Coefficient Estimates and Distortion Bounds for Rabotnov Functions with Application to Digital Image Warping

Theorem 2.1. Let $f \in \mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ be given by $f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} a_{\kappa} \zeta^{\kappa}$ . Then $$\sum_{\kappa=2}^{\infty} (\kappa - \lambda)(\varrho(\kappa - 1) + 1) \frac{\delta^{\kappa - 1} \Gamma(1 + \tau)}{\Gamma((1 + \tau)\kappa)} |a_{\kappa}| \le 1 - \lambda, \tag{2.1}$$ where $\tau \geq 0, \delta > 0, 0 \leq \varrho \leq 1, 0 \leq \lambda < 1, and \zeta \in U$ .

Theorem 2.2. Let $f \in \mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ and $f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} a_{\kappa} \zeta^{\kappa}$ . Then $$\sum_{\kappa=2}^{\infty} \left[\kappa + b - \lambda b - 1\right] \left[1 + \varrho \kappa - \varrho\right] \frac{\delta^{\kappa - 1} \Gamma(1 + \tau)}{\Gamma((1 + \tau)\kappa)} |a_{\kappa}| \le b(1 - \lambda). \tag{2.4}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , $\zeta \in U$ .

Theorem 2.3. Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,\lambda}$ and $f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} a_{\kappa} \zeta^{\kappa}$ . Then $$\sum_{\kappa=2}^{\infty} \left[ (\kappa - 1)(\mu \kappa + 1) + (1 - \lambda) \right] \frac{\delta^{\kappa - 1} \Gamma(1 + \tau)}{\Gamma((1 + \tau)\kappa)} |a_{\kappa}| \le (1 - \lambda). \tag{2.5}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , $0 \leq \lambda < 1$ , $\zeta \in U$ .

Theorem 2.4. Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,b,\lambda}$ and $f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} a_{\kappa} \zeta^{\kappa}$ . Then $$\sum_{\kappa=2}^{\infty} (\mu \kappa^2 - \mu \kappa + \kappa - 1 - \lambda b + b) \frac{\delta^{\kappa-1} \Gamma(1+\tau)}{\Gamma((1+\tau)\kappa)} |a_{\kappa}| \le b(1-\lambda).$$ (2.6) where $\tau \ge 0$ , $\delta > 0$ , $0 \le \mu \le 1$ , $0 \le \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , $\zeta \in U$ .

Corollary 2.5. (Extremal function) Let $f \in \mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ . Then $$\sum_{\kappa=2}^{\infty} (\kappa - \lambda) \left[ \varrho(\kappa - 1) + 1 \right] \frac{\delta^{\kappa - 1} \Gamma(1 + \tau)}{\Gamma((1 + \tau)\kappa)} |a_{\kappa}| \le 1 - \lambda. \tag{2.7}$$ The results is sharp for the function specified by: $$f(\zeta) = \zeta + \frac{1 - \lambda}{(\kappa - \lambda) \left[\varrho(\kappa - 1) + 1\right]} \frac{\Gamma((1 + \tau)\kappa)}{\delta^{\kappa - 1} \Gamma(1 + \tau)} \zeta^{\kappa}, \quad \kappa \ge 2.$$ (2.8)

Corollary 2.6. [19] By setting the parameters $\varrho = 0$ , $\tau = 0$ , and $\delta = 1$ in the class $\mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ , the operator reduces to the identity. Consequently, the inequality (2.7) yields the necessary condition for a function f to belong to the class of starlike functions of order $\lambda$ , represented as $\mathcal{S}^*(\lambda)$ : $$\sum_{\kappa=2}^{\infty} (\kappa - \lambda)|a_{\kappa}| \le 1 - \lambda. \tag{2.9}$$ Furthermore, by adjusting the notation to align with the results of Mustafa specifically replacing the index $\kappa$ with n, and the order parameter $\lambda$ with $\tau$ we obtain the kindred result: $$\sum_{n=2}^{\infty} (n-\tau)|a_n| \le 1 - \tau. \tag{2.10}$$

