\lambda) + 4(\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \lambda)^{2}}; \\ |\mu - 1| \geq \frac{3\gamma \kappa^{2} \tau_{\kappa} (1 + 2\lambda) + 4(\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \lambda)^{2}}{(3 + 6\lambda) |\gamma| \kappa^{2} |\tau_{\kappa}|} \end{cases}.$$

Corollary 3.2. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{BSL}_{\Sigma}^{\kappa}(\gamma, \alpha, \tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\gamma| |\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{\gamma \kappa^2 \tau_{\kappa} (1 + 2\alpha) + (\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \alpha)^2}},$$ $$|a_3| \le \frac{|\gamma| |\kappa \tau_{\kappa}| (\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \alpha)^2}{(1 + 2\alpha) [\gamma \kappa^2 \tau_{\kappa} (1 + 2\alpha) + (\kappa - (\kappa^2 + 2)\tau_{\kappa})(1 + \alpha)^2]}$$ and $$|a_{3} - \mu a_{2}^{2}| \leq \begin{cases} \frac{|\gamma| |\kappa \tau_{\kappa}|}{1 + 2\alpha}; \\ 0 \leq |\mu - 1| \leq \frac{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \alpha)^{2}}{(1 + 2\alpha) |\gamma| |\kappa^{2}| |\tau_{\kappa}|} \\ \frac{|1 - \mu| \gamma^{2} \kappa^{3} \tau_{\kappa}^{2}}{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \alpha)^{2}}; \\ |\mu - 1| \geq \frac{\gamma \kappa^{2} \tau_{\kappa} (1 + 2\alpha) + (\kappa - (\kappa^{2} + 2)\tau_{\kappa})(1 + \alpha)^{2}}{(1 + 2\alpha) |\gamma| |\kappa^{2}| |\tau_{\kappa}|} \end{cases}.$$

Corollary 3.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{HSL}_{\Sigma}^{\kappa}(\gamma, \tilde{p}_{\kappa})$ . Then $$|a_2| \leq \frac{|\gamma| |\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{3\gamma \kappa^2 \tau_{\kappa} + 4(\kappa - (\kappa^2 + 2)\tau_{\kappa})}}, \qquad |a_3| \leq \frac{4 |\gamma| |\kappa \tau_{\kappa}| (\kappa - (\kappa^2 + 2)\tau_{\kappa})}{3 [3\gamma \kappa^2 \tau_{\kappa} + 4(\kappa - (\kappa^2 + 2)\tau_{\kappa})]},$$ and $$\left|a_3 - \mu a_2^2\right| \leq \left\{ \begin{array}{ll} \frac{\left|\gamma\right| \left|\kappa\tau\right|}{3} & ; 0 \leq \left|\mu - 1\right| \leq \frac{3\gamma\kappa^2\tau_\kappa + 4(\kappa - (\kappa^2 + 2)\tau_\kappa)}{3\left|\gamma\right| \kappa^2\left|\tau_\kappa\right|} \\ \frac{\left|1 - \mu\right| \gamma^2\kappa^3\tau_\kappa^2}{3\gamma\kappa^2\tau_\kappa + 4(\kappa - (\kappa^2 + 2)\tau_\kappa)} & ; \left|\mu - 1\right| \geq \frac{3\gamma\kappa^2\tau_\kappa + 4(\kappa - (\kappa^2 + 2)\tau_\kappa)}{3\left|\gamma\right| \kappa^2\left|\tau_\kappa\right|} \end{array} \right.$$

