On Certain Classes of Bi-Bazilevic Functions Defined by q-Ruscheweyh Differential Operator

Lemma 2.3. if the function $\Omega$ in the Caratheodory class, then it can be written as $$\Omega(z) = 1 + c_1 z + c_2 z^2 + c_3 z^3 + \cdots,$$ for $z \in \mathbb{D}$ . Moreover, $|c_n| \leq 2$ for each $n \in \mathbb{N}$ . The lemma presented in the following discussion is extensively referenced in existing literature (refer to, for example, [27]) and is regarded as a foundational principle that significantly influences the research we are conducting.

Lemma 2.4. Let u and v be real numbers. Let p and q be complex numbers. If |p| < r and |q| < r, $$|(u+v)p + (u-v)q| \le \begin{cases} 2r|u|, & \text{if } |u| \ge |v| \\ 2r|v|, & \text{if } |u| \le |v|. \end{cases}$$ By selecting particular values of $\beta$ in Definition 2.1, it is possible to obtain the following subclasses. Example 1. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathfrak{B}^0(R_q^\alpha, G_\delta(x, z))$ , then the following subordinations hold $$\frac{z\left(R_q^{\alpha}f(z)\right)'}{\left(R_a^{\alpha}f(z)\right)} \prec G_{\delta}(x,z),\tag{5}$$ and $$\frac{w\left(R_q^{\alpha}g(w)\right)'}{\left(R_q^{\alpha}g(w)\right)} \prec G_{\delta}(x,w),\tag{6}$$ where the function $g(w) = f^{-1}(w)$ is given by Equation (2), the parameters 0 < q < 1, $\alpha > -1$ , $\delta \ge 0$ and $-1 \le x \le 1$ . Example 2. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathfrak{B}^1(R^{\alpha}_q, G_{\delta}(x, z))$ then the following subordinations hold $$\left(R_q^{\alpha}f(z)\right)' \prec G_{\delta}(x,z),$$ (7) and $$(R_q^{\alpha}g(w))' \prec G_{\delta}(x, w),$$ (8) where the function $g(w) = f^{-1}(w)$ is given by Equation (2), the parameters 0 < q < 1, $\alpha > -1$ , $\delta \ge 0$ and $-1 \le x \le 1$ . Moreover, as $q \to 1^-$ and taking $\alpha = 0$ , we get $R_q^0 f(z) = f(z)$ . Therefore, we get the following close-to-starlike subclasses. For more information about classes of starlike analytic functions, we refer the readers, for example, to the articles, [22], [44], [45], [46], [47], [48], [49], [50], and the references provided therein. Example 3. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathcal{B}^*(G_{\delta}(x,z))$ then the following subordinations hold $$\frac{zf'(z)}{f(z)} \prec G_{\delta}(x, z), \tag{9}$$ and $$\frac{wg'(w)}{g(w)} \prec G_{\delta}(x, w), \tag{10}$$ where the function $g(w) = f^{-1}(w)$ is given by Equation (2), the parameters $\delta \geq 0$ and $-1 \leq x \leq 1$ . In addition, by taking $\delta = 1$ in Definition 2.1, we easily obtain the following subsequent subclass. Example 4. