Third-Order Fuzzy Differential Subordination and Superordination via Generalized Mittag-Leffler Operator with Applications in Decision Making

Abstract

This article focuses on the notions of third-order fuzzy differential subordination and superordination associated with the generalized Mittag-Leffler operator. Methods emphasizing the key concept of admissible functions are implemented to investigate several third-order fuzzy differential subordination and superordination results. Sandwich-type outcomes are established based on the adopted methodology, linking the dual fuzzy theoretical frameworks. In addition, the applications of fuzzy differential subordination are discussed in the context of decision making problems. The proposed approach provides the mathematical mechanism that ensures the stability and preservation of the decision under changes in criteria and preference evaluations, highlighting the importance of the developed theory.

1. Introduction

The idea of fuzzy set theory was introduced by mathematician Lotfi A. Zadeh in 1965 [1]. It has significant applications in various scientific and technological fields and has proven to be a powerful mathematical framework for modeling systems involving uncertainty and imprecision that cannot be sufficiently modeled using classical set theory. Several studies have implemented this theory for modeling imprecise information and ranking of alternatives, as seen in [2,3].

Within the domain of geometric function theory, the technique of differential subordination has proved to be an effective tool for obtaining coefficient estimates, several new classes, and inclusion relationships for analytic functions, as discussed in [4,5,6,7,8,9]. This concept, proposed by S. S. Miller and P. T. Mocanu [10], has motivated various authors to develop generalizations in different directions.

An important extension of this concept to the fuzzy framework was introduced by Oros et al. in 2011 [11], namely fuzzy differential subordination, with the foundation of this notion being established in 2012 [12]. The investigation of this theory continues to provide interesting results in recent publications, where various operators are employed to obtain new findings, as seen in [13,14,15,16,17].

The dual idea of this theory was proposed by Atshan and Hussain as fuzzy differential superordination [18]. The emergence of this concept has significantly extended the analytical framework through the introduction of lower bounds and reverse inclusion relationships for fuzzy-valued analytic functions.

Following these contributions, several authors extended fuzzy differential subordination and superordination to second- and third-order cases. Extensive studies have been carried out on third-order fuzzy differential subordination and superordination [19] through operators associated with special functions. The generalized Bessel functions [20,21], the Liu–Srivastava operator [22] and fractional operator [23], the Hohlov operator [24], generalized Hurwitz–Lerch zeta functions [25], and various other operators [26,27,28] have all been successfully applied.

Motivated by these contributions, this study employs the generalized Mittag-Leffler operator [29], introduced by Shukla and Prajapati in 2007 [30], to third-order fuzzy differential subordination and superordination. It generalizes the corresponding first- and second-order fuzzy differential subordination and superordination results of already existing operators. Therefore, the higher-order developments discussed in this article enhance the analytical depth of the theory and facilitate the handling of uncertainty inherent in the proposed framework of the generalized Mittag-Leffler operator and third-order fuzzy differential subordination and superordination in fuzzy geometric settings. Altogether, these applications establish the operator as a powerful and impactful tool for advancing fuzzy set theory.

In view of the above-mentioned applications, the present work aims to apply the generalized Mittag-Leffler operator within the dual fuzzy theoretical framework, with the objective of establishing new results that are significant from both theoretical and practical perspectives. While the admissibility framework employed in this study follows the classical approach of differential subordination theory, the present study introduces novelty in the third-order fuzzy differential subordination and superordination framework by incorporating the generalized Mittag-Leffler operator. Such an investigation has not been addressed in the literature. The obtained results extend existing studies by providing new relationships and sufficient conditions within this framework. Furthermore, existing studies have not adequately explored the use of fuzzy differential subordination in decision making problems. These approaches, as demonstrated in [31,32,33], are static in nature and assume fixed parameters, failing to account for dynamic interactions and evolving information in real-world decision making scenarios. To address this gap, the present work focuses on the novel applications of fuzzy differential subordination in the context of decision making by offering a robust framework for dealing with uncertainty under dynamic variations. The incorporation of the generalized Mittag-Leffler operator into this framework enhances its applicability by capturing memory effects and supporting structured rankings. Overall, this research contributes significantly to the advancement of geometric function theory and opens new directions for applying fuzzy differential subordination to complex decision making problems.

We now present the following definitions that are fundamental to our investigation. The class of analytic functions on D is represented by $\mathcal{H}\left(D\right),$ where D is the open unit disk, $D=\{z:z\in \mathbb{C}$ and $|z|<1\},$ with $\overline{D}=\{z\in \mathbb{C}:|z|\le 1\},$ and $\partial D=\{z\in D:|z|=1\}.$ Important subclasses of $\mathcal{H}\left(D\right)$ are mentioned below:

$$A_m=\{f\in \mathcal{H}\left(D\right):f\left(z\right)=z+b_{m+1}{z}^{m+1}+\dots ,z\in D\}.$$

where $b\in \mathbb{C}$ and $m\in \mathbb{N}$. $A_1=A$ is known as the class of normalized analytic functions and normalization conditions are $f\left(0\right)=0,f^{\prime }\left(0\right)=1.$ The series representation of $f\left(z\right)\in A$ is given by $f\left(z\right)=z+{\sum }_{m=2}^{\infty }{b}_m{z}^m$, $(z\in D)$. We now define:

$$\mathcal{H}[b,m]=\{f\in \mathcal{H}\left(D\right):f\left(z\right)=b+b_m{z}^m+b_{m+1}{z}^{m+1}+\dots ,z\in D\},$$

where ${\mathcal{H}}_0=\mathcal{H}[0,1]$ and ${\mathcal{H}}_1=\mathcal{H}[1,1]$. The subclass S of A contains all normalized analytic univalent functions. The theory of univalent functions was systematically developed by Paul Koebe in 1907 [34].

Let $f\left(z\right)$, $g\left(z\right)$$\in \mathcal{H}\left(D\right)$. Then $f\left(z\right)$ is subordinate to $g\left(z\right),$ represented by $f\left(z\right)\prec g\left(z\right),$ if there exists a function $q\left(z\right)\in \mathcal{H}\left(D\right)$, named as a Schwarz function, such that $q\left(0\right)=0$ and $|q(z)|<1$, $\forall z\in D,$ satisfying $f\left(z\right)=g\left(q\right(z\left)\right)$. If $g\in S$ such that $f\left(z\right)\prec g\left(z\right)$, then the following conditions are satisfied:

$$f\left(0\right)=g\left(0\right) \mathrm{and} f\left(D\right)\subset g\left(D\right).$$

Convolution is an essential technique in geometric function theory that combines two analytic functions to produce a new analytic function while preserving geometric properties. Let $f\left(z\right)$ and $g\left(z\right)$$\in \mathcal{H}\left(D\right)$, where $f\left(z\right)={\sum }_{m=2}^{\infty }{b}_m{z}^m$ and $g\left(z\right)={\sum }_{m=2}^{\infty }{c}_m{z}^m$, with $b,c\in \mathbb{C}$ and $z\in D$. The convolution, or Hadamard product, of $f\left(z\right)$ and $g\left(z\right)$, denoted by $f\ast g$, is defined as

$$f\left(z\right)\ast g\left(z\right)=z+{\sum }_{m=2}^{\infty }{b}_m{c}_m{z}^m,\forall z\in D,$$

We now recall the Mittag-Leffler function, denoted by $L_{\alpha ,\beta }^{\eta ,\lambda }\left(z\right)$, defined by

$$L_{\alpha ,\beta }^{\eta ,\lambda }\left(z\right)=z+\sum _{m=2}^{\infty }{d}_m{z}^m,d_m={\displaystyle \frac{\Gamma (\eta +m\lambda )\Gamma (\alpha +\beta )}{\Gamma (\eta +\lambda )\Gamma (\beta +m\alpha )m!}},$$

(1)

with d ∈ $\mathbb{C}$ and $m\in \mathbb{N}$. The parameters $\beta ,\eta \in \mathbb{C},$$\Re \left(\alpha \right)$ > $max\{0, \Re (\lambda )-1\}$ and $\Re \left(\lambda \right)>0$. The convolution of a generalized Mittag-Leffler function with $g\left(z\right)$ ∈ $\mathcal{H}\left(D\right)$ yields the generalized Mittag-Leffler operator $L_{\alpha ,\beta }^{\eta ,\lambda }:A\to A$, given by

$$L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)=(L_{\alpha ,\beta }^{\eta ,\lambda }\ast g)\left(z\right)=z+\sum _{m=2}^{\infty }{d}_m{z}^m\ast z+\sum _{m=2}^{\infty }{c}_m{z}^m$$

$$L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)=z+\sum _{m=2}^{\infty }{\displaystyle \frac{\Gamma (\eta +m\lambda )\Gamma (\alpha +\beta )}{\Gamma (\eta +\lambda )\Gamma (\beta +m\alpha )m!}}{c}_m{z}^m,\forall z\in A.$$

(2)

The series of $g\left(z\right)$ is analytic and hence convergent in D; moreover, the Mittag-Leffler function series converges due to the boundedness of its parameters. Therefore, the resulting convolution series is well-defined and convergent.

The associated operator framework plays an important role in defining new subclasses of analytic and univalent functions. Applying this operator to normalized analytic functions helps preserve various geometric properties such as univalence, starlikeness, convexity, and close-to-convexity. Numerous papers have used different operators associated with special functions. Several existing well-known operators, including the Lommel-type [19], hypergeometric [17], and Mittag-Leffler operators with seven parameters [8], have previously been applied in fuzzy subordination theory. Compared with these, the generalized Mittag-Leffler operator with its parameterized nature allows for fine control over geometric properties through a suitable choice of the parameters $(\alpha ,\beta ,\eta ,\lambda )$, resulting in sharp coefficient bounds, distortion inequalities, and growth estimates.