Corollary 3.4. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{SL}_{\Sigma}^{\kappa}(\gamma, \tilde{p}_{\kappa})$ . Then $$|a_2| \leq \frac{|\gamma| |\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{\gamma \kappa^2 \tau_{\kappa} + (\kappa - (\kappa^2 + 2)\tau_{\kappa})}}, \quad |a_3| \leq \frac{|\gamma| |\kappa \tau_{\kappa}| |(\kappa - (\kappa^2 + 2)\tau_{\kappa}) - \gamma \kappa^2 \tau_{\kappa}|}{2\gamma \kappa^2 \tau_{\kappa} + 2(\kappa - (\kappa^2 + 2)\tau_{\kappa})}$$ ana $$\left|a_{3}-\mu a_{2}^{2}\right| \leq \begin{cases} \frac{\left|\gamma\right|\left|\kappa\tau_{\kappa}\right|}{2} & ; 0 \leq \left|\mu-1\right| \leq \frac{\gamma\kappa^{2}\tau_{\kappa}+\left(\kappa-\left(\kappa^{2}+2\right)\tau_{\kappa}\right)}{2\left|\gamma\right|\kappa^{2}\left|\tau_{\kappa}\right|} \\ \frac{\left|1-\mu\right|\gamma^{2}\kappa^{3}\tau_{\kappa}^{2}}{\gamma\kappa^{2}\tau_{\kappa}+\left(\kappa-\left(\kappa^{2}+2\right)\tau_{\kappa}\right)} & ; \left|\mu-1\right| \geq \frac{\gamma\kappa^{2}\tau_{\kappa}+\left(\kappa-\left(\kappa^{2}+2\right)\tau_{\kappa}\right)}{2\left|\gamma\right|\kappa^{2}\left|\tau_{\kappa}\right|}. \end{cases}$$ <span id="page-17-0"></span>Corollary 3.5. [16] Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{SL}_{\Sigma}^{\kappa}(\tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{\kappa - 2\tau_{\kappa}}}, \quad |a_3| \le \frac{|\kappa \tau_{\kappa}| (\kappa - 2(\kappa^2 + 1)\tau_{\kappa})}{2\kappa - 4\tau_{\kappa}}$$ and $$\left|a_3 - \mu a_2^2\right| \le \begin{cases} \frac{\left|\kappa \tau_\kappa\right|}{2} & ; 0 \le |\mu - 1| \le \frac{\kappa - 2\tau_\kappa}{2\kappa^2 \left|\tau_\kappa\right|} \\ \frac{\left|1 - \mu\right| \kappa^3 \tau_\kappa^2}{\kappa - 2\tau_\kappa} & ; |\mu - 1| \ge \frac{\kappa - 2\tau_\kappa}{2\kappa^2 \left|\tau_\kappa\right|}. \end{cases}$$ <span id="page-17-1"></span>Corollary 3.6. [16] Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{KSL}_{\Sigma}^{\kappa}(\tilde{p}_{\kappa})$ . Then $$|a_2| \le \frac{|\kappa \tau_{\kappa}| \sqrt{\kappa}}{\sqrt{2(2\kappa - (\kappa^2 + 4)\tau_{\kappa})}}, \quad |a_3| \le \frac{|\kappa \tau_{\kappa}| (\kappa - 2(\kappa^2 + 1)\tau_{\kappa})}{3(2\kappa - (\kappa^2 + 4)\tau_{\kappa})}.$$ and $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{|\kappa \tau_{\kappa}|}{6} & ; 0 \le |\mu - 1| \le \frac{2\kappa - (\kappa^2 + 4)\tau_{\kappa}}{3\kappa^2 |\tau_{\kappa}|} \\ \frac{|1 - \mu| \kappa^3 \tau_{\kappa}^2}{2(2\kappa - (\kappa^2 + 4)\tau_{\kappa})} & ; |\mu - 1| \ge \frac{2\kappa - (\kappa^2 + 4)\tau_{\kappa}}{3\kappa^2 |\tau_{\kappa}|}. \end{cases}$$ Remark 3.1. Results discussed in Corollaries 3.5 and 3.6 are coincide with bounds obtained in [16]. Also, For $\kappa = 1$ , all the results obtained are coincides with results obtained in [18].