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathfrak{S}^*(\delta,\phi)$ then the following subordinations hold $$\frac{z^{1-\beta} \left(R_q^{\alpha} f(z)\right)'}{\left(R_q^{\alpha} f(z)\right)^{1-\beta}} \prec T_n(x, z), \tag{11}$$ and $$\frac{w^{1-\beta} \left(R_q^{\alpha} g(w)\right)'}{\left(R_q^{\alpha} g(w)\right)^{1-\beta}} \prec T_n(x, w), \tag{12}$$ where the function $g(w) = f^{-1}(w)$ is given by Equation (2), the parameters $\beta \geq 0$ , 0 < q < 1, $\alpha > -1$ and $-1 \leq x \leq 1$ . Similarly, By selecting particular values of $\beta$ in Definition 2.2, it is possible to obtain the subsequent subclasses. Example 5. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathfrak{B}^0(R_q^{\alpha},\tanh)$ then the following subordinations hold $$\frac{z\left(R_q^{\alpha}f(z)\right)'}{\left(R_q^{\alpha}f(z)\right)} \prec 1 + \tanh(qz),\tag{13}$$ and $$\frac{w\left(R_q^{\alpha}g(w)\right)'}{\left(R_q^{\alpha}g(w)\right)} \prec 1 + \tanh(qw), \tag{14}$$ where the function $g(w) = f^{-1}(w)$ is given by the Equation (2), the parameters 0 < q < 1, $\alpha > -1$ , $\delta \ge 0$ and $-1 \le x \le 1$ . Example 6. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathfrak{B}^1(R_q^{\alpha},\tanh)$ then the following subordinations hold $$(R_q^{\alpha} f(z))' \prec 1 + \tanh(qz),$$ (15) and $$(R_q^{\alpha}g(w))' \prec 1 + \tanh(qw),$$ (16) where the function $g(w) = f^{-1}(w)$ is given by Equation (2), the parameters 0 < q < 1, $\alpha > -1$ , $\delta \ge 0$ and $-1 \le x \le 1$ . Moreover, as $q \to 1^-$ and taking $\alpha = 0$ , we get $R_q^0 f(z) = f(z)$ . Therefore, we get the following close-to-starlike subclasses. Example 7. Let f be a bi-univalent function that is given by Equation (1). If f belongs to the subclass $\mathfrak{B}^*(\tanh)$ then the following subordinations hold $$\frac{zf'(z)}{f(z)} \prec 1 + \tanh(qz),\tag{17}$$ and $$\frac{wg'(w)}{g(w)} \prec 1 + \tanh(qw), \tag{18}$$ where the function $g(w) = f^{-1}(w)$ is given by Equation (2), the parameters $\delta \geq 0$ and $-1 \leq x \leq 1$ . This study aims to investigate two novel classes of bi-Bazilevic functions, which are characterized by the q-Ruscheweyh operator within the open unit disk $\mathbb{D}$ . The primary goal is to derive estimates for the moduli of the initial coefficients $|a_2|$ and $|a_3|$ , which are associated with the Taylor-Maclaurin series representation of functions within these categories. Furthermore, the research addresses the Fekete-Szegö functional problem relevant to these functions, thereby contributing to a deeper understanding of their fundamental properties. Additionally, several established corollaries are provided based on the selection of parameters used in defining the specific classes under consideration.