The operator $L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)$ satisfies [35]

$$z{(L_{\alpha ,\beta }^{\eta +1,\lambda }g(z))}^{\prime }=({\displaystyle \frac{\eta +\lambda }{\lambda }})(L_{\alpha ,\beta }^{\eta +1,\lambda }g(z))-({\displaystyle \frac{\eta }{\lambda }})(L_{\alpha ,\beta }^{\eta ,\lambda }g(z))$$

(3)

and

$$\alpha z{(L_{\alpha ,\beta +1}^{\eta ,\lambda }g(z))}^{\prime }=(\alpha +\beta )(L_{\alpha ,\beta }^{\eta ,\lambda }g(z))-\beta (L_{\alpha ,\beta +1}^{\eta ,\lambda }g(z)).$$

(4)

The following definitions support the subsequent results:

Definition 1  

([36]). Let $\mathcal{Q}$ refer to the set of functions h that are analytic and univalent on $\overline{D}\setminus E\left(h\right)$, where

$$E\left(h\right)=\{\xi \in \partial D:{lim}_{z\to \xi }h\left(z\right)=\infty \},$$

with $h^{\prime }\left(\xi \right)\ne 0$ for ξ ∈ $\partial D\setminus E\left(h\right)$, and $E\left(h\right)$ is an exception set. The subclass of $\mathcal{Q}$ consists of functions that satisfy $h\left(0\right)$ = b and are represented by $\mathcal{Q}\left(b\right)$. In particular, ${\mathcal{Q}}_0=\mathcal{Q}\left(0\right)$ and ${\mathcal{Q}}_1=\mathcal{Q}\left(1\right)$.

We now define the following concepts from the theory of fuzzy differential subordination:

Definition 2  

([1]). A fuzzy set W is the collection of the ordered pairs $(y,M)$, where the first element is $y\in Y$$(Y\ne \phi )$ and the second element $M:y\to [0,1]$ is called a membership function. When M = 0, the element y has the smallest membership degree, and when M = 1, it has the largest membership degree.

Definition 3  

([11]). A function $f\left(z\right)$ is said to be fuzzy subordinate to $g\left(z\right)$, denoted by $f\left(z\right){\prec }_{F}g\left(z\right)$, when there exists a function $M:\mathbb{C}\to [0,1]$, such that

$$f\left(z_0\right)=g\left(z_0\right), z_0\in D,$$

$$M\left(f\right(z\left)\right)\le M\left(g\right(z\left)\right), \forall z\in D.$$

Definition 4  

([19]). Consider $f,g\in \mathcal{H}\left(D\right)$. Then

$$f\left(D\right)=sup(f\left(D\right),M_{f\left(D\right)})=\{f\left(z\right):0<|M_{f\left(D\right)}\left(f\left(z\right)\right)|\le 1,z\in D\},$$

(5)

and

$$g\left(D\right)=sup(g\left(D\right),M_{f\left(D\right)})=\{g\left(z\right):0<|M_{g\left(D\right)}\left(g\left(z\right)\right)|\le 1,z\in D\}.$$

(6)

Definition 5  

([36]). Let $\mathcal{U}\subset \mathbb{C}$ and $h\in \mathcal{Q}$. The class of admissible functions ${\mathrm{\Psi }}_m[\mathcal{U},h]$ consists of those $\psi :{\mathbb{C}}^4\times \overline{D}\to \mathbb{C}$ that satisfy the admissibility condition

$$\psi (t,u,v,w:\xi )\notin \mathcal{U},$$

whenever

$$t=h\left(z\right),u={\displaystyle \frac{z{h}^{\prime }\left(z\right)}{\tau }},$$

$$\Re \left\{{\displaystyle \frac{v}{u}}+1\right\}\ge \tau \Re \left\{1+{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right\},$$

$$\Re \left\{{\displaystyle \frac{w}{u}}\right\}\ge {\tau }^2\Re \left\{{\displaystyle \frac{z^2{h}^{‴}\left(z\right)}{h^{\prime }\left(z\right)}}\right\},$$

where $z\in D,$$\xi \in \partial D\setminus E\left(h\right)$ and $\tau \in \mathbb{N}\setminus \left\{1\right\}.$

Definition 6  

([37]). Let $\mathcal{U}\subset \mathbb{C}$ and $h\in \mathcal{Q}$. The class of admissible functions ${\mathrm{\Psi }}_m[\mathcal{U},h]$ consists of those $\psi :{\mathbb{C}}^4\times \overline{D}\to C$ that satisfy the admissibility condition

$$M_{\psi ({\mathbb{C}}^4\times \overline{D})}\psi (t,u,v,w;\xi )<M_{\mathcal{U}}\left(z\right),0<M_{\mathcal{U}}\left(z\right)\le 1,$$

(7)

whenever

$$t=h\left(z\right),u={\displaystyle \frac{z{h}^{\prime }\left(z\right)}{\tau }},$$

$$\Re \left\{{\displaystyle \frac{v}{u}}+1\right\}\ge \tau \Re \left\{1+{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right\},$$

$$\Re \left\{{\displaystyle \frac{w}{u}}\right\}\ge {\tau }^2\Re \left\{{\displaystyle \frac{z^2{h}^{‴}\left(z\right)}{h^{\prime }\left(z\right)}}\right\},$$

where $z\in D$, $\xi \in \partial D\setminus E\left(h\right)$ and $\tau \in \mathbb{N}\setminus \left\{1\right\}.$

Let the class of admissible functions ${\mathrm{\Psi }}_m[\mathcal{U},h]$ consist of $\psi :{\mathbb{C}}^4\times \overline{D}\to C$. Then, the third-order fuzzy differential subordination [19] with respect to a function $h\in \mathcal{H}\left(D\right)$ satisfies $p{\prec }_{F}h$ whenever p holds:

$$M_{\psi ({\mathbb{C}}^4\times \overline{D})}\psi (p\left(z\right),z{p}^{\prime }\left(z\right),z^2{p}^{″}\left(z\right),z^3{p}^{‴}\left(z\right);z)\le M_{\omega \left(D\right)}\omega \left(z\right)$$

i.e.,

$$\psi (p\left(z\right),z{p}^{\prime }\left(z\right),z^2{p}^{″}\left(z\right),z^3{p}^{‴}\left(z\right);z){\prec }_F\omega \left(z\right),z\in D.$$

(8)

where $\omega \in \mathcal{H}\left(D\right)$ and p is called the solution of the fuzzy third-order differential subordination (8). A univalent function $\tilde{h}$ is said to be the fuzzy best dominant function for all fuzzy dominants h associated with (8) if it satisfies $\tilde{h}{\prec }_{F}h$, and $\tilde{h}$ is unique.

In the context of the fuzzy third-order differential superordination [37], let $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h]$ and $h\in \mathcal{H}\left(D\right)$ such that $h{\prec }_{F}p$ with $p\in \mathcal{H}\left(D\right)$ whenever p satisfies

$$M_{\omega \left(D\right)}\omega \left(z\right)\le M_{\psi ({\mathbb{C}}^4\times \overline{D})}\psi (p\left(z\right),z{p}^{\prime }\left(z\right),z^2{p}^{″}\left(z\right),z^3{p}^{‴}\left(z\right);z), z\in D$$

i.e.,

$$\omega \left(z\right){\prec }_F\psi (p\left(z\right),z{p}^{\prime }\left(z\right),z^2{p}^{″}\left(z\right),z^3{p}^{‴}\left(z\right);z),$$

(9)

where $\omega$ ∈ $\mathcal{H}\left(D\right)$ and p represents a solution of third-order fuzzy differential superordination (9). $\tilde{h}$ is called the best fuzzy subordinant for all fuzzy subordinants h of (9) if satisfies $h\left(z\right){\prec }_F\tilde{h}\left(z\right).$

2. A Set of Lemmas

This section presents two important lemmas that are crucial for obtaining new outcomes in the subsequent sections, proven in [38] and [37], respectively.

Lemma 1  

([38] (Theorem 3.4)). Let the membership function $M:\mathbb{C}\to [0,1]$, $\varphi \in \mathcal{H}[b,m]$, and $h\in \mathcal{Q}\left(b\right)$ satisfy

$$\Re \left\{{\displaystyle \frac{\xi h^{″}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|{\displaystyle \frac{z{\varphi }^{\prime }\left(z\right)}{h^{\prime }\left(\xi \right)}}\right|\le \tau ,$$

where $z\in D$, $\xi \in \partial D\setminus E\left(h\right)$, and $\tau \in \mathbb{N}\setminus \left\{1\right\}.$ Consider the set $\mathcal{U}\subset \mathbb{C}$ and assume that $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h]$ is an admissible function such that

$$\left\{\psi (\varphi \left(z\right),z{\varphi }^{\prime }\left(z\right),z^2{\varphi }^{″}\left(z\right),z^3{\varphi }^{‴}\left(z\right);z)\right\}\subset \mathcal{U},$$

then, $\varphi \left(z\right)$ is said to be fuzzy subordinate to $h\left(z\right)$, given by

$$\varphi \left(z\right){\prec }_{F}h\left(z\right) or M_{\varphi \left(D\right)}\varphi \left(z\right)\le M_{h\left(D\right)}h\left(z\right).$$

Lemma 2  

([37] (Theorem 1)). Let $\mathcal{U}\in \mathbb{C}$, $h\in \mathcal{H}[b,m]$. Consider $\varphi \in \mathcal{Q}\left(b\right)$, $\varphi \left(0\right)$ = $h\left(0\right)$ = $b,$$\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h]$, $M:\mathbb{C}\to [0,1]$ represented by $M\left(z\right)={\displaystyle \frac{1}{1+|z|}}$, satisfying

$$\Re \left\{{\displaystyle \frac{\xi h^{″}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|{\displaystyle \frac{z{\varphi }^{\prime }\left(z\right)}{h^{\prime }\left(\xi \right)}}\right|\le \tau ,$$

with $z\in D$, $\xi \in \partial D\setminus E\left(h\right)$, and $\tau \in \mathbb{N}\setminus \left\{1\right\}.$

If $\varphi \left(z\right)$ and $\psi (\varphi \left(z\right),z{\varphi }^{\prime }\left(z\right),z^2{\varphi }^{″}\left(z\right),z^3{\varphi }^{‴}\left(z\right);z)\subset \mathcal{U}$ are univalent in D, then

$$\mathcal{U}\subset \{\psi (\varphi \left(z\right),z{\varphi }^{\prime }\left(z\right),z^2{\varphi }^{″}\left(z\right),z^3{\varphi }^{‴}\left(z\right);z):z\in D\},$$

that is,

$${\displaystyle \frac{1}{1+|z|}}\le {\displaystyle \frac{1}{1+|\psi (\varphi \left(z\right),z{\varphi }^{\prime }\left(z\right),z^2{\varphi }^{″}\left(z\right),z^3{\varphi }^{‴}\left(z\right);z)|}},$$

Consequently,

$$h\left(z\right){\prec }_F\varphi \left(z\right)or{\displaystyle \frac{1}{1+|h(z)|}}\le {\displaystyle \frac{1}{1+|\varphi (z)|}}(z\in D).$$

3. Main Results

Third-Order Fuzzy Differential Subordination Results

Certain new third-order fuzzy differential subordinations are derived in this section associated with $L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)$ by defining suitable classes of admissible functions. Throughout the study, unless stated otherwise, we suppose that $\eta ,\beta >2$ and $\alpha ,\lambda >0.$ These restrictions are imposed to ensure that the derived expressions remain well-defined. In particular, the condition $\eta >2$ prevents the denominators appearing in the proofs from vanishing.