Corollary 2.7. [19] By the setting the parameter $\varrho = 1$ , $\tau = 0$ and $\delta = 1$ in $f \in \mathcal{R}^{\varrho,\lambda}_{\tau,\delta}$ , the operator reduces to the identity. Consequently, the inequality 2.7 reduces to the sufficient condition for a function f to be in the class of convex function of order $\lambda$ , denoted by $\kappa(\lambda)$ $$\sum_{\kappa=2}^{\infty} \kappa(\kappa - \lambda)|a_{\kappa}| \le 1 - \lambda. \tag{2.11}$$ Replace the parameter $\tau = \lambda$ , $n = \kappa$ , $\delta = \varrho$ the outcome obtained a corollary kindred to the result Silverman. $$\sum_{n=2}^{\infty} n(n-\tau)|a_n| \le 1 - \tau.$$

Corollary 2.8. (Extremal function) Let $f \in \mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ . Then $$\sum_{\kappa=2}^{\infty} [\kappa + b - \lambda b - 1] [1 + \varrho \kappa - \varrho] \frac{\delta^{\kappa - 1} \Gamma(1 + \tau)}{\Gamma((1 + \tau)\kappa)} |a_{\kappa}| \le b(1 - \lambda). \tag{2.12}$$ The outcome is sharp for the function specified by: $$f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \frac{(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]} \zeta^{\kappa}.$$ (2.13)

Corollary 2.9. [2, 26] By the setting the parameter b=1 in $f \in \mathcal{R}^{\varrho,1,\lambda}_{\tau,\delta}$ . Then derive a sufficient requirement for a function to belong to the class. $\mathcal{R}^{\varrho,\lambda}_{\tau,\delta}$ $$f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \frac{1 - \lambda \Gamma((1+\tau)\kappa)}{\delta^{\kappa-1} \Gamma(1+\tau)[\kappa-\lambda][1 + \varrho\kappa - \varrho]} \zeta^{\kappa}.$$ Replace the parameter $\tau = \lambda$ , $n = \kappa$ , $\delta = \varrho$ the outcome obtained a corollary kindred to the result Silverman. $$\sum_{n=2}^{\infty} (n-\tau)[1+\delta(n-1)]|a_n| \le 1-\tau.$$

Corollary 2.10. (Extremal function) Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,\lambda}$ . Then $$\sum_{\kappa=2}^{\infty} [\mu\kappa(\kappa-1) + \kappa - \lambda] \frac{\delta^{\kappa-1}\Gamma(1+\tau)}{\Gamma((1+\tau)\kappa)} |a_{\kappa}| \le 1 - \lambda.$$ The outcome is sharp for the function specified by: <span id="page-7-0"></span> $$f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]} \zeta^{\kappa}.$$ (2.14)

Corollary 2.11. [28] By setting the parameters $\tau = 0$ and $\delta = 1$ in (2.14), the outcome obtained a corollary kindred to the results. Replace the parameter $\lambda = \delta, \mu = \tau$ the outcome obtained a corollary kindred to the result, $$\sum_{\kappa=2}^{\infty} (\kappa - 1)(\tau \kappa + 1) + (1 - \delta)|a_{\kappa}| \le 1 - \delta.$$

Corollary 2.12. (Extremal function) Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,b,\lambda}$ . Then $$\sum_{\kappa=2}^{\infty} (\mu \kappa^2 - \mu \kappa + \kappa - 1 - \lambda b + b) \frac{\delta^{\kappa-1} \Gamma(1+\tau)}{\Gamma((1+\tau)\kappa)} |a_{\kappa}| \le b(1-\lambda).$$ The outcome is sharp for the function specified by: <span id="page-7-1"></span> $$f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \frac{(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)\,(\mu\kappa^2 - \mu\kappa + \kappa - 1 - \lambda b + b)} \zeta^{\kappa}.$$ (2.15)

Corollary 2.13. [16] By setting the parameters $\tau = 0$ , b = 1 and $\delta = 1$ in (2.15), the outcome obtained a corollary kindred to the results. Replace the parameter $\lambda = \delta, \mu = \tau$ the outcome obtained a corollary kindred to the result, $$\sum_{\kappa=2}^{\infty} (\kappa - 1)(\tau \kappa + 1) + (1 - \delta)|a_{\kappa}| \le 1 - \delta.$$