Theorem 3.1. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^{\beta}(R_q^{\alpha}, G_{\delta}(x, z))$ , then the following inequalities hold: $$|a_{2}| \leq \frac{4\delta^{3/2}|x|^{3/2}}{\sqrt{\left|4x^{2}\delta^{2}\left(2(\beta+2)\psi_{3}+\right.\right.}},$$ $$\left.\sqrt{(\beta-1)(\beta+2)\psi_{2}^{2}-2(\beta+1)^{2}\psi_{2}^{2}g_{2}^{\delta}(x)\right|}$$ (19) and $$|a_3| \le \frac{2\delta|x|}{(\beta+2)\psi_3} + \frac{4\delta^2 x^2}{(\beta+1)^2 \psi_2^2}.$$ (20) Proof. Le f be a bi-univalent function that belongs to the class $\mathcal{B}(\beta, \delta, R_n^{\alpha}, \beta(t))$ . Consulting the Definition 2.1 and Subordination Principle, two Schwarz functions u(z) and v(w) can be identified within the open unit disk $\mathbb D$ such that $$\frac{z^{1-\beta} \left(R_q^{\alpha} f(z)\right)'}{\left(R_q^{\alpha} f(z)\right)^{1-\beta}} = G_{\delta}(x, u(z)), \tag{21}$$ and $$\frac{w^{1-\beta} \left(R_q^{\alpha} g(w)\right)'}{\left(R_q^{\alpha} g(w)\right)^{1-\beta}} = G_{\delta}(x, v(w)). \tag{22}$$ now, comparing the coefficients of both sides of Equation (21) and Equation (22), we obtain the following set of equations $$(\beta + 1)\psi_2 a_2 = g_1^{\delta}(x)u_1, \tag{23}$$ $$\frac{(\beta-1)(\beta+2)}{2}\psi_2^2 a_2^2 + (\beta+2)\psi_3 a_3$$ $$= g_1^{\delta}(x)u_2 + g_2^{\delta}(x)u_1^2,$$ (24) $$-(\beta+1)\psi_2 a_2 = g_1^{\delta}(x)v_1, \tag{25}$$ and $$\left(2(\beta+2)\psi_3 + \frac{(\beta-1)(\beta+2)}{2}\psi_2^2\right)a_2^2 - (\beta+2)\psi_3 a_3 = g_1^{\delta}(x)v_2 + g_2^{\delta}(x)v_1^2.$$ (26) Therefore, using Equation (23) and Equation (25), we easily derive the subsequent equation $$a_2 = \frac{g_1^{\delta}(x)u_1}{(\beta + 1)\psi_2} = \frac{-g_1^{\delta}(x)v_1}{(\beta + 1)\psi_2}$$ (27) Moreover, we easily get the following equation $$2(\beta+1)^2\psi_2^2a_2^2 = \left(g_1^{\delta}(x)\right)^2(u_1^2+v_1^2). \tag{28}$$ On one hand, adding Equation (24) to Equation (26), we derive the subsequent equation $$((\beta - 1)(\beta + 2)\psi_2^2 + 2(\beta + 2)\psi_3) a_2^2 = g_1^{\delta}(x)(u_2 + v_2) + g_2^{\delta}(x)(u_1^2 + v_1^2).$$ (29) Therefore, using equation (28), the last equation can be written as $$((\beta - 1)(\beta + 2)\psi_2^2 + 2(\beta + 2)\psi_3) a_2^2$$ $$= g_1^{\delta}(x)(u_2 + v_2) + \frac{2g_2^{\delta}(x)(\beta + 1)^2\psi_2^2}{(g_1^{\delta}(x))^2} a_2^2.$$ (30) Equivalently, the last equation can be written as $$a_2^2 = \frac{\left(g_1^{\delta}(x)\right)^3 \left(u_2 + v_2\right)}{\left\{ \left[ (\beta - 1)(\beta + 2)\psi_2^2 + 2(\beta + 2)\psi_3 \right] \left(g_1^{\delta}(x)\right)^2 \right\}}.$$ $$\left\{ -2g_2^{\delta}(x)(\beta + 1)^2 \psi_2^2 \right\}.$$ (31) Therefore, considering Equation (4) and then using the constraints $|u_2| \leq 1$ and $|v_2| \leq 1$ , , we easily get the desired estimation of $|a_2|$ presented in Inequality (19). Secondly, our objective is to ascertain the coefficient estimate for $|a_3|$ . By substituting Equation (26) into Equation (24), we can formulate the following equation. $$2(\beta+2)\psi_3 a_3 - 2(\beta+2)\psi_3 a_2^2$$ = $g_1^{\delta}(x)(u_2-v_2) + g_2^{\delta}(u_1^2-v_1^2).$ (32) Now, using Equation (27), we get $u_1 = -v_1$ . Consequently, the last equation can be expressed in the following manner $$a_3 = \frac{g_1^{\delta}(x)(u_2 - v_2)}{2(\beta + 2)\psi_3} + a_2^2.$$ (33) Moreover, consulting Equation (28), Equation (33) can be written as follows $$a_3 = \frac{g_1^{\delta}(x)(u_2 - v_2)}{2(\beta + 2)\psi_3} + \frac{\left(g_1^{\delta}(x)\right)^2 (u_1^2 + v_1^2)}{2(\beta + 1)^2 \psi_2^2}$$ (34) Finally, considering the Equation (4) then using the constraints $|u_j| \leq 1$ and $|v_j| \leq 1$ for j = 1, 2, the last equation provides the required bound for $|a_3|$ , as indicated by Inequality (20). Therefore, the proof of Theorem 3.1 is now complete. The subsequent theorem establishes the estimates concerning the modulus of the initial coefficients of functions that are part of the class $\mathcal{B}^{\beta}(R_q^{\alpha}, \tanh)$ . The methods employed in its proof are akin to those utilized in the proof of Theorem 3.1.

