Third-Order Fuzzy Subordination and Superordination on Analytic Functions on Punctured Unit Disk

Abstract

This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best subordinant for the third-order fuzzy differential subordinations and superordinations, respectively. The investigation concludes with the assertion of sandwich-type theorems connecting the conclusions of the studies conducted using the particular methods of the theories of the third-order fuzzy differential subordination and superordination, respectively.

1. Introduction

The present work focuses on advancing the recently started study of the dual theories associated with the newly introduced generalizations of third-order fuzzy differential subordination and superordination. The motivation of the research is to use the recently developed techniques that are specific to the two dual fuzzy theories for obtaining new outcomes related to third-order fuzzy differential subordination and superordination and to further connect the dual results through sandwich-type results, as familiar when investigating dual theories in the field of geometric function theory.

In 2024, third-order fuzzy differential subordination and superordination were proposed [1,2] in an attempt to improve upon the second-order fuzzy differential subordination and superordination ideas that had already been well established. Many classical third-order differential subordination and superordination results are currently being obtained that could be enhanced to fit the new fuzzy differential subordination and superordination theories. For instance, third-order differential subordinations and superordinations are obtained through a fractional differential operator in [3], a seven-parameter Mittag–Leffler operator in [4], and a convolution operator in [5]. The motivation behind introducing the concepts of third-order fuzzy differential subordination and superordination, as explained in the articles first revealing them [1,2], was to develop the concepts of third-order fuzzy differential subordination and superordination by broadening current ideas related to second-order fuzzy differential subordination and superordination. Since different types of operators added into the studies on third-order differential subordination and superordination theories enhanced the relevance of the outcomes, the use of such operators is considered beneficial for developments concerning the particular forms of third-order differential subordination and superordination when they are fuzzy in nature. This is the motivation behind the study presented in this paper. Hence, to obtain the new results of the present investigation, in this work, a new convolution operator is introduced and applied in the particular fuzzy contexts of the two dual theories of third-order fuzzy differential subordination and superordination.

Similar to how the classical theory of third-order differential subordination [6] broadened the theory of second-order differential subordination [7,8], the concepts associated with third-order fuzzy differential subordination theory emerged recently in [1], carrying on the idea of improving the theory of second-order fuzzy differential subordination [2]. In their work [1], the authors presented the main ideas of the third-order fuzzy subordination extension as well as the key concepts to be used for studies related to this specific type of fuzzy differential subordination theory, such as the admissible functions class and some fundamental theorems. Motivated by the intention of moving forward with the advancement of the theory of second-order fuzzy differential superordination [9] using the fundamental pattern set for the fuzzy differential superordination theory [10], the dual concept associated with third-order fuzzy differential superordination theory also recently emerged [11]. In [11], the authors studied the dual problem of third-order fuzzy differential superordination by introducing the fundamental ideas that characterize this concept, the admissible functions class, and certain basic theorems that are to be used for further development on the topic.

The present investigation follows the pattern set for second-order fuzzy differential subordination and superordination. This new type of differential subordination and superordination originally emerged motivated by the idea seen in many fields of mathematics that have tried to incorporate the fuzzy set notion introduced by Lotfi A. Zadeh [12]. The use of different types of operators produced the first published studies on the development of fuzzy differential subordination and superordination theories [13,14,15]. The application of different operators in fuzzy differential subordination and superordination theories continues to generate multiple new results at this time, as evidenced in recently published articles such as [16,17,18]. The class of meromorphic functions has also been recently targeted by recent investigations on third-order differential subordinations as seen in [19] and on fuzzy differential subordination, as in [20].

The present investigation considers a new linear operator introduced by the convolution applied to the operator ${\mathcal{L}}_{p,\varkappa }^{\varrho }(\nu ,\mathfrak{s};\tau )$ introduced by El-Ashwah and Bulboacă [21] by using Hurwitz–Lerch Zeta function and the Gaussian hypergeometric function. The use of the new operator is inspired by the research presented above showing how the use of different types of operators has enhanced our knowledge regarding second-order fuzzy differential subordination and superordination theories. It must also be considered that the line of research involving third-order fuzzy differential subordination and superordination is new and needs to be explored. Hence, the addition of operators to the investigations must prove beneficial for the development of new fuzzy dual theories. It is shown that the new operator generalizes previously investigated operators. The purpose of this endeavor is to establish third-order fuzzy differential subordination and superordination properties of meromorphic multivalent functions by applying this new operator.

Since only papers introducing the concepts [1,2] have been published so far on third-order fuzzy differential subordination and superordination dual concepts, the present paper is essential for the development of these newly emerging topics. The previously established results used for this investigation are recalled in the next section, and the new third-order fuzzy differential subordination and superordination findings are revealed in the section that follows.