Theorem 3.1. Let the functions $f_1$ and $f_{\kappa}$ be defined by $f_1(\zeta) = \zeta$ and $$f_{\kappa}(\zeta) = \zeta + \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)\left((\kappa-\lambda)(\varrho(\kappa-1)+1)\right)}\zeta^{\kappa}, \quad \text{for } \kappa \ge 2.$$ A function $f(\zeta)$ is in the subclass $\mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ if and only if it can be expressed in the form $$f(\zeta) = \sum_{\kappa=1}^{\infty} \eta_{\kappa} f_{\kappa}(\zeta),$$  Figure 6: Mapping the complex plane with the Rabotnov function where $\eta_{\kappa} \geq 0$ and $\sum_{\kappa=1}^{\infty} \eta_{\kappa} = 1$ .

Theorem 3.2. Let $$f_1(\zeta) = \zeta, \quad f_{\kappa}(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \eta_{\kappa} \frac{(1-\lambda)b\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]} \zeta^{\kappa}, \quad \kappa \ge 2.$$ Then $f \in \mathcal{R}^{\varrho,b,\lambda}_{\tau,\delta}$ strictly in the event that $$f(\zeta) = \sum_{\kappa=1}^{\infty} \eta_{\kappa} f_{\kappa}(\zeta),$$ where $\eta_{\kappa} > 0$ and $\sum_{\kappa=1}^{\infty} \eta_{\kappa} = 1$ .

Theorem 3.3. Let $$f_1(\zeta) = \zeta, \quad f_{\kappa}(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \eta_{\kappa} \frac{(1-\lambda)b\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]} \zeta^{\kappa}, \quad \kappa \ge 2.$$ Then $f \in \mathcal{R}^{\mu,\lambda}_{\tau,\delta}$ strictly in the event that $$f(\zeta) = \sum_{\kappa=1}^{\infty} \eta_{\kappa} f_{\kappa}(\zeta),$$ where $\eta_{\kappa} > 0$ and $\sum_{\kappa=1}^{\infty} \eta_{\kappa} = 1$ .

Theorem 3.4. Let $$f_1(\zeta) = \zeta, \quad f_{\kappa}(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} \eta_{\kappa} \frac{(1-\lambda)b\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)(\mu\kappa^2 - \mu\kappa + \kappa - 1 - \lambda b + b)} \zeta^{\kappa}, \quad \kappa \ge 2.$$ Then $f \in \mathcal{R}^{\mu,b,\lambda}_{\tau,\delta}$ strictly in the event that $$f(\zeta) = \sum_{\kappa=1}^{\infty} \eta_{\kappa} f_{\kappa}(\zeta),$$ where $\eta_{\kappa} > 0$ and $\sum_{\kappa=1}^{\infty} \eta_{\kappa} = 1$ .

Theorem 4.1. Let consider the function f be in the defined subclass $\mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ . Then f is convex of order $\gamma$ , where $0 \leq \gamma < 1$ , in the disk $|\zeta| < R_1$ , where $$R_1 := \inf_{\kappa \ge 2} \left( \frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)\left((\kappa-\lambda)(\varrho(\kappa-1)+1)\right)}{\kappa(\kappa-\gamma)(1-\lambda)\Gamma((1+\tau)\kappa)} \right)^{\frac{1}{\kappa-1}}.$$ The result is sharp.

Theorem 4.2. Let $f \in \mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ . Then the function f is convex of order $\gamma$ in $|\zeta| < R_{1a}$ , where $$R_{1a} := \inf\left(\frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]}{\kappa(\kappa-\gamma)(1-\lambda)b\Gamma((1+\tau)\kappa)}\right)^{\frac{1}{\kappa-1}}, (\kappa \ge 2), \quad (4.1)$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , $\zeta \in U$ .

Theorem 4.3. Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,\lambda}$ . Then the function f is convex of order $\gamma$ in $|\zeta| < R_{1b}$ , where $$R_{1b} := \inf\left(\frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]}{\kappa(\kappa-\gamma)(1-\lambda)\Gamma((1+\tau)\kappa)}\right)^{\frac{1}{\kappa-1}}, (\kappa \ge 2), \tag{4.2}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , $0 \leq \lambda < 1$ , $\zeta \in U$ .