Definition 7.  

Let $\mathcal{U}\subset \mathbb{C}$ and $h\in {\mathcal{Q}}_0\cap {\mathcal{H}}_0$. The class of admissible functions ${\mathrm{\Psi }}_m[\mathcal{U},h]$ consists of functions $\psi :{\mathbb{C}}^4\times D\to \mathbb{C}$, satisfying

$$\psi (\theta ,\kappa ,\rho ,\sigma ;z)\notin \mathcal{U},$$

(10)

whenever

$$\theta =h\left(\xi \right),\kappa ={\displaystyle \frac{-\lambda \tau \xi h^{\prime }\left(\xi \right)+(\eta +\lambda )h\left(\xi \right)}{\eta }},$$

(11)

$$\Re \{{\displaystyle \frac{\rho \eta (\eta -1)-\theta (\eta +\lambda )(\eta +\lambda -1)}{\lambda \left(\right(\eta +\lambda )\theta -\eta \kappa )}}+{\displaystyle \frac{2\eta +2\lambda -1}{\lambda }}\}\ge \tau \Re \left\{{\displaystyle \frac{\xi h^{″}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}+1\right\},$$

(12)

and

$$\begin{array}{cc}\hfill & \Re \{{\displaystyle \frac{-\sigma \eta (\eta -1)(\eta -2)+\theta (\eta +\lambda )(\eta +\lambda -1)(\lambda -2\eta +1)+3\rho \eta {(\eta -1)}^2}{{\lambda }^2\left((\eta +\lambda )\theta -\eta \kappa \right)}}\hfill \\ \hfill & +{\displaystyle \frac{3{\eta }^2-3\eta -{\lambda }^2+1}{{\lambda }^2}}\}\ge {\tau }^2\Re \left\{{\displaystyle \frac{{\xi }^2{h}^{‴}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}\right\}.\hfill \end{array}$$

(13)

where $z\in D$, $\xi \in \partial D\setminus E\left(h\right)$, and $\tau \in \mathbb{N}\setminus \left\{1\right\}$.

The above definition illustrates that $\psi \notin \mathcal{U}$ preserves the fuzzy subordination condition. When the input values $\theta ,\kappa ,\rho ,\sigma$ satisfy conditions (11)–(13), the output of $\psi$ must stay outside $\mathcal{U}$; otherwise, the fuzzy subordination condition would be violated. Further discussion is provided in Example 1.

Theorem 1.  

Suppose $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h].$ If $g\in A$ and $h\in {\mathcal{Q}}_0\cap {\mathcal{H}}_0$ satisfy

$$\Re \left\{{\displaystyle \frac{\xi h^{″}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}\right\}\ge 0,|{\displaystyle \frac{\eta +\lambda }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)|\le \tau |h^{\prime }\left(\xi \right)|,$$

(14)

and

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\le M_{\mathcal{U}\left(z\right)},$$

$$\tau \in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial D\setminus E\left(h\right), and z\in D.$$

Then

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_{F}h\left(z\right)(z\in D),$$

(15)

or

$$M_{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(D\right)}\left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\right)\le M_{h\left(D\right)}h\left(z\right).$$

Proof. 

Define

$$f\left(z\right)=L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)$$

(16)

Using (3) and (16), we get

$$L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)={\displaystyle \frac{(\eta +\lambda )f\left(z\right)-\lambda z{f}^{\prime }\left(z\right)}{\eta }}$$

(17)

Taking the derivative on both sides of (17) and then using (3) with η replaced by $\eta -1$, we get

$$\left({\displaystyle \frac{\eta -1+\lambda }{\lambda }}\right)L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)-\left({\displaystyle \frac{\eta -1}{\lambda }}\right)L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)={\displaystyle \frac{\eta z{f}^{\prime }\left(z\right)-\lambda z^2{f}^{″}\left(z\right)}{\eta }}$$

Simplifying for $L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)$, we obtain

$$L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)={\displaystyle \frac{{\lambda }^2{z}^2{f}^{″}\left(z\right)-\lambda (2\eta +\lambda -1)z{f}^{\prime }\left(z\right)+(\lambda +\eta )(\lambda +\eta -1)f\left(z\right)}{\eta (\eta -1)}}$$

(18)

Similarly, differentiation on both sides of (18) and utilizing (3) with η replaced by $\eta -2$ yields

$$\begin{array}{cc}\hfill L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)& ={\displaystyle \frac{-{\lambda }^3{z}^3{f}^{‴}\left(z\right)+3{\lambda }^2(\eta -1)z^2{f}^{″}\left(z\right)}{\eta (\eta -1)(\eta -2)}}\hfill \\ \hfill & +{\displaystyle \frac{\lambda \left(-{\lambda }^2+3\lambda -3{\eta }^2+6\eta -3\lambda \eta -2\right)z{f}^{\prime }\left(z\right)+(\eta +\lambda )(\eta +\lambda -1)(\eta +\lambda -2)f\left(z\right)}{\eta (\eta -1)(\eta -2)}}.\hfill \end{array}$$

(19)

Now, we express the transformation of ${\mathbb{C}}^4$ to $\mathbb{C}$ by

$$\theta (t,u,v,w)=t,\kappa (t,u,v,w)={\displaystyle \frac{(\eta +\lambda )t-\lambda u}{\eta }},$$

(20)

$$\rho (t,u,v,w)={\displaystyle \frac{{\lambda }^{2}v-u\lambda (2\eta +\lambda -1)+t(\lambda +\eta )(\lambda +\eta -1)}{\eta (\eta -1)}},$$

(21)

and

$$\begin{array}{cc}\hfill \sigma (t,u,v,w)& ={\displaystyle \frac{-{\lambda }^{3}w+3{\lambda }^2(\eta -1)v}{\eta (\eta -1)(\eta -2)}}\hfill \\ \hfill & +{\displaystyle \frac{\lambda \left(-{\lambda }^2+3\lambda -3{\eta }^2+6\eta -3\lambda \eta -2\right)u+(\eta +\lambda )(\eta +\lambda -1)(\eta +\lambda -2)t}{\eta (\eta -1)(\eta -2)}}.\hfill \end{array}$$

(22)

The transformation from $(t,u,v,w)$ to $(\theta ,\kappa ,\rho ,\sigma )$ defined in (20)–(22) is vital to apply Lemma 1 and acts as a link between the generalized Mittag-Leffler operator and fuzzy third-order differential subordination. The conditions (16)–(19) represent different forms of the generalized Mittag-Leffler operator $L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)$, $L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)$, $L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)$ in terms of $f\left(z\right)=L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)$ and its derivatives $z{f}^{\prime }\left(z\right),z^2{f}^{″}\left(z\right),$ and $z^3{f}^{‴}\left(z\right).$ Thus, we have

$$\begin{array}{cc}\hfill & \hfill \psi (t,u,v,w;z)=\psi (\theta ,\kappa ,\rho ,\sigma ;z)\hfill \\ \hfill & =\psi (t,{\displaystyle \frac{(\eta +\lambda )t-\lambda u}{\eta }},{\displaystyle \frac{{\lambda }^{2}v-\lambda u(2\eta +\lambda -1)+t(\lambda +\eta )(\lambda +\eta -1)}{\eta (\eta -1)}},\hfill \\ \hfill & {\displaystyle \frac{-{\lambda }^{3}w+3{\lambda }^2(\eta -1)v+\lambda (-{\lambda }^2+3\lambda -3{\eta }^2+6\eta -3\lambda \eta -2)u+(\eta +\lambda )(\eta +\lambda -1)(\eta +\lambda -2)t}{\eta (\eta -1)(\eta -2)}};z)\hfill \end{array}$$

(23)

By using conditions (16)–(23) and Lemma 1, we obtain

$$\psi (f\left(z\right),z{f}^{\prime }\left(z\right),z^2{f}^{″}\left(z\right),z^3{f}^{‴}\left(z\right);z)=\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z).$$

(24)

Hence, (15) leads to

$$M_{\psi (C^4\times D)}\left(\psi (f\left(z\right),z{f}^{\prime }\left(z\right),z^2{f}^{″}\left(z\right),z^3{f}^{‴}\left(z\right);z)\right)\le M_{\mathcal{U}}\left(z\right).$$

(25)

Additionally, applying (20)–(22), we have

$${\displaystyle \frac{v}{u}}+1={\displaystyle \frac{v+u}{u}}={\displaystyle \frac{\rho \eta (\eta -1)+u\lambda (2\eta +\lambda -1)-\theta (\lambda +\eta )(\lambda +\eta -1)+u{\lambda }^2}{u{\lambda }^2}},$$

$${\displaystyle \frac{v}{u}}+1={\displaystyle \frac{\rho \eta (\eta -1)-\theta (\eta +\lambda )(\eta +\lambda -1)}{\lambda \left(\right(\eta +\lambda )\theta -\eta \kappa )}}+{\displaystyle \frac{2\eta +2\lambda -1}{\lambda }},$$