2. Materials and Methods

Denote by $\mathcal{H}\left(\mathfrak{D}\right)$ the class of analytic functions in $\mathfrak{D},$ with $\mathfrak{D}=\{\tau \in \mathbb{C}:\left|\tau \right|<1\}$ being the unit disk of the complex plane with $\overline{\mathfrak{D}}=\{\tau \in \mathbb{C}:\left|\tau \right|\le 1\}$ and $\partial \mathfrak{D}=\{\tau \in \mathbb{C}:\left|\tau \right|=1\}.$ The known subclasses of $\mathcal{H}\left(\mathfrak{D}\right)$ are of utmost importance for this study:

$${\mathcal{A}}_{\kappa }=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{D}\right):\mathfrak{f}\left(\tau \right)=\tau +a_{\kappa +1}{\tau }^{\kappa +1}+\cdots \tau \in \mathfrak{D}\right\},$$

where ${\mathcal{A}}_1=\mathcal{A}=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{D}\right):\mathfrak{f}\left(\tau \right)=\tau +a_2{\tau }^2+\cdots ,\tau \in \mathfrak{D}\right\},$

$$\mathcal{H}[a,\kappa ]=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{D}\right):\mathfrak{f}\left(\tau \right)=a+a_{\kappa }{\tau }^{\kappa }+a_{\kappa +1}{\tau }^{\kappa +1}+\cdots ,\tau \in \mathfrak{D}\right\},$$

where ${\mathcal{H}}_0=\mathcal{H}[0,1],$ with $a\in \mathbb{C}$ and $\kappa \in \mathbb{N}=\{1,2,3,\cdots \},$ and

$$\mathfrak{K}=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{D}\right):Re\left({\displaystyle \frac{\tau {\mathfrak{f}}^{″}\left(\tau \right)}{{\mathfrak{f}}^{\prime }\left(\tau \right)}}+1\right)>0,{\mathfrak{f}}^{\prime }\left(0\right)\ne 0,\tau \in \mathfrak{D}\right\}$$

is the class of normalized convex functions.

Furthermore,

$$\mathcal{S}=\left\{\mathfrak{f}\in \mathcal{A}:\mathfrak{f}\left(\tau \right)=\tau +a_2{\tau }^2+\cdots ,\mathfrak{f}\left(0\right)=0,{\mathfrak{f}}^{\prime }\left(0\right)=1,\mathfrak{f}\mathrm{is}\mathrm{univalent}\mathrm{for}\tau \in \mathfrak{D}\right\}.$$

With $\mathfrak{f},\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right),\mathfrak{f}$ is subordinate to $\mathfrak{g}$ or $\mathfrak{g}$ is superordinate to $\mathfrak{f}$, written $\mathfrak{f}\prec \mathfrak{g}$ or $\mathfrak{f}\left(\tau \right)\prec \mathfrak{g}\left(\tau \right),$ if there exists a Schwarz function $\omega \in \mathcal{H}\left(\mathfrak{D}\right)$ satisfying $\omega \left(0\right)=0$ and $\left|\omega \left(\tau \right)\right|<1,$ $\tau \in \mathfrak{D}$ such that $\mathfrak{f}\left(\tau \right)=\mathfrak{g}\left(\omega \right(\tau \left)\right).$ If $\mathfrak{g}\in S$, then $\mathfrak{f}\prec \mathfrak{g}\iff \mathfrak{f}\left(0\right)\prec \mathfrak{g}\left(0\right)$ and $\mathfrak{f}\left(\mathfrak{D}\right)\prec \mathfrak{g}\left(\mathfrak{D}\right).$

Meromorphic multivalent functions $\mathfrak{f}$ are gathered under the symbol ${\Sigma }_p$ defined as follows:

$$\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{a}_{\kappa }{\tau }^{\kappa }(p\in \mathbb{N},\tau \in {\mathfrak{D}}^{\ast }=\mathfrak{D}\setminus \left\{0\right\}),$$

(1)

with ${\mathfrak{D}}^{\ast }$ the punctured unit disc ${\mathfrak{D}}^{\ast }=\{\tau \in \mathbb{C}:0<\left|\tau \right|<1\}.$

Having two functions ${\mathfrak{f}}_{\iota }\in {\Sigma }_p(\iota =1,2)$ written as

$${\mathfrak{f}}_{\iota }\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{a}_{\kappa ,\iota }{\tau }^{\kappa },$$

we define the convolution of ${\mathfrak{f}}_1\left(\tau \right)$ and ${\mathfrak{f}}_2\left(\tau \right)$ as

$$({\mathfrak{f}}_1\ast {\mathfrak{f}}_2)\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{a}_{\kappa ,1}{a}_{\kappa ,2}{\tau }^{\kappa }=({\mathfrak{f}}_2\ast {\mathfrak{f}}_1)\left(\tau \right).$$

The investigation presented in this paper is initiated with the use of the operator ${\mathcal{L}}_{p,\varkappa }^{\varrho }$ [21] defined by using Hurwitz–Lerch Zeta function as follows:

$${\mathcal{L}}_{\varkappa ,p}^{\varrho }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +p}}\right)}^{\varrho }{a}_{\kappa }{\tau }^{\kappa },$$

(2)

$$(\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}=\left\{0,-1,-2,\dots \right\};\varrho \in \mathbb{C},\tau \in {\mathfrak{D}}^{\ast }).$$