Theorem 3.2. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^{\beta}(R_q^{\alpha}, \tanh)$ , then the following inequalities hold $$|a_2| \le \frac{\sqrt{2}q}{\sqrt{qK + 2(\beta + 1)^2\psi_2^2}},$$ (35) and $$|a_3| \le \frac{q}{(\beta+2)\psi_3} + \frac{2q^2}{qK + 2(\beta+1)^2\psi_2^2},$$ (36) where $$K = 2(\beta + 2)\psi_3 + (\beta - 1)(\beta + 2)\psi_2^2.$$

Corollary 3.3. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^0(R_q^{\alpha}, G_{\delta}(x, z))$ , then the following inequalities hold: $$|a_2| \le \frac{2(\delta|x|)^{3/2}}{\sqrt{|8x^2\delta^2\psi_3 + \psi_2^2\delta(2x^2(1-\delta)) - 1|}}$$ and $$|a_3| \le \frac{\delta|x|}{\psi_3} + \frac{4\delta^2 x^2}{\psi_2^2}.$$

Corollary 3.4. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^1(R_q^{\alpha}, G_{\delta}(x, z))$ , then the following inequalities hold: $$|a_2| \le \frac{\sqrt{2} (\delta|x|)^{3/2}}{\sqrt{|3x^2\delta^2\psi_3 - \psi_2^2 g_2^{\delta}(x)|}},$$ and $$|a_3| \le \frac{2\delta|x|}{3\psi_3} + \frac{\delta^2 x^2}{\psi_2^2}.$$

Corollary 3.5. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^*(G_{\delta}(x,z))$ , then the following inequalities hold: $$|a_2| \le \frac{2(\delta|x|)^{3/2}}{\sqrt{|2x^2\delta(3\delta+1)-\delta|}}$$ and $$|a_3| \le \delta|x| + 4\delta^2 x^2.$$

Corollary 3.6. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathcal{B}^{\beta}(R_q^{\alpha}, T_n(x, z))$ , then the following inequalities hold: $$|a_2| \le \frac{4|x|^{3/2}}{\sqrt{\left|4x^2\left(2(\beta+2)\psi_3 + (\beta-1)(\beta+2)\psi_2^2\right)\right|}},$$ $$-2(\beta+1)^2\psi_2^2g_2^{\delta}(x)$$ and $$|a_3| \le \frac{2|x|}{(\beta+2)\psi_3} + \frac{4x^2}{(\beta+1)^2\psi_2^2}.$$ The following corollaries are directly obtained from Theorem 3.2, contingent upon the conditions specified in the earlier examples. The techniques employed in deriving these corollaries closely mirror those applied in the proof of aforementioned theorem, which is the rationale behind our decision to exclude the detailed proofs.

Corollary 3.7. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^0(R_q^{\alpha},\tanh)$ , then the following inequalities hold $$|a_2| \le \frac{q}{\sqrt{2q\psi_3 + (1-q)\psi_2^2}},$$ and $$|a_3| \le \frac{q}{2\psi_3} + \frac{q^2}{2q\psi_3 + (1-q)\psi_2^2}.$$

Corollary 3.8. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^1(R_q^{\alpha},\tanh)$ , then the following inequalities hold $$|a_2| \le \frac{q}{\sqrt{3q\psi_3 + 4\psi_2^2}},$$ and $$|a_3| \le \frac{q}{3\psi_3} + \frac{q^2}{3q\psi_3 + 4\psi_2^2}.$$

Corollary 3.9. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $S^*(\tanh)$ , then the following inequalities hold $$|a_2| \le \frac{1}{\sqrt{2}}, \quad and \quad |a_3| \le 1.$$