Theorem 4.4. Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,b,\lambda}$ . Then the function f is convex of order $\gamma$ in $|\zeta| < R_{1c}$ , where $$R_{1c} := inf \left( \frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)(\mu\kappa^2 - \mu\kappa + \kappa - 1 - \lambda b + b)}{\kappa(\kappa - \gamma)(1-\lambda)b\Gamma((1+\tau)\kappa)} \right)^{\frac{1}{\kappa-1}}, (\kappa \ge 2), \quad (4.3)$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , $\zeta \in U$ .

Theorem 4.5. Let the function f be in the defined subclass $\mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ . Then f is starlike of order $\gamma$ , where $0 \leq \gamma < 1$ , in the disk $|\zeta| < R_2$ , where $$R_2 := \inf_{\kappa \ge 2} \left( \frac{(1 - \gamma)\delta^{\kappa - 1}\Gamma(1 + \tau) \left( (\kappa - \lambda)(\varrho(\kappa - 1) + 1) \right)}{(\kappa - \gamma)(1 - \lambda)\Gamma((1 + \tau)\kappa)} \right)^{\frac{1}{\kappa - 1}}, (\kappa \ge 2).$$ The result is sharp.

Theorem 4.6. Let $f \in \mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ . Then f is starlike of order $\gamma$ in $|\zeta| < R_{2a}$ , where $$R_{2a} := \inf\left(\frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]}{(\kappa-\gamma)((1-\lambda)b\Gamma((1+\tau)\kappa)}\right)^{\frac{1}{\kappa-1}}, (\kappa \ge 2), \quad (4.4)$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , $\zeta \in U$ .

Theorem 4.7. Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,\lambda}$ . Then f is starlike of order $\gamma$ in $|\zeta| < R_{2b}$ , where $$R_{2b} := inf\left(\frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]}{(\kappa-\gamma)((1-\lambda)\Gamma((1+\tau)\kappa)}\right)^{\frac{1}{\kappa-1}}, (\kappa \ge 2), \tag{4.5}$$ where $\tau > 0$ , $\delta > 0$ , $0 < \mu < 1$ , $0 < \lambda < 1$ , $\zeta \in U$ .

Theorem 4.8. Let $f \in \mathcal{R}_{\tau,\delta}^{\mu,b,\lambda}$ . Then f is starlike of order $\gamma$ in $|\zeta| < R_{2c}$ , where $$R_{2c} := \inf\left(\frac{(1-\gamma)\delta^{\kappa-1}\Gamma(1+\tau)(\mu\kappa^2 - \mu\kappa + \kappa - 1 - \lambda b + b)}{(\kappa - \gamma)((1-\lambda)b\Gamma((1+\tau)\kappa)}\right)^{\frac{1}{\kappa-1}}, (\kappa \ge 2), \quad (4.6)$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , $\zeta \in U$ .

Theorem 5.1. Let $f(\zeta)$ belongs to subclass $\mathcal{R}^{\varrho,\lambda}_{\tau,\delta}$ . Then for $|\zeta| = r^*$ , we have $$r^ - \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]} (r^)^2 \le |f(\zeta)|$$ $$\le r^ + \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]} (r^)^2,$$ (5.1) where $\tau > 0$ , $\delta > 0$ , $0 < \rho < 1$ , $0 < \lambda < 1$ , $\zeta \in U$ . Proof. Since $$|a_{\kappa}| \leq \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]}$$ $$f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} a_{\kappa}\zeta^{\kappa}$$ $$|f(\zeta)| \leq r^ + \sum_{\kappa=2}^{\infty} a_{\kappa}(r^)^{\kappa}.$$ Since $$|f(\zeta)| \le r^ + \left(\sum_{\kappa=2}^{\infty} \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]}\right) (r^)^{\kappa}.$$ $$|f(\zeta)| \le r^ + \frac{(1-\lambda)\Gamma(1+\tau)\kappa}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]} (r^)^2.$$ Similarly $$|f(\zeta)| \ge r^ - \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}(\Gamma(1+\tau)[(\kappa-\lambda)(\rho(\kappa-1)+1)]} (r^)^2.$$

Theorem 5.2. Let the function $f(\zeta)$ belong to the subclass $\mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ . Then for $|\zeta| = r^*$ , the following sharp bounds hold: $$|f(\zeta)| \ge r^ - \frac{(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]} (r^)^2, |f(\zeta)| \le r^ + \frac{(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]} (r^)^2,$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ and $b \in \mathbb{C} - \{0\}$ .