It was proved in [21] that for all $\mathfrak{f}\in {\Sigma }_p,\tau ,t_i\in {\mathfrak{D}}^{\ast }(i=1,2,3,\dots ,\kappa ),\kappa \in \mathbb{N}$ and $\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-},$ the following relations are true:

$${\mathcal{L}}_{1,p}^{\mathfrak{0}}\mathfrak{f}\left(\tau \right)=\mathfrak{f}\left(\tau \right)\mathrm{and}{\mathcal{L}}_{\varkappa ,p}^{\mathfrak{0}}\mathfrak{f}\left(\tau \right)=\mathfrak{f}\left(\tau \right);$$

$${\mathcal{L}}_{1,p}^1\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^{p+1}}}{\int }_0^{\tau }{t}_1^p\mathfrak{f}\left(t_1\right)d{t}_1;$$

$${\mathcal{L}}_{1,p}^2\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^{p+1}}}{\int }_0^{\tau }{\displaystyle \frac{1}{t_1}}{\int }_0^{t_1}{t}_2^p\mathfrak{f}\left(t_2\right)d{t}_{2}d{t}_1;$$

            ⋮

$${\mathcal{L}}_{1,p}^{\kappa }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^{p+1}}}{\int }_0^{\tau }{\displaystyle \frac{1}{t_1}}{\int }_0^{t_1}{\displaystyle \frac{1}{t_2}}{\int }_0^{t_2}\cdots {\displaystyle \frac{1}{t_{\kappa -1}}}{\int }_0^{t_{\kappa -1}}{t}_{\kappa }^p\mathfrak{f}\left(t_{\kappa }\right)d{t}_{\kappa }d{t}_{\kappa -1}\dots d{t}_{2}d{t}_1;$$

            ⋮

$${\mathcal{L}}_{\varkappa ,p}^{\kappa }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{{\varkappa }^{\kappa }}{{\tau }^{\varkappa +p}}}{\int }_0^{\tau }{\displaystyle \frac{1}{t_1}}{\int }_0^{t_1}{\displaystyle \frac{1}{t_2}}{\int }_0^{t_2}\cdots {\displaystyle \frac{1}{t_{\kappa -1}}}{\int }_0^{t_{\kappa -1}}{t}_{\kappa }^{\varkappa +p-1}\mathfrak{f}\left(t_{\kappa }\right)d{t}_{\kappa }d{t}_{\kappa -1}\dots d{t}_{2}d{t}_1.$$

In order to introduce the new operator, another intermediate tool is needed. The following function is developed:

$${\Psi }_p(\upsilon ,\mathfrak{s};\tau )={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{\displaystyle \frac{{\left(\upsilon \right)}_{\kappa +p}}{{\left(\mathfrak{s}\right)}_{\kappa +p}}}{\tau }^{\kappa }(\upsilon \in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \left\{0\right\};\mathfrak{s}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};\tau \in {\mathfrak{D}}^{\ast }),$$

(3)

where ${\left(\zeta \right)}_{\kappa }$ is the Pochhammer symbol defined as follows:

$${\left(\zeta \right)}_{\kappa }={\displaystyle \frac{\Gamma (\zeta +\kappa )}{\Gamma \left(\zeta \right)}}=\left\{\begin{array}{cc}1& (\kappa =0)\\ \zeta (\zeta +1)\dots (\zeta +\kappa +1)& (\kappa \in \mathbb{N}).\end{array}\right.$$

It can be seen that

$${\Psi }_p(\upsilon ,\mathfrak{s};\tau )={\displaystyle \frac{1}{{\tau }^p}}{}_{2}F_1(\upsilon ,1;\mathfrak{s};\tau ),$$

where

$${}_{2}F_1(\upsilon ,v;\mathfrak{s};\tau )=\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\left(\upsilon \right)}_{\kappa }{\left(v\right)}_{\kappa }}{{\left(\mathfrak{s}\right)}_{\kappa }{\left(1\right)}_{\kappa }}}{\tau }^{\kappa },\upsilon ,v,\mathfrak{s}\in \mathbb{C}\mathrm{and}\varkappa ,\mathfrak{s}\notin {\mathbb{Z}}_0^{-};\tau \in \mathfrak{D})$$

is the Gaussian hypergeometric function.

Using the definition for the convolution of meromorphic multivalent functions and the definition of the operator ${\mathcal{L}}_{\varkappa ,p}^{\varrho }\mathfrak{f}\left(\tau \right)$ given in (2), by letting

$${\mathcal{L}}_{\varkappa ,p}^{\varrho }\left(\tau \right)\ast {\mathfrak{k}}_{\varkappa ,p}^{\varrho }\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p(1-\tau )}},$$

we have

$${\mathfrak{k}}_{\varkappa ,p}^{\varrho }\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{\left({\displaystyle \frac{\kappa +\varkappa +p}{\varkappa }}\right)}^{\varrho }{\tau }^{\kappa }.$$

As an application of the operator ${\mathfrak{k}}_{\varkappa ,p}^{\varrho }\left(\tau \right)$, a new operator ${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )$ is introduced by

$${\mathfrak{k}}_{\varkappa ,p}^{\varrho }\left(\tau \right)\ast {\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )={\Psi }_p(\upsilon ,\mathfrak{s};\tau )(\tau \in {\mathfrak{D}}^{\ast }).$$