Definition 1.1. A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ belongs to the class $WSL_{\Sigma}^{\kappa}(\gamma, \lambda, \alpha, \tilde{p}_{\kappa})$ , $\gamma \in \mathbb{C}\setminus\{0\}$ , $\alpha \geq 0$ and $\lambda \geq 0$ , if the following conditions are satisfied: $$1 + \frac{1}{\gamma} \left( (1 - \alpha + 2\lambda) \frac{f(z)}{z} + (\alpha - 2\lambda) f'(z) + \lambda z f''(z) - 1 \right) \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D},$$ $$(1.5)$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} \left( (1 - \alpha + 2\lambda) \frac{g(w)}{w} + (\alpha - 2\lambda) g'(w) + \lambda w g''(w) - 1 \right) \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D},$$ $$(1.6)$$ where $$\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$$ . It is interesting to note that the special values of $\alpha$ , $\gamma$ and $\lambda$ lead the class $WSL_{\Sigma}^{\kappa}(\gamma, \lambda, \alpha, \tilde{p}_{\kappa})$ to various subclasses, as following illustrations: (1) For $\alpha = 1 + 2\lambda$ , we get the class $\mathcal{WSL}^{\kappa}_{\Sigma}(\gamma, \lambda, 1 + 2\lambda, \tilde{p}_{\kappa}) \equiv \mathcal{FSL}^{\kappa}_{\Sigma}(\gamma, \lambda, \tilde{p}_{\kappa})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $\mathcal{FSL}^{\kappa}_{\Sigma}(\gamma,\lambda,\tilde{p}_{\kappa})$ , if the following conditions $$1 + \frac{1}{\gamma} (f'(z) + \lambda z f''(z) - 1) \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} (g'(w) + \lambda w g''(w) - 1) \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D},$$ hold, where $$\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$$ . (2) For $\lambda = 0$ , we obtain the class $\mathcal{WSL}^{\kappa}_{\Sigma}(\gamma, 0, \alpha, \tilde{p}_{\kappa}) \equiv \mathcal{BSL}^{\kappa}_{\Sigma}(\gamma, \alpha, \tilde{p}_{\kappa})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $\mathcal{BSL}^{\kappa}_{\Sigma}(\gamma, \alpha, \tilde{p}_{\kappa})$ , if the following conditions $$1 + \frac{1}{\gamma} \left( (1 - \alpha) \frac{f(z)}{z} + \alpha f'(z) - 1 \right) \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} \left( (1 - \alpha) \frac{g(w)}{w} + \alpha g'(w) - 1 \right) \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D}$$ hold, where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ . (3) For $\lambda = 0$ and $\alpha = 1$ , we have the class $\mathcal{WSL}^{\kappa}_{\Sigma}(\gamma, 0, 1, \tilde{p}_{\kappa}) \equiv \mathcal{HSL}^{\kappa}_{\Sigma}(\gamma, \tilde{p}_{\kappa})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $\mathcal{HSL}^{\kappa}_{\Sigma}(\gamma, \tilde{p}_{\kappa})$ , if the following conditions $$1 + \frac{1}{\gamma} (f'(z) - 1) \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} \left( g'(w) - 1 \right) \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D}$$ hold, where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ .

Definition 1.2. A function $f \in \Sigma$ of the form (1.1) belongs to the class $\mathcal{RSL}^{\kappa}_{\Sigma}(\gamma, \lambda, \tilde{p}_{\kappa})$ , $\gamma \in \mathbb{C} \setminus \{0\}$ and $\lambda \geq 0$ , if the following conditions are satisfied: $$1 + \frac{1}{\gamma} \left( \frac{z^{1-\lambda} f'(z)}{(f(z))^{1-\lambda}} - 1 \right) \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ (1.7) and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} \left( \frac{w^{1-\lambda} g'(w)}{(g(w))^{1-\lambda}} - 1 \right) \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D}$$ (1.8) where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ . (1) For $\lambda = 0$ , we let the class $\mathcal{RSL}^{\kappa}_{\Sigma}(\gamma, 0, \tilde{p}_{\kappa}) \equiv \mathcal{SL}^{\kappa}_{\Sigma}(\gamma, \tilde{p}_{\kappa})$ . A function $f \in \Sigma$ of the form (1.1) is said to be in $\mathcal{SL}^{\kappa}_{\Sigma}(\gamma, \tilde{p}_{\kappa})$ , if the following conditions $$1 + \frac{1}{\gamma} \left( \frac{zf'(z)}{f(z)} - 1 \right) \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{1}{\gamma} \left( \frac{wg'(w)}{g(w)} - 1 \right) \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D}$$ hold, where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ . Remark 1.1. For $\gamma = 1$ the class $\mathcal{SL}^{\kappa}_{\Sigma}(1, \tilde{p}_{\kappa}) \equiv \mathcal{SL}^{\kappa}_{\Sigma}(\tilde{p}_{\kappa})$ was introduced and studied Güney et al. [16] and for $\kappa = 1$ the class was studied by Güney et al. [15]. (2) For $\lambda = 1$ , we have $\mathcal{RSL}^{\kappa}_{\Sigma}(\gamma, 1, \tilde{p}_{\kappa}) \equiv \mathcal{HSL}^{\kappa}_{\Sigma}(\gamma, \tilde{p}_{\kappa})$ .