(26)

and

$${\displaystyle \frac{w}{u}}={\displaystyle \frac{-\sigma \eta (\eta -1)(\eta -2)+3\eta {(\eta -1)}^2\rho +u\lambda (3{\eta }^2-3\eta -{\lambda }^2+1)+\theta (\eta +\lambda )(\eta +\lambda -1)(\eta +\lambda -2)}{{\lambda }^2((\eta +\lambda )\theta -\eta \kappa )}},$$

After simplification, the expression becomes

$${\displaystyle \frac{w}{u}}={\displaystyle \frac{-\sigma \eta (\eta -1)(\eta -2)+\theta (\eta +\lambda )(\eta +\lambda -1)(\lambda -2\eta +1)+3\rho \eta {(\eta -1)}^2}{{\lambda }^2((\eta +\lambda )\theta -\eta \kappa )}}+{\displaystyle \frac{3{\eta }^2-3\eta -{\lambda }^2+1}{{\lambda }^2}}.$$

(27)

As a result, the $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h]$ admissibility condition in Definition 7 coincides with the $\psi \in {\mathrm{\Psi }}_2[\mathcal{U},h]$ admissibility condition in Definition 5. Hence, by using (14) and Lemma 1, we obtain $M_{f\left(D\right)}f\left(z\right)\le M_{h\left(D\right)}h\left(z\right)$ or, equivalently, $M_{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(D\right)}\left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\right)\le M_{h\left(D\right)}h\left(z\right),$ i.e.,

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_{F}h\left(z\right),$$

This completes the proof. □

Example 1.  

Consider ψ = $\kappa -\theta$ in Theorem 1. Now, we verify the admissibility condition from Definition 7. Substituting κ and θ from (11) into $\psi =\kappa -\theta$ gives

$$\psi =\kappa -\theta ={\displaystyle \frac{\lambda }{\eta }}(h\left(\xi \right)-\tau \xi h^{\prime }\left(\xi \right))$$

Since $h\in {\mathcal{Q}}_0\cap {\mathcal{H}}_0$ satisfies (14) and conditions (12) and (13) hold, Definition 7 implies $\psi \notin \mathcal{U}$. Therefore, $\psi \in [\mathcal{U},h]$ and the hypothesis of Theorem 1 is satisfied.

Now using (20), we obtain

$$\begin{array}{cc}\hfill \psi & ={\displaystyle \frac{(\eta +\lambda )t-\lambda u}{\eta }}-t\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\lambda (t-u)}{\eta }}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\lambda t+\eta L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)-\lambda t-\eta t}{\eta }}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & =L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)-L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right).\hfill \end{array}$$

(31)

which is analytic in D.

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi \left(L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)-L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\right)\le M_{h\left(D\right)}h\left(z\right),$$

Consequently, by Theorem 1, we get

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_{F}h\left(z\right)(z\in D).$$

(32)

or

$$M_{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(D\right)}\left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\right)\le M_{h\left(D\right)}h\left(z\right).$$

The following result is established when h behaves in an unknown manner on $\partial D.$

Corollary 1.  

Assume $\mathcal{U}\subset \mathbb{C}$ and consider $h\in S$ with $h\left(0\right)=1$. Let $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h_n]$ for fixed $n\in (0,1)$ when $h_n\left(z\right)=h\left(nz\right).$ If $g\in A$ and $h_n$ satisfy

$$\Re \left\{{\displaystyle \frac{\xi h_n^{″}\left(\xi \right)}{h_n^{\prime }\left(\xi \right)}}\right\}\ge 0,|{\displaystyle \frac{\eta +\lambda }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)|\le \tau |h_n^{\prime }\left(\xi \right)|,$$

(33)

$$(\xi \in \partial D\setminus E\left(h_n\right);z\in D),$$

and

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\le M_{\mathcal{U}\left(z\right)},$$

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_{F}h\left(z\right).$$

Proof. 

From Theorem 1, we get

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_F{h}_n\left(z\right).$$

Corollary 1 directly follows from

$$h_n\left(z\right){\prec }_{F}h\left(z\right) (z\in D).$$

□

The class ${\mathrm{\Psi }}_m[\mathcal{H}\left(D\right),h]$ is denoted as ${\mathrm{\Psi }}_m[\omega ,q]$ if $\mathcal{U}\ne \mathbb{C}$ is a simply connected domain and $\mathcal{U}$ = $\omega \left(D\right)$ for a suitable conformal mapping $\omega \left(D\right)$ of D onto $\mathcal{U}.$

The subsequent theorems are direct results of Theorem 1 and Corollary 1.

Theorem 2.  

Let $\psi \in {\mathrm{\Psi }}_m[\omega ,h].$ If $g\in A$ and $h\in {\mathcal{Q}}_0$ satisfy condition (14), then

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\le M_{\omega \left(\mathcal{U}\right)}\omega \left(z\right)$$

(34)

implies that

$$M_{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(D\right)}\left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\right)\le M_{\omega \left(\mathcal{U}\right)}\omega \left(z\right),$$

i.e.,

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_F\omega \left(z\right) (z\in D).$$

Corollary 2.  

Suppose $\mathcal{U}\subset \mathbb{C}$ and $h\in S$ with $h\left(0\right)=1.$ Consider $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},h_n]$ for certain $n\in (0,1)$ when $h_n\left(z\right)$ = $h\left(nz\right).$ If $g\in A$ and $h_n$ satisfy

$$\Re \left\{{\displaystyle \frac{\xi h_n^{″}\left(\xi \right)}{h_n^{\prime }\left(\xi \right)}}\right\}\ge 0,|{\displaystyle \frac{\eta +\lambda }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)|\le \tau |h_n^{\prime }\left(\xi \right)|,$$

(35)

$$(\xi \in \partial D\setminus E\left(h_n\right);z\in D),$$

and

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\le M_{\omega \left(\mathcal{U}\right)}\omega \left(z\right),$$

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_{F}h\left(z\right).$$

The next theorem provides the fuzzy best dominant of the fuzzy differential subordination (15) or (34).

Theorem 3.  

Consider $\omega \left(z\right)$ ∈ $\mathcal{H}\left(D\right)$. Consider $\psi :{\mathbb{C}}^4\times D\to \mathbb{C}$, where ψ is expressed by (23). Assume

$$\psi (h\left(z\right),z{h}^{\prime }\left(z\right),z^2{h}^{″}\left(z\right),z^3{h}^{‴}\left(z\right);z)=\omega \left(z\right)$$

(36)

has a solution $h\left(z\right)\in {\mathcal{Q}}_0\cap {\mathcal{H}}_0$, satisfying the conditions in (14). If $g\in A$ satisfies condition (33) and

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\le M_{\omega }\left(z\right)$$

is analytic in D, then

$$L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_{F}h\left(z\right),$$

and $h\left(z\right)$ is said to be the best fuzzy dominant.

Proof. 

By using Theorem 1, we identify that h is a dominant of (34). h satisfies (36), hence it is also a solution of (34). Thus, the best fuzzy dominant is h since all dominants dominate $h$. □

The class of ${\mathrm{\Psi }}_m[\mathcal{U},h]$ of admissible functions, represented by ${\mathrm{\Psi }}_m[\mathcal{U},R]$, is mentioned below in view of Definition 7 and when $h\left(z\right)=Rz$$(R>0).$

Definition 8.  

Assume that $\mathcal{U}\in \mathbb{C}$ and $R>0.$ The class ${\mathrm{\Psi }}_m[\mathcal{U},R]$ of admissible functions includes the functions $\psi :{\mathbb{C}}^4\times D\to \mathbb{C}$ such that

$$\psi (R{e}^{i\gamma },{\displaystyle \frac{(-\lambda \tau +\eta +\lambda )R{e}^{i\gamma }}{\eta }},{\displaystyle \frac{{\lambda }^{2}L+[-\tau \lambda (2\eta +\lambda -1)+(\eta +\lambda )(\eta +\lambda -1)]R{e}^{i\gamma }}{\eta (\eta -1)}},$$

$${\displaystyle \frac{-{\lambda }^{3}N+3{\lambda }^2(\eta -1)L+[\tau \lambda (-{\lambda }^2+3\lambda -3{\eta }^2+6\eta -3\eta \lambda -2)+(\eta +\lambda )(\eta +\lambda -1)(\eta +\lambda -2)]R{e}^{i\gamma }}{\eta (\eta -1)(\eta -2)}})\notin \mathcal{U}$$

whenever $z\in D,$ $\gamma \in \mathbb{R}$ and

$$\Re \left(L{e}^{-i\gamma }\right)\geqq (\tau -1)\tau R and \Re \left(N{e}^{-i\gamma }\right)\geqq 0 (\forall \gamma \in \mathbb{R};\tau \in \mathbb{N}\setminus \{1\left\}\right).$$

Corollary 3.  

Let $\psi \in {\mathrm{\Psi }}_m[\mathcal{U},R].$ If $g\in A,$ then

$$|{\displaystyle \frac{\eta +\lambda }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)|\le \tau R.$$

If

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\le M_{\mathcal{U}\left(z\right)},$$

$$\tau \in \mathbb{N}\setminus \left\{1\right\},z\in \partial D\setminus E\left(h\right), and z\in D,$$

then

$$|L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)|{\prec }_{F}R(z\in D),$$

or

$$M_{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(D\right)}|L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)|\le M_{h\left(D\right)}R,$$

where $\tau \ge 2,$ $R>0$, and $z\in D.$

Definition 9.  