(4)

The linear operator ${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau ):{\Sigma }_p\to {\Sigma }_p,$ is defined by

$${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)={\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\ast \mathfrak{f}\left(\tau \right)(\varkappa ,\mathfrak{s}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};\upsilon \in {\mathbb{C}}^{\ast };\varrho \in \mathbb{C},\tau \in {\mathfrak{D}}^{\ast }),$$

where the series expansion for $\varkappa ,\mathfrak{s}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};\upsilon \in {\mathbb{C}}^{\ast };\varrho \in \mathbb{C};\tau \in {\mathfrak{D}}^{\ast }$ with $\mathfrak{f}$ given by (1) is expressed as

$${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p}}+\sum _{\kappa =1-p}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +p}}\right)}^{\varrho }{\displaystyle \frac{{\left(\upsilon \right)}_{\kappa +p}}{{\left(\mathfrak{s}\right)}_{\kappa +p}}}{a}_{\kappa }{\tau }^{\kappa }.$$

(5)

The operator ${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )$ satisfies

$$\tau {\left({\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=\varkappa {\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-(\varkappa +p){\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)$$

(6)

and

$$\tau {\left({\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=\upsilon {\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-(\upsilon +p){\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\upsilon \in \mathbb{C}\setminus \{-1\}).$$

(7)

We note that

(i)

${\mathfrak{k}}_{\varkappa ,p}^1(1,1;\tau )\mathfrak{f}\left(\tau \right)=F_{\varkappa }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{\varkappa }{{\tau }^{\varkappa +p}}}\underset{0}{\overset{\tau }{\int }}{t}^{\varkappa +p+1}\mathfrak{f}\left(t\right)\varkappa t,\left(\varkappa >0\right)$ (see Miller and Mocanu [22], p. 389);

(ii)

${\mathfrak{k}}_{1,p}^{\varrho }(1,1;\tau )\mathfrak{f}\left(\tau \right)=P^{\varrho }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{{\tau }^p\Gamma \left(\varrho \right)}}\underset{0}{\overset{\tau }{\int }}{\left(log{\displaystyle \frac{\tau }{t}}\right)}^{\varrho -1}{t}^p\mathfrak{f}\left(t\right)\varkappa t$, $\left(\varrho >0\right)$ (see Aqlan et al. [23]);

(iii)

${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(1,1;\tau )\mathfrak{f}\left(\tau \right)=J_{\varkappa ,p}^{\varrho }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{{\alpha }^{\varrho }}{{\tau }^{\varkappa +p}\Gamma \left(\varrho \right)}}\underset{0}{\overset{\tau }{\int }}{\left(log{\displaystyle \frac{\tau }{t}}\right)}^{\varrho -1}{t}^{\varkappa +p-1}\mathfrak{f}\left(t\right)\varkappa t,\left(\varkappa ,\varrho >0\right)$ (see El-Ashwah and Aouf [24]);

(iv)

${\mathfrak{k}}_{\varkappa ,1}^{\varrho }(1,1;\tau )\mathfrak{f}\left(\tau \right)={\mathcal{L}}_{\varkappa }^{\varrho }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{\tau }}+{\displaystyle \sum _{\kappa =0}^{\infty }}{\left({\displaystyle \frac{\varkappa }{\kappa +1+\varkappa }}\right)}^{\varrho }{a}_{\kappa }{\tau }^{\kappa },\left(\varkappa \in {\mathbb{C}}^{\ast },\varrho \in \mathbb{C}\right)$ (see El-Ashwah [25]).

The following notion is well known in third-order differential subordination theory.

Definition 1

([6], p. 441, Definition 2). Denote by $\mathfrak{Q}$ the set of all functions $\mathfrak{q}$ that are analytic and injective on $\overline{\mathfrak{D}}\setminus E\left(\mathfrak{q}\right)$ where

$$E\left(\mathfrak{q}\right)=\left\{\xi \in \partial \mathfrak{D}:\underset{\tau \to \xi }{lim}\mathfrak{q}\left(\tau \right)=\infty \right\}$$

and that are such that $min\left|{\mathfrak{q}}^{\prime }\left(\xi \right)\right|=\rho >0$ for $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$. $E\left(\mathfrak{q}\right)$ is called the exception set, and the subclass of $\mathfrak{Q}$ when $\mathfrak{q}\left(0\right)=a$ is denoted by $\mathfrak{Q}\left(a\right)$.

Intended to generalize the classical concept of subordination, fuzzy subordination was initiated using the following definitions in [2]:

Definition 2

([2]). A fuzzy subset of X is a pair $\left(\upsilon ,{\mathcal{F}}_{\upsilon }\right)$, where ${\mathcal{F}}_{\upsilon }:X\to \left[0,1\right]$ is known as the membership function of the fuzzy set $\left(\upsilon ,{\mathcal{F}}_{\upsilon }\right)$ and $\upsilon =\left\{\mathrm{Y}\in X:0<{\mathcal{F}}_{\upsilon }\left(\mathrm{Y}\right)\le 1\right\}=\mathit{supp}\left(\upsilon ,{\mathcal{F}}_{\upsilon }\right)$ is called the support of the fuzzy subset $\left(\upsilon ,{\mathcal{F}}_{\upsilon }\right)$.