Theorem 4.1. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathcal{B}^{\beta}(R_q^{\alpha}, G_{\delta}(x, z))$ , then for a real number $\zeta$ and $\delta \neq 0$ the following inequality holds $$|a_{3} - \zeta a_{2}^{2}| \leq \begin{cases} \frac{2\delta|x|}{(\beta+2)\psi_{3}}, & if \ |1 - \zeta| \leq |\mathcal{Y}| \\ \frac{8\delta^{3}|x^{3}||1 - \zeta|}{|2\delta^{2}x^{2}K - B^{2}g_{2}^{\delta}(x)|}, & if \ |1 - \zeta| \geq |\mathcal{Y}|, \end{cases}$$ $$(52)$$ where $$K = 2(\beta+2)\psi_3 + (\beta-1)(\beta+2)\psi_2^2, \quad B = (\beta+1)\psi_2, \quad and$$ $$\mathcal{Y} = \frac{2x^2\delta(K-B^2) - B^2(2x^2+1)}{4x\delta(\beta+2)\psi_2}.$$

Corollary 4.2. Let f be a bi-univalent function that is given by Equation (1). If the function f is obeying the Subordination conditions (5) and (6), then for a real number $\zeta$ and $\delta \neq 0$ the following holds $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{\delta|x|}{2\psi_3}, & \text{if } |1 - \zeta| \le |\mathcal{Y}_1| \\ \frac{8\delta^3|x^3||1 - \zeta||g_1^\delta(x)|^3}{|\mathcal{A}|}, & \text{if } |1 - \zeta| \ge |\mathcal{Y}_1|, \end{cases}$$ where $$\mathcal{A} = 2\delta^2 x^2 (16\psi_3 - 3\psi_2^2) + \delta\psi_2^2 (1 - 2x^2)),$$ $$\mathcal{Y}_1 = \frac{2x^2 \delta((4\psi_3 - 3\psi_2^2) - \psi_2^2 (2x^2 + 1))}{8x\delta\psi_2}.$$

Corollary 4.3. Let f be a bi-univalent function that is given by Equation (1). If the function f is obeying the Subordination conditions (7) and (8), then for a real number $\zeta$ and $\delta \neq 0$ the following holds $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{2\delta|x|}{3\psi_3}, & \text{if } |1 - \zeta| \le |\mathcal{Y}_2| \\ \frac{2\delta^3|x^3||1 - \zeta|}{|3\delta^2 x^2 \psi_3 - \psi_2^2 \sigma_2^\delta(x)|}, & \text{if } |1 - \zeta| \ge |\mathcal{Y}_2|, \end{cases}$$ where $$\mathcal{Y}_2 = \frac{x^2 \delta(3\psi_3 - 2\psi_2^2) - \psi_2^2(2x^2 + 1)}{3x\delta\psi_3}.$$

Corollary 4.4. Let f be a bi-univalent function that is given by Equation (1). If the function f is obeying the Subordination conditions (9) and (10), then for a real number $\zeta$ and $\delta \neq 0$ the following holds $$|a_3 - \zeta a_2^2| \le \begin{cases} \delta |x|, & \text{if } |1 - \zeta| \le |\frac{1 - 2x^2(1 - \delta)}{8x\delta}| \\ \frac{8\delta^2 |x^3||1 - \zeta|}{|\delta^2 x^2 - 2x^2 + 1|}, & \text{if } |1 - \zeta| \ge |\frac{1 - 2x^2(1 - \delta)}{8x\delta}|. \end{cases}$$ In the upcoming theorem, we present the Fekete-Szegö inequality specifically for functions that are part of the designated class $\mathcal{B}^{\beta}(R_q^{\alpha}, \tanh)$ , which contains the bi-Bazilevic functions defined through the q-Ruscheweyh differential operator and subordinate to q-analogue of hyperbolic tangent function. The proof of this Theorem uses similar techniques as that in the proof of Theorem 3.1.