Consider that $\mathcal{U}\in \mathbb{C}$ and $h\in {\mathcal{Q}}_1\cap {\mathcal{H}}_1.$ The class of admissible functions ${\tilde{\mathrm{\Psi }}}_m[\mathcal{U},h]$ comprises functions $\psi :{\mathbb{C}}^4\times D\to \mathbb{C}$, which satisfy

$$\psi (\theta ,\kappa ,\rho ,\sigma ;z)\notin \mathcal{U}$$

if

$$\theta =h\left(\xi \right),\kappa ={\displaystyle \frac{\eta h\left(\xi \right)-\tau \lambda \xi h^{\prime }\left(\xi \right)}{\eta }},$$

$$\Re \left\{{\displaystyle \frac{\rho \eta (\eta -1)-\theta \eta (\eta -1)}{\lambda (\eta \theta -\eta \kappa )}}+{\displaystyle \frac{2\eta -1}{\lambda }}\right\}\le \tau \Re \left\{{\displaystyle \frac{\xi h^{″}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}+1\right\},$$

$$\Re \left\{\begin{array}{cc}& {\displaystyle \frac{-\eta (\eta -1)(\eta -2)\sigma +3\eta (\eta -1)(\eta -\lambda -1)\rho +\eta (\eta -1)(3\lambda -2\eta +1)\theta }{{\lambda }^2(\eta \theta -\eta \kappa )}}\hfill \\ \hfill +& {\displaystyle \frac{3{\eta }^2-3\eta +2{\lambda }^2+3\lambda -6\eta \lambda +1}{{\lambda }^2}}\hfill \end{array}\right\}\le {\tau }^2\Re \left\{{\displaystyle \frac{{\xi }^2{h}^{‴}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}\right\},$$

where $\eta \in \mathbb{C}\setminus {\mathbb{Z}}^{-},$ $\tau \in \mathbb{N}\setminus \left\{1\right\},$$\xi \in \partial D\setminus E\left(h\right)$, and $z\in D.$

Theorem 4.  

Consider ψ ∈ ${\tilde{\mathrm{\Psi }}}_m[\mathcal{U},h].$ Let $g\in A$ and $h\in {\mathcal{Q}}_1$ satisfy

$$\Re \left\{{\displaystyle \frac{\xi h^{″}\left(\xi \right)}{h^{\prime }\left(\xi \right)}}\right\}\ge 0,|{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)|\le \tau |z{h}^{\prime }\left(\xi \right)|,$$

(37)

and

$$M_{\psi ({\mathbb{C}}^4\times D)}\left(\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right);z\in D\right)\le M_{\mathcal{U}}\left(z\right),$$

(38)

then

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}{\prec }_{F}h\left(z\right)(z\in D).$$

Proof. 

Let

$$f\left(z\right)={\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}$$

(39)

Utilizing (3) and (39), we have

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}}={\displaystyle \frac{-\lambda z{f}^{\prime }\left(z\right)+\eta f\left(z\right)}{\eta }}$$

Taking the derivative on both sides of the above equation and then using (3) with η replaced by $\eta -1$, we get

$$({\displaystyle \frac{\eta -1+\lambda }{\lambda }}){\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}}-({\displaystyle \frac{\eta -1}{\lambda }}){\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}}={\displaystyle \frac{-\lambda z^2{f}^{″}\left(z\right)+(\eta -2\lambda )z{f}^{\prime }\left(z\right)+\eta f\left(z\right)}{\eta }}$$

Further simplifications provide

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}}={\displaystyle \frac{{\lambda }^2{z}^2{f}^{″}\left(z\right)+\lambda (\lambda -2\eta +1)z{f}^{\prime }\left(z\right)+\eta (\eta -1)f\left(z\right)}{\eta (\eta -1)}}$$

Similarly, differentiation on both sides of the above equation and utilizing (3) with η replaced by $\eta -2$ yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}}& ={\displaystyle \frac{-{\lambda }^3{z}^3{f}^{‴}\left(z\right)+3{\lambda }^2(\eta -\lambda -1)z^2{f}^{″}\left(z\right)}{\eta (\eta -1)(\eta -2)}}\hfill \\ \hfill & +{\displaystyle \frac{-\lambda (3{\eta }^2-6\eta +{\lambda }^2+3\lambda -3\eta \lambda +2)z{f}^{\prime }\left(z\right)+\eta (\eta -1)(\eta -2)f\left(z\right)}{\eta (\eta -1)(\eta -2)}}.\hfill \end{array}$$

(40)

We now introduce the transformation of ${\mathbb{C}}^4$ to $\mathbb{C}$ by

$$\theta (t,u,v,w)=t,\kappa (t,u,v,w)={\displaystyle \frac{-\lambda u+\eta t}{\eta }},$$

$$\rho (t,u,v,w)={\displaystyle \frac{{\lambda }^{2}v+\lambda (\lambda -2\eta +1)u+\eta (\eta -1)t}{\eta (\eta -1)}},$$

$$\mathrm{and}$$

$$\sigma (t,u,v,w)={\displaystyle \frac{-{\lambda }^{3}w+3{\lambda }^2(\eta -\lambda -1)v-\lambda (3{\eta }^2-6\eta +{\lambda }^2+3\lambda -3\eta \lambda +2)u+\eta (\eta -1)(\eta -2)t}{\eta (\eta -1)(\eta -2)}}.$$

The transformation defined above converts the generalized Mittag-Leffler expressions into the standard fuzzy third-order differential subordination form, making Lemma 1 applicable. Hence, we have

$$\begin{array}{cc}\hfill & \hfill \psi (t,u,v,w;z)=\psi (\theta ,\kappa ,\rho ,\sigma ;z)\hfill \\ \hfill & =\psi (t,{\displaystyle \frac{-\lambda u+\eta t}{\eta }},{\displaystyle \frac{{\lambda }^{2}v+\lambda (\lambda -2\eta +1)u+\eta (\eta -1)t}{\eta (\eta -1)}},\hfill \\ \hfill & {\displaystyle \frac{-{\lambda }^{3}w+3{\lambda }^2(\eta -\lambda -1)v-\lambda (3{\eta }^2-6\eta +{\lambda }^2+3\lambda -3\eta \lambda +2)u+\eta (\eta -1)(\eta -2)t}{\eta (\eta -1)(\eta -2)}};z)\hfill \end{array}$$

(41)

By employing Lemma 1 and conditions (39)–(41), we have the following expression:

$$\psi \left(f\left(z\right),z{f}^{\prime }\left(z\right),z^2{f}^{″}\left(z\right),z^3{f}^{‴}\left(z\right);z\right)=\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right).$$

(42)

Hence, (42) leads to

$$M_{\psi ({\mathbb{C}}^4\times D)}\left(\psi (f\left(z\right),z{f}^{\prime }\left(z\right),z^2{f}^{″}\left(z\right),z^3{f}^{‴}\left(z\right);z\right)\le M_{\mathcal{U}}\left(z\right).$$

Using (41), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{v}{u}}+1& ={\displaystyle \frac{\rho \eta (\eta -1)-\lambda u(\lambda -2\eta +1)-\theta \eta (\eta -1)+u{\lambda }^2}{u{\lambda }^2}}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\displaystyle \frac{v}{u}}+1& ={\displaystyle \frac{\eta (\eta -1)\rho -\eta (\eta -1)\theta }{\lambda (\eta \theta -\eta \kappa )}}+{\displaystyle \frac{2\eta -1}{\lambda }}\hfill \end{array}$$

(44)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{w}{u}}& ={\displaystyle \frac{3v{\lambda }^2(\eta -\lambda -1)-\lambda u(3{\eta }^2-6\eta +{\lambda }^2+3\lambda -3\eta \lambda +2)}{u{\lambda }^3}}\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{\theta \eta (\eta -1)(\eta -2)-\sigma \eta (\eta -1)(\eta -2)}{u{\lambda }^3}}\hfill \end{array}$$

(46)

After simplification, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{w}{u}}& ={\displaystyle \frac{1}{{\lambda }^2(\eta \theta -\eta \kappa )}}[-\eta (\eta -1)(\eta -2)\sigma +3\eta (\eta -1)(\eta -\lambda -1)\rho \hfill \\ \hfill & +\eta (\eta -1)(3\lambda -2\eta +1)\theta ]+{\displaystyle \frac{1}{{\lambda }^2}}\left[3{\eta }^2-3\eta +2{\lambda }^2+3\lambda -6\eta \lambda +1\right].\hfill \end{array}$$

Accordingly, the $\psi \in \tilde{\mathrm{\Psi }}[\mathcal{U},h]$ admissibility condition in Definition 9 coincides with the $\psi \in {\mathrm{\Psi }}_2[\mathcal{U},h]$ admissibility condition in Definition 5. Hence by using Lemma 1 and (37), we obtain $M_{f\left(D\right)}f\left(z\right)\le M_{h\left(D\right)}h\left(z\right)$ or, equivalently, $M_{{\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(D\right)}{z}}}\left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}\right)\le M_{h\left(D\right)}h\left(z\right),$

i.e.,

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}{\prec }_{F}h\left(z\right),$$

Hence, this completes the proof. □

Example 2.  

Consider ψ = $\rho -\theta$ in Theorem 4. The expression $\rho -\theta$ satisfies admissibility conditions in Definition 9. Since $h\in {\mathcal{Q}}_1$ satisfies (37), Definition 9 indicates $\psi \notin \mathcal{U}.$ Accordingly, $\psi \in [\mathcal{U},h]$ and the hypothesis of Theorem 4 is satisfied.

Now, using (41), we have

$$\begin{array}{cc}\hfill \psi (\theta ,\kappa ,\rho ,\sigma ;z)=\rho -\theta & ={\displaystyle \frac{{\lambda }^{2}v+\lambda (\lambda -2\eta +1)u+\eta (\eta -1)t}{\eta (\eta -1)}}-t\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{{\lambda }^{2}v+\lambda (\lambda -2\eta +1)u}{\eta (\eta -1)}}\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\eta (\eta -1)\rho -\lambda (\lambda -2\eta +1)u-\eta (\eta -1)t+\lambda (\lambda -2\eta +1)u}{\eta (\eta -1)}}\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}}-t\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}}-{\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}.\hfill \end{array}$$

(51)

which is analytic in D and

$$M_{\psi ({\mathbb{C}}^4\times D)}\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}}-{\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}\right)\le M_{\mathcal{U}}\left(z\right),$$

Consequently,

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}{\prec }_{F}h\left(z\right)(z\in D).$$

(52)

The class ${\mathrm{\Psi }}_m^{\prime }[\omega \left(D\right),h]$ is written as ${\mathrm{\Psi }}_m^{\prime }[\omega ,h]$ when $\mathcal{U}\ne \mathbb{C}$ is a simply connected domain and $\mathcal{U}=\omega \left(D\right)$ for a suitable conformal mapping $\omega \left(z\right)$ of D onto $\mathcal{U}.$ The direct consequence of Theorem 4 is now stated below.