Theorem 5.3. Let $f(\zeta)$ belongs to subclass $\mathcal{R}^{\mu,\lambda}_{\tau,\delta}$ . Then for $|\zeta| = r^*$ , we have $$r^ - \frac{(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]} (r^)^2 \le |f(\zeta)| \le r^ + \frac{(1-\lambda)\Gamma(1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)([\mu\kappa(\kappa-1)+\kappa-\lambda])} (r^)^2,$$ (5.2) where $\tau \ge 0$ , $\delta > 0$ , $0 \le \mu \le 1$ , $0 \le \lambda < 1$ , $\zeta \in U$ .

Theorem 5.4. Let the function $f(\zeta)$ belong to the subclass $\mathcal{R}^{\mu,b,\lambda}_{\tau,\delta}$ . Then for $|\zeta| = r$ , the following sharp bounds hold: $$|f(\zeta)| \ge r - \frac{(1-\lambda)b\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)\left[(\kappa-1)(\mu\kappa+1) + b(1-\lambda)\right]} (r^*)^2,$$ $$|f(\zeta)| \le r + \frac{(1-\lambda)b\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)\left[(\kappa-1)(\mu\kappa+1) + b(1-\lambda)\right]} (r^*)^2,$$ where $\tau \ge 0$ , $\delta > 0$ , $0 \le \mu \le 1$ , $0 \le \lambda < 1$ , and $b \in \mathbb{C} - \{0\}$ .

Theorem 5.5. (Distortion Theorem). Let $f(\zeta)$ belong to subclass $\mathcal{R}^{\varrho,\lambda}_{\tau,\delta}$ . Then for $|\zeta| = r^*$ , we have $$1 - \frac{2(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]}r^ \le |f'(\zeta)| \le 1 + \frac{2(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]}r^,$$ (5.3) where $\tau \ge 0, \delta > 0, \ 0 \le \varrho \le 1, \ 0 \le \lambda < 1, \ \zeta \in U$ . Proof. Since $$a_{\kappa} \leq \frac{1 - \lambda \Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]}$$ $$f(\zeta) = \zeta + \sum_{\kappa=2}^{\infty} a_{\kappa} \zeta^{\kappa}$$ $$|f'(\zeta)| \leq 1 + \sum_{\kappa=2}^{\infty} \kappa a_{\kappa} |\zeta|^{\kappa-1}.$$ $$|f'(\zeta)| \leq 1 + \frac{2(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]} r^{*}.$$ Similarly $$|f'(\zeta)| \ge 1 - \frac{2(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[(\kappa-\lambda)(\varrho(\kappa-1)+1)]} r^*.$$

Theorem 5.6. Let the function $f(\zeta)$ belong to the subclass $\mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ . Then for $|\zeta| = r$ , the following sharp bounds hold: $$|f'(\zeta)| \ge 1 - \frac{2(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]}r^*,$$ $$|f'(\zeta)| \le 1 + \frac{2(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\kappa+b-\lambda b-1][1+\varrho\kappa-\varrho]}r^*,$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ , and $b \in \mathbb{C} - \{0\}$ .

Theorem 5.7. Let the function $f(\zeta)$ belong to the subclass $\mathcal{R}^{\mu,\lambda}_{\tau,\delta}$ . Then for $|\zeta| = r$ , the following sharp bounds hold: $$|f'(\zeta)| \ge 1 - \frac{2(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]}r^*,$$ $$|f'(\zeta)| \le 1 + \frac{2(1-\lambda)\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)[\mu\kappa(\kappa-1)+\kappa-\lambda]}r^*,$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , and $0 \leq \lambda < 1$ .