Theorem 4.5. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^{\beta}(R_q^{\alpha}, \tanh)$ , then for a real number $\zeta$ and $\delta \neq 0$ the following inequality holds $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{q}{2(\beta + 2)\psi_3}, & \text{if } |1 - \zeta| \le \mathcal{B} \\ \frac{2q^2|1 - \zeta|}{qK + 2(\beta + 1)^2\psi_2^2}, & \text{if } |1 - \zeta| \ge \mathcal{B}, \end{cases}$$ (53) where $$\mathcal{B} = \frac{qK + 2(\beta + 1)^2 \psi_2^2}{2q(\beta + 2)\psi_3},$$ $$K = 2(\beta + 2)\psi_3 + (\beta - 1)(\beta + 2)\psi_2^2$$ .

Corollary 4.6. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^0(R_q^{\alpha},\tanh)$ , then the following inequalities hold $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{q}{4\psi_3}, & \text{if } \zeta \in [\zeta_1, \zeta_2] \\ \frac{q^2|1-\zeta|}{2q\psi_3 + (1-q)\psi_2^2}, & \text{if } \zeta \notin [\zeta_1, \zeta_2], \end{cases}$$ where $$\zeta_1 = \frac{(q-1)\psi_2^2}{2q\psi_3}$$ and $\zeta_2 = 2 - \zeta_1$ .

Corollary 4.7. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $\mathfrak{B}^1(R_q^{\alpha},\tanh)$ , then the following inequalities hold $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{q}{6\psi_3}, & \text{if } \zeta \in [\zeta_3, \zeta_4] \\ \frac{q^2|1-\zeta|}{3q\psi_3 + 4\psi_2^2}, & \text{if } \zeta \notin [\zeta_3, \zeta_4], \end{cases}$$ where $$\zeta_3 = \frac{-4\psi_2^2}{3q\psi_3}$$ and $\zeta_4 = 2 - \zeta_3$ .

Corollary 4.8. Let f be a bi-univalent function that is given by Equation (1). If the function f belongs to the class $S^*(\tanh)$ , then the following inequalities hold $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{1}{4}, & \text{if } \zeta \in [0, 2]\\ \frac{|1 - \zeta|}{2}, & \text{if } \zeta \notin [0, 2]. \end{cases}$$

Definition 2.1. A bi-univalent function f that is given by Equtaion (1) is said to be in the class $\mathfrak{B}^{\beta}(R_q^{\alpha}, G_{\delta}(x, z))$ if the following subordinations hold: $$\frac{z^{1-\beta} \left(R_q^{\alpha} f(z)\right)'}{\left(R_q^{\alpha} f(z)\right)^{1-\beta}} \prec G_{\delta}(x,z),$$ and $$\frac{w^{1-\beta} \left(R_q^{\alpha} g(w)\right)'}{\left(R_q^{\alpha} g(w)\right)^{1-\beta}} \prec G_{\delta}(x, w),$$ where the function $g(w) = f^{-1}(w)$ is given by the Equation (2), the parameters $\beta \geq 0$ , 0 < q < 1, $\alpha > -1$ , $\delta \geq 0$ and $-1 \leq x \leq 1$ . The second class is comprised of bi-Bazilevic functions characterized by the q-Ruscheweyh differential operator associated with the q-analogue hyperbolic functions. We denote this class as $\mathcal{B}^{\beta}(R_q^{\alpha}, G_{\delta}(t, z))$ , and we next provide a formal definition for this class.

Definition 2.2. A bi-univalent function f that is given by Equtaion (1) is said to be in the class $\mathfrak{B}^{\beta}(R_q^{\alpha}, \tanh)$ if the following subordinations hold: $$\frac{z^{1-\beta} \left(R_q^{\alpha} f(z)\right)'}{\left(R_q^{\alpha} f(z)\right)^{1-\beta}} \prec 1 + \tanh(qz),$$ and $$\frac{w^{1-\beta} \left(R_q^{\alpha} g(w)\right)'}{\left(R_q^{\alpha} g(w)\right)^{1-\beta}} \prec 1 + \tanh(qw),$$ where the function $g(w) = f^{-1}(w)$ is given by the Equation (2), the parameters $\beta \geq 0$ , 0 < q < 1, and $\alpha > -1$ . The following lemma, extensively elaborated upon in existing literature (refer to, for example, [27]), represents well-established principles that hold significant importance for our presenting research.