Corollary 4.  

Consider $\psi \in {\tilde{\mathrm{\Psi }}}_m[\omega ,h]$. If $g\in A$ and $h\in {\mathcal{Q}}_1$ satisfy condition (37), then we have

$${\displaystyle M_{\psi ({\mathbb{C}}^4\times D)}\left(\psi \left(\frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z},\frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z},\frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z},\frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z};z\right):z\in D\right)\le M_{\mathcal{U}}\left(z\right),}$$

which implies that

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}{\prec }_{F}h\left(z\right)(z\in D).}$$

The class $\psi \in {\tilde{\mathrm{\Psi }}}_m[\mathcal{U},R]$ is now defined below in view of Definition 7 and when $h\left(z\right)=1+Rz$$(R>0)$.

Definition 10.  

Consider $\mathcal{U}\in \mathbb{C}$ and $R>0.$ The class $\psi \in {\tilde{\mathrm{\Psi }}}_m[\mathcal{U},R]$ of admissible functions comprises functions $\psi :{\mathbb{C}}^4\times D\to \mathbb{C}$ such that

$$\psi (1+R{e}^{i\gamma },1+{\displaystyle \frac{(-\lambda \tau +\eta )R{e}^{i\gamma }}{\eta }},1+{\displaystyle \frac{{\lambda }^{2}L+[\lambda (\lambda -2\eta +1)\tau +\eta (\eta -1)]R{e}^{i\gamma }}{\eta (\eta -1)}},$$

$$1+{\displaystyle \frac{-{\lambda }^{3}N+3{\lambda }^2(\eta -\lambda -1)L+[-\lambda (3{\eta }^2-6\eta +{\lambda }^2+3\lambda -3\eta \lambda +2)\rho +\eta (\eta -1)(\eta -2)]R{e}^{i\gamma }}{\eta (\eta -1)(\eta -2)}})\notin \mathcal{U}$$

whenever $z\in D,$$\gamma \in \mathbb{R},$ and

$$\Re \left(L{e}^{-i\gamma }\right)\geqq (\tau -1)\tau R and \Re \left(N{e}^{-i\gamma }\right)\geqq 0 (\forall \gamma \in \mathbb{R}; \tau \in N\setminus \{1\left\}\right).$$

Corollary 5.  

Let $\psi \in {\tilde{\mathrm{\Psi }}}_m[\mathcal{U},R]$. If $g\in A$ satisfies

$$\left|{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)\right|\le \tau R$$

If

$$M_{\psi ({\mathbb{C}}^4\times D)}\left(\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right):z\in D\right)\le M_{\mathcal{U}}\left(z\right),$$

then

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}{\prec }_{F}1+Rz(z\in D),$$

where $\tau \in \mathbb{N}\setminus \left\{1\right\}$, $R>0$, and $z\in D$.

In particular, if

$$\mathcal{U}=h\left(z\right)=\{w:|w-1|<R(R>0\left)\right\},$$

then the class ${\tilde{\mathrm{\Psi }}}_m[\mathcal{U},R]$ is denoted by ${\tilde{\mathrm{\Psi }}}_m\left[R\right].$

4. Third-Order Fuzzy Differential Superordination Results

This section provides certain third-order fuzzy differential superordination results and for this purpose, the following definition is introduced here. The parameter ℓ used here is independent of the parameter $\tau$ used in the previous section.

Definition 11.  

Let $\mathcal{U}\subset \mathbb{C}$ and $h\in {\mathcal{H}}_0$ with $h^{\prime }\left(z\right)\ne 0$. Let $\psi :{\mathbb{C}}^4\times \overline{D}\to \mathbb{C}$ that satisfy

$$\psi (\theta ,\kappa ,\rho ,\sigma ;\xi )\in \mathcal{U}$$

if

$$\theta =h\left(z\right),\kappa ={\displaystyle \frac{-\lambda z{h}^{\prime }\left(z\right)+\mathsf{\ell }(\eta -1)h\left(z\right)}{\mathsf{\ell }\eta }},$$

$$\Re \left\{{\displaystyle \frac{\rho \eta (\eta -1)-\theta (\eta +\lambda )(\eta +\lambda -1)}{\lambda \left((\eta +\lambda )\theta -\eta \kappa \right)}}+{\displaystyle \frac{2\eta +2\lambda -1}{\lambda }}\right\}\le {\displaystyle \frac{1}{\mathsf{\ell }}}\Re \left\{{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}+1\right\},$$

$$\begin{array}{cc}\hfill & \Re \{{\displaystyle \frac{-\sigma \eta (\eta -1)(\eta -2)+\theta (\eta +\lambda )(\eta +\lambda -1)(\lambda -2\eta +1)+3\rho \eta {(\eta -1)}^2}{{\lambda }^2\left((\eta +\lambda )\theta -\eta \kappa \right)}}\hfill \\ & +{\displaystyle \frac{3{\eta }^2-3\eta -{\lambda }^2+1}{{\lambda }^2}}\}\le {\displaystyle \frac{1}{{\mathsf{\ell }}^2}}\Re \left\{{\displaystyle \frac{z^2{h}^{‴}\left(z\right)}{h^{\prime }\left(z\right)}}\right\}.\hfill \end{array}$$

where $\xi \in \partial D,$$z\in D,$ and $\mathsf{\ell }\in \mathbb{N}\setminus \left\{1\right\}$ constitute the class of admissible functions represented as ${\mathrm{\Psi }}_m^{\prime }[\mathcal{U},h].$

This definition presents the admissible conditions for fuzzy superordination, featuring reversed inequalities compared to subordination. When input values meet the given conditions, the requirement $\psi$ ∈ $\mathcal{U}$ ensures that h acts as a fuzzy subordinant.

Theorem 5.  

Assume $\psi \in {\mathrm{\Psi }}_m^{\prime }[\mathcal{U},h].$ Also, consider $g\in A$ and $L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)$ ∈ ${\mathcal{Q}}_0$ that satisfy

$$\Re \left\{{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right\}\ge 0,|{\displaystyle \frac{\eta +\lambda }{\lambda }}{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)-{\displaystyle \frac{\eta }{\lambda }}{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)|\le \mathsf{\ell }|h^{\prime }\left(\xi \right)|,$$

(53)

and

$$\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)$$

is univalent in D, then

$$M_{\mathcal{U}}\left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\left(\psi \left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z\right):z\in D\right)$$

leads to

$$h\left(z\right){\prec }_F{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)(z\in D).$$

(54)

Proof. 

Consider $f\left(z\right)$ to be defined by (16) and ψ be represented by (23). Since $\psi \in {\mathrm{\Psi }}_m^{\prime }[\mathcal{U},h],$ (24) and (54) give

$$M_{\mathcal{U}}\left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\left(\psi \left(f\left(z\right),z{f}^{\prime }\left(z\right),z^2{f}^{″}\left(z\right),z^3{f}^{″\prime }\left(z\right);z\right):z\in D\right).$$

(55)

From (23), it follows that the $\psi \in {\mathrm{\Psi }}_m^{\prime }[\mathcal{U},h]$ admissibility condition in Definition 11 coincides with the admissibility condition for ψ in Definition 7. Consequently, by using the expressions in (53) and Lemma 2, we have

$$h\left(z\right){\prec }_{F}f\left(z\right),$$

(56)

i.e.,

$$h\left(z\right){\prec }_F{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)(z\in D).$$

(57)

The proof is now complete. □

The next theorem is deduced from Theorem 5 by employing the methods that are similar to those in the preceding section.

Theorem 6.  

Consider $\psi \in {\mathrm{\Psi }}_m^{\prime }[\omega ,h],$ where ω is analytic in D. If $g\in A$ and $L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)$ ∈ ${\mathcal{Q}}_0$ holds the conditions in (53) and

$$\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)$$

(58)

is univalent in D, then

$$M_{\omega \left(\mathcal{U}\right)}\omega \left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\psi \left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right):z\right),$$

which implies

$$h\left(z\right){\prec }_F,L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)(z\in D).$$

(59)

The subsequent theorem for a well-chosen $\psi$ provides the best fuzzy subordinant of (59).

Theorem 7.  

Let $\omega \in \mathcal{H}\left(D\right)$, $\psi :{\mathbb{C}}^4\times \overline{D}\to \mathbb{C}$, and ψ be expressed by (23). Let the differential equation

$$\psi \left(h\left(z\right),z{h}^{\prime }\left(z\right),z^2{h}^{″}\left(z\right),z^3{h}^{‴}\left(z\right);z\right)=\omega \left(z\right)$$

have a solution $h\left(z\right)\in {\mathcal{Q}}_0.$ If $g\in A$ and $L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\in {\mathcal{Q}}_0$ satisfies condition (53) and

$$\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)$$

is univalent in D, then

$$M_{\omega \left(\mathcal{U}\right)}\omega \left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)$$

leads to

$$h\left(z\right){\prec }_F{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right) (z\in D).$$

and $h\left(z\right)$ is said to be the best fuzzy subordinant.

Proof. 

By utilizing Theorem 6, we observe that $h\left(z\right)$ acts as a fuzzy subordinant of the given differential superordination. Moreover, h satisfies the differential equation, given by

$$\psi (h\left(z\right),z{h}^{\prime }\left(z\right),z^2{h}^{″}\left(z\right),z^3{h}^{‴}\left(z\right);z)=\omega \left(z\right),$$

Consequently, h represents the best fuzzy subordinant, since every other fuzzy subordinant is subordinate to it. □

Next, we introduce a new definition for the class of admissible functions, ${\tilde{\mathrm{\Psi }}}_m^{\prime }[\mathcal{U},h]$.

Definition 12.  