Definition 1.3. A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ belongs to the class $\mathcal{SLB}^{\kappa}_{\Sigma}(\lambda; \tilde{p}_{\kappa}), \lambda \geq 1$ , if the following conditions are satisfied: $$\frac{z\left[f'(z)\right]^{\lambda}}{f(z)} \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ $$\tag{1.9}$$ and for $g(w) = f^{-1}(w)$ $$\frac{w\left[g'(w)\right]^{\lambda}}{g(w)} \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^{2} w^{2}}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^{2} w^{2}}, \ w \in \mathbb{D},\tag{1.10}$$ where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ . (1) For $\lambda = 1$ , we have the class $\mathcal{SLB}_{\Sigma}^{\kappa}(1; \tilde{p}_{\kappa}) \equiv \mathcal{SL}_{\Sigma}^{\kappa}(\tilde{p}_{\kappa})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $\mathcal{SL}^{\kappa}_{\Sigma}(\tilde{p}_{\kappa})$ , if the following conditions $$\frac{zf'(z)}{f(z)} \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$\frac{wg'(w)}{g(w)} \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D},$$ hold, where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ .

Definition 1.4. A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ belongs to the class $\mathcal{PSL}^{\kappa}_{\Sigma}(\lambda; \tilde{p}_{\kappa}), 0 \leq \lambda \leq 1$ , if the following conditions are satisfied: $$\frac{zf'(z) + \lambda z^2 f''(z)}{(1 - \lambda)f(z) + \lambda z f'(z)}$$ and for $g(w) = f^{-1}(w)$ $$\frac{wf'(w) + \lambda w^2 g''(w)}{(1 - \lambda)g(w) + \lambda w g'(w)} \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D},$$ (1.12) where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ . - (1) For $\lambda = 0$ , we have the class $\mathcal{PSL}^{\kappa}_{\Sigma}(0; \tilde{p}_{\kappa}) \equiv \mathcal{SL}^{\kappa}_{\Sigma}(\tilde{p}_{\kappa})$ . - (2) For $\lambda = 1$ , we have the class $\mathcal{PSL}_{\Sigma}^{\kappa}(1; \tilde{p}_{\kappa}) \equiv \mathcal{KSL}_{\Sigma}^{\kappa}(\tilde{p}_{\kappa})$ . A function $f \in \Sigma$ of the form $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$ is said to be in $KSL_{\Sigma}^{\kappa}(\tilde{p}_{\kappa})$ , if the following conditions $$1 + \frac{z^2 f''(z)}{f'(z)} \prec \widetilde{p_{\kappa}(z)} = \frac{1 + \tau_{\kappa}^2 z^2}{1 - \kappa \tau_{\kappa} z - \tau_{\kappa}^2 z^2}, \ z \in \mathbb{D}$$ and for $g(w) = f^{-1}(w)$ $$1 + \frac{w^2 g''(w)}{g'(w)} \prec \widetilde{p_{\kappa}(w)} = \frac{1 + \tau_{\kappa}^2 w^2}{1 - \kappa \tau_{\kappa} w - \tau_{\kappa}^2 w^2}, \ w \in \mathbb{D},$$ hold, where $\tau_{\kappa} = \frac{\kappa - \sqrt{\kappa^2 + 4}}{2}$ . Remark 1.2. For $\gamma = 0$ , $\mathcal{PSL}^{\kappa}_{\Sigma}(0, \tilde{p}_{\kappa}) \equiv \mathcal{SL}^{\kappa}_{\Sigma}(\tilde{p}_{\kappa})$ and $\gamma = 1$ , $\mathcal{PSL}^{\kappa}_{\Sigma}(1, \tilde{p}_{\kappa}) \equiv \mathcal{KSL}^{\kappa}_{\Sigma}(\tilde{p}_{\kappa})$ the classes were introduced and studied Güney et al. [16] and for $\kappa = 1$ the class was studied by Güney et al. [15]. In order to prove our results for the function in the classes $\mathcal{WSL}^{\kappa}_{\Sigma}(\gamma, \lambda, \alpha, \tilde{p}_{\kappa}), \mathcal{SLB}^{\kappa}_{\Sigma}(\lambda; \tilde{p}_{\kappa})$ and $\mathcal{PSL}^{\kappa}_{\Sigma}(\lambda; \tilde{p}_{\kappa})$ , we need the following lemma.