Let $\mathfrak{f},\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right).$

$$\mathfrak{f}\left(\mathfrak{D}\right)=\mathrm{supp}(\mathfrak{f}\left(\mathfrak{D}\right),{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{D}\right)})=\{\mathfrak{f}\left(\tau \right):0<\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le 1,\tau \in \mathfrak{D}\},$$

(8)

and

$$\mathfrak{g}\left(\mathfrak{D}\right)=\mathrm{supp}(\mathfrak{g}\left(\mathfrak{D}\right),{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{D}\right)})=\{\mathfrak{g}\left(\tau \right):0<\left|{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{D}\right)}\left(\mathfrak{g}\left(\tau \right)\right)\right|\le 1,\tau \in \mathfrak{D}\}.$$

(9)

Definition 3

([2]). Let ${\tau }_0\in \mathfrak{D}$ and $\mathfrak{f}$, $\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right).$ $\mathfrak{f}$ is fuzzy subordinate to $\mathfrak{g}$ and written as $\mathfrak{f}{\prec }_{\mathcal{F}}\mathfrak{g}\mathit{or}\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{g}\left(\tau \right)$ if

$$\mathfrak{f}\left({\tau }_0\right)=\mathfrak{g}\left({\tau }_0\right)\mathit{and}\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{D}\right)}\left(\mathfrak{g}\left(\tau \right)\right)\right|,\tau \in \mathfrak{D}.$$

Proposition 1

([2]). Let ${\tau }_0\in \mathfrak{D}$ and $\mathfrak{f}$, $\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right).$ If $\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{g}\left(\tau \right),\tau \in \mathfrak{D},$ then

$$\begin{array}{ccc}\hfill \left(\mathfrak{i}\right)\mathfrak{f}\left({\tau }_0\right)& =& \mathfrak{g}\left({\tau }_0\right);\hfill \\ \hfill \left(\mathfrak{ii}\right)\mathfrak{f}\left(\mathfrak{D}\right)& \subseteq & \mathfrak{g}\left(\mathfrak{D}\right)\mathit{and}\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{D}\right)}\left(\mathfrak{g}\left(\tau \right)\right)\right|,\tau \in \mathfrak{D},\hfill \end{array}$$

where $\mathfrak{f}\left(\mathfrak{D}\right)$ and $\mathfrak{g}\left(\mathfrak{D}\right)$ are given by (8) and (9), respectively.

Next, the class of admissible functions is defined as follows:

Definition 4

([6]). Let $\Omega \subset \mathbb{C}$ be a function, $\mathfrak{q}\in \mathfrak{Q}$ and $\mathit{\ell }\ge 2$. Functions $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{D}}\to \mathbb{C},$ are called admissible functions and belong to the so-called class of admissible functions denoted by ${\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ if the admissibility condition

$$\psi (r,s,t,u;\xi )\notin \Omega ,$$

is satisfied when

$$r=\mathfrak{q}\left(\tau \right),s={\displaystyle \frac{\tau {\mathfrak{q}}^{\prime }\left(\tau \right)}{\mathit{\ell }}},$$

$$Re\left\{{\displaystyle \frac{t}{s}}+1\right\}\ge \mathit{\ell }Re\left\{1+{\displaystyle \frac{\tau {\mathfrak{q}}^{\prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

$$Re\left\{{\displaystyle \frac{u}{s}}\right\}\ge {\mathit{\ell }}^{2}Re\left\{{\displaystyle \frac{{\tau }^2{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

where $\tau \in \mathfrak{D}$, $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$ and $\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}.$

Further, the notions of fuzzy best dominant and subordinant are recalled, introduced as extensions of the well-known notions of best dominant and best subordinant from the classical theories of classical differential subordination and superordination, respectively.

Definition 5

([11], Definition 5). Let Ω be a set in $\mathbb{C}$, $\mathfrak{q}\in \mathfrak{Q},\mathit{\ell }\ge 2.$ Denote by ${\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ the set of functions $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{D}}\to \mathbb{C},$ satisfying the condition

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \overline{\mathfrak{D}})}\psi (r,s,t,u;\xi )<{\mathcal{F}}_{\Omega }\left(\tau \right),0<{\mathcal{F}}_{\Omega }\left(\tau \right)\le 1,$$

(10)

where

$$r=\mathfrak{q}\left(\tau \right),s={\displaystyle \frac{\tau {\mathfrak{q}}^{\prime }\left(\tau \right)}{\mathit{\ell }}},$$

$$Re\left\{{\displaystyle \frac{t}{s}}+1\right\}\le {\displaystyle \frac{1}{\mathit{\ell }}}Re\left\{1+{\displaystyle \frac{\tau {\mathfrak{q}}^{\prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

$$Re\left\{{\displaystyle \frac{u}{s}}\right\}\le {\displaystyle \frac{1}{{\mathit{\ell }}^2}}Re\left\{{\displaystyle \frac{{\tau }^2{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

where $\tau \in \mathfrak{D}$, $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$ and $\mathit{\ell }\ge \kappa \ge 2$. Condition (10) is called the admissibility condition.