Let $\mathcal{U}\subset \mathbb{C}$ and $h\in {\mathcal{H}}_1$ with $h^{\prime }\left(z\right)\ne 0$. Consider $\psi :{\mathbb{C}}^4\times \overline{D}\to \mathbb{C}$ that satisfies

$$\psi (\theta ,\kappa ,\rho ,\sigma ;\xi )\in \mathcal{U}$$

if

$$\theta =h\left(z\right),\kappa ={\displaystyle \frac{-\lambda z{h}^{\prime }\left(z\right)+\mathsf{\ell }\eta h\left(z\right)}{\mathsf{\ell }\eta }},$$

$$\Re \left\{{\displaystyle \frac{\eta (\eta -1)\rho -\eta (\eta -1)\theta }{\lambda (\eta \theta -\eta \kappa )}}+{\displaystyle \frac{2\eta -1}{\lambda }}\right\}\le {\displaystyle \frac{1}{\mathsf{\ell }}}\Re \left\{{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}+1\right\},$$

$$\begin{array}{cc}\hfill & \Re \{{\displaystyle \frac{-\eta (\eta -1)(\eta -2)\sigma +3\eta (\eta -1)(\eta -\lambda -1)\rho +\eta (\eta -1)(3\lambda -2\eta +1)\theta }{{\lambda }^2(\eta \theta -\eta \kappa )}}\hfill \\ & +{\displaystyle \frac{3{\eta }^2-3\eta +2{\lambda }^2+3\lambda -6\eta \lambda +1}{{\lambda }^2}}\}\le {\displaystyle \frac{1}{{\mathsf{\ell }}^2}}\Re \left\{{\displaystyle \frac{z^2{h}^{‴}\left(z\right)}{h^{\prime }\left(z\right)}}\right\}.\hfill \end{array}$$

where $\xi \in \partial D,$$z\in D,$ and $\mathsf{\ell }\in \mathbb{N}\setminus \left\{1\right\}$ constitute the class of admissible functions represented by ${\tilde{\mathrm{\Psi }}}_m^{\prime }[\mathcal{U},h].$

Whenever $\mathcal{U}\ne \mathbb{C}$ is a simple connected domain and $\mathcal{U}=\omega \left(D\right)$ for a suitable choice of conformal mapping of $\omega \left(z\right)$ of D onto $\mathcal{U}$, then the class ${\tilde{\mathrm{\Psi }}}_m^{\prime }[\omega \left(D\right),h]$ = ${\tilde{\mathrm{\Psi }}}_m^{\prime }[\omega ,h]$. The subsequent theorem is deduced from Theorem 7 by following methods involving the approach used in the previous section.

Theorem 8.  

Let $\omega \in \mathcal{H}\left(D\right)$ and $\psi :{\mathbb{C}}^4\times \overline{D}\to \mathbb{C}$, and ψ is given by (23). Let the differential equation

$$\psi \left(h\left(z\right),z{h}^{\prime }\left(z\right),z^2{h}^{″}\left(z\right),z^3{h}^{‴}\left(z\right);z\right)=\omega \left(z\right)$$

have a solution $h\left(z\right)\in {\mathcal{Q}}_1$. If $g\in A$ and ${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}\in {\mathcal{Q}}_1$ satisfy the conditions in (33) and

$$\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right)$$

is univalent in D, then

$$M_{\omega \left(\mathcal{U}\right)}\omega \left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right)$$

yields

$$h\left(z\right){\prec }_F{\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},(z\in D),$$

and h represents the best fuzzy subordinant.

Proof. 

The proof is similar to the proof of Theorem 4. By applying Theorem 7 together with the relations involving ${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}}$,${\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}}$,${\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}}$,${\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}}$ derived in Theorem 4, the required result follows. Furthermore, Theorem 7 identifies h as the best fuzzy subordinant. □

4.1. Sandwich-Type Results

The following section presents two sandwich-type outcomes. A sandwich-type result yielded by combining Theorems 1 and 5 is mentioned below:

Theorem 9.  

Let the functions ${\omega }_1$, $h_1$$\in \mathcal{H}\left(D\right)$. Furthermore, assume ${\omega }_2$ to be univalent in D, $h_2\in {\mathcal{Q}}_0$ with $h_1\left(0\right)=h_2\left(0\right)=1$ and $\psi \in {\mathrm{\Psi }}_m[{\omega }_2,h_2]$∩${\mathrm{\Psi }}_m^{\prime }[{\omega }_1,h_1].$ If $g\in A$ and $L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)\in {\mathcal{Q}}_0\cap {\mathcal{H}}_0$ and

$$\psi (L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z)\in S,$$

with conditions (14) and (53) being satisfied, then

$${\displaystyle M_{{\omega }_1\left(\mathcal{U}\right)}{\omega }_1\left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\psi \left(L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right),L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right);z\right)\le M_{{\omega }_2\left(\mathcal{U}\right)}{\omega }_2\left(z\right)}$$

implies that

$${\omega }_1\left(z\right){\prec }_F{L}_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right){\prec }_F{\omega }_2\left(z\right).$$

We get the following result by combining Theorems 4 and 8.

Theorem 10.  

Consider the functions ${\tilde{\omega }}_1,{\tilde{h}}_1\in \mathcal{H}\left(D\right)$. Assume ${\tilde{\omega }}_2\in S$, ${\tilde{h}}_2\in {\mathcal{Q}}_1$ with ${\tilde{h}}_1\left(0\right)={\tilde{h}}_2\left(0\right)=1$, and $\psi \in {\tilde{\mathrm{\Psi }}}_m[{\tilde{\omega }}_2,{\tilde{h}}_2]\cap {\tilde{\mathrm{\Psi }}}_m^{\prime }[{\tilde{\omega }}_1,{\tilde{h}}_1]$. If $g\in A$ and $L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)\in {\mathcal{Q}}_1\cap {\mathcal{H}}_1$, and

$$\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right)\in S,$$

with conditions (14) and (53) being satisfied, then

$$M_{{\tilde{\omega }}_1\left(\mathcal{U}\right)}{\tilde{\omega }}_1\left(z\right)\le M_{\psi ({\mathbb{C}}^4\times D)}\psi \left({\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }g\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }g\left(z\right)}{z}};z\right)\le M_{{\tilde{\omega }}_2\left(\mathcal{U}\right)}{\tilde{\omega }}_2\left(z\right)$$

Consequently

$$\widetilde{{\omega }_1}\left(z\right){\prec }_F{L}_{\alpha ,\beta }^{\eta ,\lambda }g\left(z\right){\prec }_F\widetilde{{\omega }_2}\left(z\right).$$

4.2. Application of Fuzzy Differential Subordination in Decision Making

Fuzzy differential subordination has significant importance in decision making, as it offers a comprehensive mathematical framework for addressing uncertainty, imprecision, and vagueness in decision-making problems [39]. This framework enables decision variables to evolve within well-defined fuzzy bounds and thus ensures the consistency of all admissible decisions with the imposed fuzzy constraints. Moreover, the concept of fuzzy extremal functions is central to identifying optimal decision outcomes under uncertainty. Fuzzy differential subordination is applicable to real-world systems where data are inherently imprecise, such as finance, control systems, and image processing.

Recent studies on fuzzy systems have shown significant developments in neural networks, control, and decision-support systems. Techniques such as non-linear fractional-order type 3 fuzzy controllers [40] have been successfully applied in fuzzy control to enhance the accuracy and robustness of path-tracking performance in autonomous vehicles. Another recent study introduced a type 2 fuzzy structural control system [41] which provides improved vibration control and stability for engineering structures under uncertainty. Additionally, a hybrid decision making model has been developed to support strategic decision making in agricultural supply chains [42]. These developments highlight the practical importance of fuzzy techniques in addressing complex decision-making problems.

Motivated by these developments, the present study provides a rigorous mathematical framework through fuzzy differential subordination theory that ensures reliable and robust decision making under uncertainty. Following examples demonstrate the applicability of fuzzy differential subordination in decision making.

Example 3  

(Job Candidate Selection). We consider a decision-making problem involving the selection of a sales manager under the fuzzy goal and constraints. Consider a company that has to select one sales manager from a set of four candidates, W = $\{E,F,,G,H\}$. The decision is based on fuzzy goal G (Best qualification) and two fuzzy constraints $C_1$ (Low salary demand) and $C_2$ (Close residence). Membership degrees are obtained through standard normalization and by using the available quantitative data (test scores, salary figures, distance in km).

Table 1. Fuzzy evaluation of candidates with respect to goal G and constraints.

Candidate	Goal G	Constraint ${\mathit{C}}_1$	Constraint ${\mathit{C}}_2$
E	0.9	0.419	0.1
F	0.6	0.5	0.9
G	0.8	0.719	0.7
H	0.6	0.875	1.0

presents these values, which represent the degree to which each candidate satisfies the corresponding fuzzy goal or constraint and lie in the interval [0,1]. We now find a classical fuzzy decision by using the Bellman–Zadeh principle [2], and the score for each candidate is given by

$$F_y\left(0\right)=min\{{\mu }_{G_y},{\mu }_{{C_1}_y},{\mu }_{{C_2}_y}\},y\in W,$$

(60)

By using the values from the given table, we get fuzzy decision for each candidate as $F_E\left(0\right)=0.1,$$F_F\left(0\right)=0.5,$$F_G\left(0\right)=0.7,$ and $F_H\left(0\right)=0.6.$ Hence, the highest membership degree corresponds to G, confirming G as the best choice for the company. The ranking among all candidates is given by

$$G>H>F>E.$$

The Bellman–Zadeh approach identifies this optimal decision at a fixed point of the decision space, representing a static evaluation. In real-world decision problems, decisions rarely remain static, such as market conditions, customer preferences, and resource availability, which may change over a period of time. Fuzzy differential subordination addresses this problem by providing the dynamic mechanism that ensures the stability of decision under uncertainty. We now introduce t as an evaluation parameter, allowing membership degrees to change over time. We take a suitable fuzzy linear model that is given by

$$g_{j_y}\left(t\right)={\mu }_{j_y}+h_{j_y}.t,t\ge 0$$

(61)

with $y\in W,$$j\in \{G,C_1,C_2\},$ and $h_{j_y}$ is the rate of change of satisfaction, providing time-dependent fuzzy membership. Consider $\psi \left(t\right)$ = $\alpha +\gamma t,(\alpha \in (0,1),\gamma >0)$ to be a fuzzy membership function representing the company’s benchmark satisfaction profile, and the implementation of fuzzy differential subordination gives

$$g_{j_y}\left(t\right){\prec }_F\psi \left(t\right)$$

(62)