Theorem 5.8. Let the function $f(\zeta)$ belong to the subclass $\mathcal{R}_{\tau,\delta}^{\mu,b,\lambda}$ . Then for $|\zeta| = r$ , the following sharp bounds hold: $$|f'(\zeta)| \ge 1 - \frac{2(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)(\mu\kappa^2 - \mu\kappa + \kappa - 1 - \lambda b + b)}r^*,$$ $$|f'(\zeta)| \le 1 + \frac{2(1-\lambda)b\,\Gamma((1+\tau)\kappa)}{\delta^{\kappa-1}\Gamma(1+\tau)(\mu\kappa^2 - \mu\kappa + \kappa - 1 - \lambda b + b)}r^*,$$ where $\tau \ge 0$ , $\delta > 0$ , $0 \le \mu \le 1$ , $0 \le \lambda < 1$ , and $b \in \mathbb{C} - \{0\}$ .

Definition 1.1. A function $f \in \mathcal{A}$ of the type (1.1) is classified into the class $\mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ if it fulfills the subsequent requirement: $$\Re\left\{\frac{\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))' + \varrho\zeta^2(\mathcal{R}_{\tau,\delta}f(\zeta))''}{\varrho\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))' + (1 - \varrho)\mathcal{R}_{\tau,\delta}f(\zeta)}\right\} > \lambda,\tag{1.4}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ , and $\zeta \in U$ . Remark 1.2. For the special cases where $\varrho = 0$ , the class $\mathcal{R}^{\varrho,\lambda}_{\tau,\delta}$ reduces to the starlike functions of order $\lambda$ associated with the Rabotnov operator, denoted $\mathcal{S}^*(\lambda)[19]$ . When $\varrho = 1$ , it reduces to the convex functions of order $\lambda$ , denoted $\mathcal{K}(\lambda)$ .

Definition 1.3. A function $f \in \mathcal{A}$ of the type (1.1) is classified into the class $\mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ if it fulfills the subsequent requirement: $$\Re\left(1 + \frac{1}{b} \left[ \frac{\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))' + \varrho\zeta^2(\mathcal{R}_{\tau,\delta}f(\zeta))''}{\varrho\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))' + (1 - \varrho)\mathcal{R}_{\tau,\delta}f(\zeta)} - 1 \right] \right) > \lambda, \tag{1.5}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \varrho \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , and $\zeta \in U$ . Remark 1.4. [2] Setting b = 1 in (1.5) reduces the class $\mathcal{R}_{\tau,\delta}^{\varrho,b,\lambda}$ to the class $\mathcal{R}_{\tau,\delta}^{\varrho,\lambda}$ defined in Definition 1.1.

Definition 1.5. A function $f \in \mathcal{A}$ having the form (1.1) is said to belong to the class $\mathcal{R}_{\tau,\delta}^{\mu,\lambda}$ if the following condition holds: $$\Re\left(\frac{\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))' + \mu\zeta^2(\mathcal{R}_{\tau,\delta}f(\zeta))''}{\mathcal{R}_{\tau,\delta}f(\zeta)}\right) > \lambda,\tag{1.6}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , $0 \leq \lambda < 1$ , and $\zeta \in U$ . Remark 1.6. [2, 26] For $\mu = 0$ , the condition reduces to $\Re(\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))'/(\mathcal{R}_{\tau,\delta}f(\zeta))) > \lambda$ , which corresponds to starlikeness. For $\mu = 1$ , it relates to convexity conditions involving the operator.

Definition 1.7. A function $f \in \mathcal{A}$ having the form (1.1) is said to belong to the class $\mathcal{R}_{\tau,\delta}^{\mu,b,\lambda}$ if the following condition holds: <span id="page-3-0"></span> $$\Re\left(1 + \frac{1}{b} \left\lceil \frac{\zeta(\mathcal{R}_{\tau,\delta}f(\zeta))' + \mu\zeta^2(\mathcal{R}_{\tau,\delta}f(\zeta))''}{\mathcal{R}_{\tau,\delta}f(\zeta)} - 1 \right\rceil \right) > \lambda, \tag{1.7}$$ where $\tau \geq 0$ , $\delta > 0$ , $0 \leq \mu \leq 1$ , $0 \leq \lambda < 1$ , $b \in \mathbb{C} - \{0\}$ , and $\zeta \in U$ . Motivated by recent developments in special functions [28, 12, 13], this study investigates sufficient conditions for the univalence of these subclasses. The primary contributions include establishing criteria for starlikeness and convexity, deriving sharp coefficient inequalities, and applying these findings to signal processing contexts.