Definition 6

([1], Definition 3.2). Consider $\psi :{\mathbb{C}}^4\times \mathfrak{D}⟶\mathbb{C}$, $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ and let $\lambda \in \mathcal{H}\left(\mathfrak{D}\right).$ A function $\omega \in \mathcal{H}\left(\mathfrak{D}\right)$ is called a solution for a third-order fuzzy differential subordination if it satisfies

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right),{\tau }^3{\omega }^{\prime \prime \prime }\left(\tau \right);\tau )\right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),$$

i.e.,

$$\psi (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right),{\tau }^3{\omega }^{\prime \prime \prime }\left(\tau \right);\tau ){\prec }_{\mathcal{F}}\lambda \left(\tau \right).$$

(11)

A function $\mathfrak{q}\in \mathcal{H}\left(\mathfrak{D}\right)$ is said to be a dominant of the solutions of a third-order fuzzy differential subordination if it satisfies the fuzzy subordination $\omega {\prec }_{\mathcal{F}}\mathfrak{q}$ for all solutions $\omega$ of the third-order fuzzy differential subordination (11). The fuzzy best dominant is a dominant $\tilde{\mathfrak{q}}$ that satisfies the fuzzy subordination $\tilde{\mathfrak{q}}{\prec }_{\mathcal{F}}\mathfrak{q}$ for all dominants $\mathfrak{q}$ of (11).

Definition 7

([11], Definition 6). $\left(i\right)$ Consider $\psi :{\mathbb{C}}^4\times \mathfrak{D}⟶\mathbb{C}$, $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ and let $\lambda \in \mathcal{H}\left(\mathfrak{D}\right).$ A function $\omega \in \mathcal{H}\left(\mathfrak{D}\right)$ satisfying

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right),{\tau }^3{\omega }^{\prime \prime \prime }\left(\tau \right);\tau )\right),\tau \in \mathfrak{D},$$

i.e.,

$$\lambda \left(\tau \right){\prec }_{\mathcal{F}}\psi (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right),{\tau }^3{\omega }^{\prime \prime \prime }\left(\tau \right);\tau ),$$

(12)

is called the solution of fuzzy differential superordination (12).

$\left(ii\right)$ Fuzzy differential superordination (12) is called third-order fuzzy differential superordination and a univalent function $\mathfrak{q}$ is called a fuzzy subordinant of the third-order fuzzy differential superordination (12) if $\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}\omega \left(\tau \right)$, or ${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)\le {\mathcal{F}}_{\omega \left(\mathfrak{D}\right)}\left(\omega \left(\tau \right)\right),\tau \in \mathfrak{D}.$

$\left(iii\right)$ A fuzzy subordinant $\tilde{\mathfrak{q}}$ that is univalent and satisfies $\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}\tilde{\mathfrak{q}}\left(\tau \right)$ or is equivalently written ${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)\le {\mathcal{F}}_{\tilde{\mathfrak{q}}\left(\mathfrak{D}\right)}\left(\tilde{\mathfrak{q}}\left(\tau \right)\right),\tau \in \mathfrak{D}$ for all fuzzy subordinants $\mathfrak{q}$ of the third-order fuzzy differential superordination (12) is called the fuzzy best subordinant of the third-order fuzzy differential superordination (12).

The following established results will be used in the proof of the original results contained in the following two sections.

Lemma 1

([1], Theorem 3.4). Let $p\in \mathcal{H}[a,\kappa ]$ with $\kappa \ge 2$. Consider $\mathcal{F}:\mathbb{C}\to [0,1]$ and a function $\mathfrak{q}\in \mathfrak{Q}\left(a\right)$ for which

$$Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|{\displaystyle \frac{\tau p^{\prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right|\le \mathit{\ell },$$

where $\tau \in \mathfrak{D}$, $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$ and $\mathit{\ell }\ge \kappa$. If Ω is a set in $\mathbb{C}$, $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ satisfying

$$\left\{\psi (p\left(\tau \right),\tau p^{\prime }\left(\tau \right),{\tau }^2{p}^{\prime \prime }\left(\tau \right),{\tau }^3{p}^{\prime \prime \prime }\left(\tau \right);\tau )\right\}\subset \Omega ,$$

then

$$p\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right)\mathit{or}{\mathcal{F}}_{p\left(\mathfrak{D}\right)}p\left(\tau \right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right).$$