The above condition is satisfied if and only if ${\mu }_{j_y}\le \alpha$ and $h_{j_y}\le \gamma$, restricting the growth behavior of decisions within a stable fuzzy domain. We now express an analytic form to represent our decision data that facilitates the verification of subordination conditions by applying the classical tools from geometric function theory, given by

$$g_y\left(z\right)=z+\mu G_y{z}^2+\mu C_1{z}^3+\mu C_2{z}^4,z\in D,$$

where $g_y\left(z\right)\in A$ and satisfies the normalization conditions $g_y\left(0\right)=0$ and $g_y^{\prime }\left(0\right)=1$, as required by Theorem 1. The analytic form of optimal static decision G is as follows:

$$C:g\left(z\right)=z+0.8z^2+0.719z^3+0.7z^4,z\in D,$$

(63)

Now, we employ the generalized Mittag-Leffler operator here to enhance the flexibility of the decision model. The operator provides a more refined control of the decision evaluation within the fuzzy differential framework. Applying the generalized Mittag-Leffler operator to $g_y\left(z\right),$ we obtain the result denoted by $f_y\left(z\right)\in A.$

$$f_y\left(z\right)=L_{\alpha ,\beta }^{\eta +1,\lambda }{g}_y\left(z\right),z\in D,$$

(64)

Theorem 1 applies third-order fuzzy differential subordination to the generalized Mittag-Leffler operator using the relations (17)–(19), which express $L_{\alpha ,\beta }^{\eta ,\lambda }{g}_y\left(z\right),L_{\alpha ,\beta }^{\eta -1,\lambda }{g}_y\left(z\right)$ and $L_{\alpha ,\beta }^{\eta -2,\lambda }{g}_y\left(z\right)$ in terms of $f_y\left(z\right)=L_{\alpha ,\beta }^{\eta +1,\lambda }{g}_y\left(z\right)$ and its derivatives up to the third order. This transformation allows for dominance evaluation of the decision function. Let $h\left(z\right)\in A$ be the dominant function given by

$$h\left(z\right)={\displaystyle \frac{z}{1-z}},\Re \left\{{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right\}\ge 0,$$

(65)

The above dominant function is specifically chosen as it satisfies (14) and bounds the decision function, thereby guaranteeing the stability of decisions in the fuzzy decision making process. To define the permissible range for Equation (63), we choose an admissible function defined in Example 1 of Theorem 1.

$$\psi (\theta ,\kappa ,\rho ,\sigma ;z)=\kappa -\theta$$

It is verified by the calculations in (28)–(32) that Definition 7 is satisfied, as already discussed in Example 1. Since all required conditions are satisfied, applying Theorem 1 yields

$$L_{\alpha ,\beta }^{\eta +1,\lambda }{g}_y\left(z\right){\prec }_F{\displaystyle \frac{z}{1-z}},\forall y\in W.$$

(66)

Thus, Theorem 1 provides the mathematical guarantee that the decision functions of all candidates remain inside the fuzzy dominant region when applying the generalized Mittag-Leffler operator and third-order fuzzy differential subordination constraints. Hence, the relative ranking is preserved and formulated as

$$f_E\left(z\right){\prec }_F{f}_F\left(z\right){\prec }_F{f}_H\left(z\right){\prec }_F{f}_G\left(z\right).$$

(67)

Hence, candidate G remains the best and most stable choice for the company.

The application of the generalized Mittag-Leffler operator together with Theorem 1 ensures that the analytic formulation of all candidate decision functions fulfill a third-order differential subordination relative to a common dominant function. This guarantees the stability of the ranking, bounded evolution, and robustness of decisions in the presence of uncertainty and dynamic changes. Hence, this example not only validates the practical significance of the theoretical results discussed in this paper but also highlights fuzzy differential subordination as a tool for extending static fuzzy decision models toward more structured and analytically manageable formulations.

Example 4  

(Supplier Selection). We now consider a supplier selection problem to further extend and illustrate the proposed approach in decision making under uncertainty. This is a well-known multi-criteria decision making problem [43] in which an organization must select a suitable supplier from four alternatives, represented by W = $\{S_1,S_2,S_3,S_4\}$, taking into account multi-evaluation criteria. The membership values for each supplier for a fuzzy goal G (Product quality), and the constraints $C_1$ (Delivery reliability) and $C_2$ (Cost effectiveness) are provided below (see

Table 2. Fuzzy evaluation of suppliers with respect to goal G and constraints.

Supplier	Goal G	Constraint ${\mathit{C}}_1$	Constraint ${\mathit{C}}_2$
$S_1$	0.79	0.9	0.86
$S_2$	0.9	0.5	0.762
$S_3$	0.6	0.719	0.7
$S_4$	0.78	0.42	0.9

).

These values represent the degree of satisfaction. By utilizing Equation (60), we obtain a static fuzzy decision for each candidate as $F_{S_1}\left(0\right)=0.79$, $F_{S_2}\left(0\right)=0.5$, $F_{S_3}\left(0\right)=0.6$, and $F_{S_4}\left(0\right)=0.42$. The ranking of all supplier alternatives confirms $S_1$ as an optimal choice for an organization, presented as

$$S_1>S_3>S_2>S_4.$$

This shows $S_1$ to be an optimal choice; however, this decision is static and not considering any dynamic variations. So, we now convert this static fuzzy decision to a dynamic one by employing (61) and (62). In order to employ Theorem 4, we need to represent the analytic form for $S_1$, defined by

$$S_1:g\left(z\right)=z+0.79z^2+0.9z^3+0.86z^4,z\in D,$$

(68)

Hence $S_1\in A$, as required by Theorem 4. We now consider third-order fuzzy differential subordination for the normalized form of the generalized Mittag-Leffler operator discussed in Theorem 4 that expresses ${\displaystyle \frac{L_{\alpha ,\beta }^{\eta ,\lambda }{g}_y\left(z\right)}{z}},{\displaystyle \frac{L_{\alpha ,\beta }^{\eta -1,\lambda }{g}_y\left(z\right)}{z}}$ and ${\displaystyle \frac{L_{\alpha ,\beta }^{\eta -2,\lambda }{g}_y\left(z\right)}{z}}$ in terms of $f_y\left(z\right)={\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }{g}_y\left(z\right)}{z}}$ and its derivatives up to the third order. We now introduce a dominant function that satisfies the condition (37) required by Theorem 4:

$$h\left(z\right)={\displaystyle \frac{z}{{(1-z)}^2}},\Re \left\{{\displaystyle \frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}}\right\}\ge 0,$$

(69)

The admissible function $\psi (\theta ,\kappa ,\rho ,\sigma :z)=\rho -\theta$ is taken from Example 2. It satisfies Definition 9 and verifies Theorem 4 in (51). We now apply Theorem 4 since all assumptions meet the criteria.

$${\displaystyle \frac{L_{\alpha ,\beta }^{\eta +1,\lambda }{g}_y\left(z\right)}{z}}{\prec }_F{\displaystyle \frac{z}{{(1-z)}^2}},\forall y\in W.$$

(70)

Thus, Theorem 4 facilitates the stable ranking guarantee when applying third-order fuzzy differential subordination and the generalized Mittag-Leffler operator constraints. Therefore, the relative ranking of all suppliers is preserved and expressed as

$$f_{S_4}\left(z\right){\prec }_F{f}_{S_2}\left(z\right){\prec }_F{f}_{S_3}\left(z\right){\prec }_F{f}_{S_1}\left(z\right).$$

(71)

It follows that $S_1$ is an optimal and stable choice.

This example validates the importance of a fuzzy differential subordination approach in a critical decision making supplier selection problem in the presence of vagueness. The stability of the decision ensures that small changes in data do not lead to significant changes in supplier ranking, thereby providing robustness to the decision making process. The implementation of the generalized Mittag-Leffler operator integrated with Theorem 4 leads to consistent decision outcomes in the presence of prescribed constraints. Overall, the proposed framework connects the theoretical results of third-order fuzzy differential subordination with practical decision making and serves as a useful tool for managing complex evaluation processes. This approach is flexible and can be extended to other real-world applications involving uncertain and dynamic environments.

The two applications discussed here illustrate the strong applicability of the theoretical results developed earlier in the section on the third-order fuzzy differential subordination results. The candidate selection problem utilizes Theorem 1 with the admissible function $\kappa -\theta$, while Theorem 4 is employed in the supplier selection problem by considering the admissible function $\rho -\theta$. In both cases, the generalized Mittag-Leffler operator converts static decision data into analytic functions which satisfy the assumptions of Theorems 1 and 4. The dominant functions are chosen accordingly to meet the criteria defined in both theorems. Therefore, the defined framework in both examples serves as a powerful mathematical tool for decision making under uncertainty, with ranking stability guaranteed by the theorems.

5. Conclusions

In this study, numerous findings of third-order fuzzy differential subordination and superordination are obtained via the generalized Mittag-Leffler operator. Various inclusion relationships and sharp results are deduced by defining suitable admissibility conditions. Some examples are presented to illustrate the applicability of our findings. The obtained results not only serve as the extension of existing works from second-order cases but also highlight the significance of higher-order fuzzy differential techniques. A notable feature of the present work is the implementation of fuzzy differential subordination together with the incorporation of the generalized Mittag-Leffler operator in decision making. The operator is highly suitable for handling imprecision due to its flexible structure. The proposed mechanism ensures the robustness and stability of decision outcomes under uncertainty, showing the practical relevance of theoretical findings.

The presented work enriches the theory of fuzzy differential subordination. It provides a solid foundation and new directions for further investigation in higher-order fuzzy analysis. Future research may extend these investigations to other fractional or integral operators, as well as to numerous applications in control theory, optimization, and multi-criteria decision making under uncertainty.