Lemma 2

([11], Theorem 1). Let $\Omega \in \mathbb{C},\mathfrak{q}\in \mathcal{H}[a,\kappa ],\kappa \ge 2$, a function $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}],\mathcal{F}:\mathbb{C}\to [0,1]$ given by $F\left(\tau \right)={\displaystyle \frac{1}{1+\left|\tau \right|}}$, and $p\in \mathfrak{Q}\left(a\right),p\left(0\right)=\mathfrak{q}\left(0\right)=a,$ satisfying

$$Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|{\displaystyle \frac{\tau p^{\prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right|\le \mathit{\ell },$$

where $\tau \in \mathfrak{D}$, $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$, and $\mathit{\ell }\ge \kappa \ge 2$; if $p\left(\tau \right)$ and $\psi (p\left(\tau \right),\tau p^{\prime }\left(\tau \right),{\tau }^2{p}^{\prime \prime }\left(\tau \right),{\tau }^3{p}^{\prime \prime \prime }\left(\tau \right);\tau )$ are univalent in $\mathfrak{D}$, then

$$\Omega \subset \left\{\psi (p\left(\tau \right),\tau p^{\prime }\left(\tau \right),{\tau }^2{p}^{\prime \prime }\left(\tau \right),{\tau }^3{p}^{\prime \prime \prime }\left(\tau \right);\tau ):\tau \in \mathfrak{D}\right\},$$

or equivalently,

$${\displaystyle \frac{1}{1+\left|\tau \right|}}\le {\displaystyle \frac{1}{1+\left|\psi (p\left(\tau \right),\tau p^{\prime }\left(\tau \right),{\tau }^2{p}^{\prime \prime }\left(\tau \right),{\tau }^3{p}^{\prime \prime \prime }\left(\tau \right);\tau \right|}},$$

implying that

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}p\left(\tau \right)\mathit{or}{\displaystyle \frac{1}{1+\left|\mathfrak{q}\left(\tau \right)\right|}}\le {\displaystyle \frac{1}{1+\left|p\left(\tau \right)\right|}}\left(\tau \in \mathfrak{D}\right).$$

With this research context established, the following two sections describe the new third-order fuzzy differential subordination and superordination results, respectively.

3. Results

3.1. Third-Order Fuzzy Differential Subordination Advancements

Throughout the study, unless otherwise indicated, we will assume that $\varkappa ,\mathfrak{s}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};\upsilon \in {\mathbb{C}}^{\ast }$, and $\varrho \in \mathbb{C}$.

This section presents new results on third-order fuzzy differential subordination. The following definition of the class of admissible functions is used for this purpose:

Definition 8.

Let $\Omega \subset \mathbb{C}$ and $\mathfrak{q}\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$. The functions $\psi :{\mathbb{C}}^4\times \mathfrak{D}\to \mathbb{C}$ belong to the class of admissible functions ${\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ if the admissibility condition

$$\psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)\notin \Omega$$

(13)

is satisfied whenever

$$\alpha =\mathfrak{q}\left(\xi \right),\beta ={\displaystyle \frac{\mathit{\ell }\xi {\mathfrak{q}}^{\prime }\left(\xi \right)+\varkappa \mathfrak{q}\left(\xi \right)}{\varkappa }},$$

(14)

$$Re\left\{{\displaystyle \frac{\varkappa \left(\gamma +\alpha -2\beta \right)}{\beta -\alpha }}\right\}\ge \mathit{\ell }Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}+1\right\},$$

(15)

$$Re\left\{{\displaystyle \frac{1}{\beta -\alpha }}\left[{\varkappa }^2\left(\delta -\alpha \right)+3\varkappa \left(\varkappa +1\right)\left(2\beta -\alpha -\gamma \right)\right]-3{\varkappa }^2+2\right\}\ge {\mathit{\ell }}^{2}Re\left\{{\displaystyle \frac{{\xi }^2{\mathfrak{q}}^{\prime \prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\},$$

(16)

where $\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$, and $\tau \in \mathfrak{D}$.

The first new finding is stated and proved as Theorem 1 below. The third-order fuzzy dominant of certain third-order fuzzy differential subordinations involving the operator presented in Relation (5) applied to meromorphic multivalent functions can be obtained using the following theorem.

Theorem 1.

Let $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in {\Sigma }_p$ and $\mathfrak{q}\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le \mathit{\ell }\left|{\displaystyle \frac{{\mathfrak{q}}^{\prime }\left(\xi \right)}{\varkappa }}\right|,$$

(17)

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$$\left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\le {\mathcal{F}}_{\Omega }\left(\tau \right),$$

(18)

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)\mathit{and}\tau \in \mathfrak{D},$$

which implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right)(\tau \in \mathfrak{D}).$$

(19)

or

$${\mathcal{F}}_{{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\mathfrak{D}\right)}\left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right).$$

Proof. 

Consider $\mathfrak{g}\left(\tau \right)\in \mathcal{H}\left(\mathfrak{D}\right)$ as follows:

$$\mathfrak{g}\left(\tau \right)={\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right).$$

(20)

By differentiating (20) and using the property of the operator ${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )$ given in (6), we obtain

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )={\displaystyle \frac{\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\varkappa \mathfrak{g}\left(\tau \right)}{\varkappa }}.$$

(21)

Similarly, by differentiating (21) and using the property seen in (6) alongside Relation (20), we write

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)={\displaystyle \frac{{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right)+\left(1+2\varkappa \right)\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+{\varkappa }^2\mathfrak{g}\left(\tau \right)}{{\varkappa }^2}}.$$

(22)

Further, by differentiating (22) and using the property seen in (6) together with the previously obtained Relations (21) and (22), we obtain

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)={\displaystyle \frac{{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right)+3\left[1+\varkappa \right]{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right)+\left[3{\varkappa }^2+3\varkappa +1\right]\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+{\varkappa }^3\mathfrak{g}\left(\tau \right)}{{\varkappa }^3}}.$$

(23)

Next, the transformation from ${\mathbb{C}}^4$ to $\mathbb{C}$ is expressed as

$$\alpha (t,u,v,w)=t,\beta (t,u,v,w)={\displaystyle \frac{u+\varkappa t}{\varkappa }},$$

(24)

$$\gamma (t,u,v,w)={\displaystyle \frac{v+\left(1+2\varkappa \right)u+{\varkappa }^{2}t}{{\varkappa }^2}},$$

(25)

and

$$\delta (t,u,v,w)={\displaystyle \frac{w+3\left(1+\varkappa \right)v+\left(3{\varkappa }^2+3\varkappa +1\right)u+{\varkappa }^{3}t}{{\varkappa }^3}}.$$

(26)

Let

$$\begin{array}{ccc}\hfill \psi \left(t,u,v,w\right)& =& \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)\hfill \\ & =& \psi \left(t,\frac{u+\varkappa t}{\varkappa },\frac{v+\left(1+2\varkappa \right)u+{\varkappa }^{2}t}{{\varkappa }^2},\frac{w+3\left(1+\varkappa \right)v+\left(3{\varkappa }^2+3\varkappa +1\right)u+{\varkappa }^{3}t}{{\varkappa }^3};\tau \right)\hfill \end{array}$$

(27)

Using Lemma 1 and (20)–(27), we have

$$\psi \left(\mathfrak{g}\left(\tau \right),\tau {\mathfrak{g}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right);\tau \right)$$

$$=\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right).$$

(28)

Hence, (18) leads to

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi \left(\mathfrak{g}\left(\tau \right),\tau {\mathfrak{g}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right);\tau \right)\right)\le {\mathcal{F}}_{\Omega }\left(\tau \right).$$

(29)

Additionally, applying (24)–(26), we obtain

$${\displaystyle \frac{v}{u}}+1=\varkappa \left(\frac{\gamma +\alpha -2\beta }{\beta -\alpha }\right),$$

(30)

and

$${\displaystyle \frac{w}{u}}={\displaystyle \frac{1}{\beta -\alpha }}\left[{\varkappa }^2\left(\delta -\alpha \right)+3\varkappa \left(\varkappa +1\right)\left(2\beta -\alpha -\gamma \right)\right]-3{\varkappa }^2+2.$$

(31)

Thus, for $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$, the admissibility condition given by Definition 8 is equivalent to the one given for $\psi \in {\Psi }_{\upsilon }[\Omega ,\mathfrak{q}]$ in Definition 4. Consequently, by utilizing (17) and Lemma 1, we deduce ${\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{D}\right)}\mathfrak{g}\left(\tau \right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)$ or, equivalently,

$${\mathcal{F}}_{{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\mathfrak{D}\right)}\left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right),$$

i.e.,

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right).$$

Hence, the proof is completed. □

Example 1.

By taking $\psi =1+{\displaystyle \frac{\beta }{\alpha }},$ $\varrho =-1$ and $p=\upsilon =\mathfrak{s}=1$ in Theorem 1, we obtain that

$$\begin{array}{ccc}\hfill \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)& =& 2+{\displaystyle \frac{1}{\varkappa }}\left({\displaystyle \frac{\tau {\mathfrak{g}}^{\prime }\left(\tau \right)}{\mathfrak{g}\left(\tau \right)}}\right)\hfill \\ & =& 2+{\displaystyle \frac{1}{\varkappa }}\left({\displaystyle \frac{{\left(\tau \mathfrak{f}\left(\tau \right)\right)}^{\prime }}{\mathfrak{f}\left(\tau \right)}}\right)\hfill \\ & =& 2+{\displaystyle \frac{1}{\varkappa }}\left({\displaystyle \frac{\tau {\mathfrak{f}}^{\prime }\left(\tau \right)}{\mathfrak{f}\left(\tau \right)}}+1\right)\hfill \end{array}$$

is analytic in $\mathfrak{D}$ and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left(2+{\displaystyle \frac{1}{\varkappa }}\left({\displaystyle \frac{\tau {\mathfrak{f}}^{\prime }\left(\tau \right)}{\mathfrak{f}\left(\tau \right)}}+1\right)\right)\le {\mathcal{F}}_{\Omega }\left(\tau \right),$$

so

$$\tau \mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right)(\tau \in \mathfrak{D}).$$

or

$${\mathcal{F}}_{\tau \mathfrak{f}\left(\mathfrak{D}\right)}\left(\tau \mathfrak{f}\left(\tau \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right).$$

When the behavior of $\mathfrak{q}$ on $\partial \mathfrak{D}$ is unknown, Theorem 1 immediately leads to the following result.

Corollary 1.

Let $\Omega \subset \mathbb{C}$ and let $\mathfrak{q}\in S$ with $\mathfrak{q}\left(0\right)=1$. Let $\psi \in {\Psi }_{\kappa }[\Omega ,{\mathfrak{q}}_{\rho }]$ for certain $\rho \in (0,1)$ when ${\mathfrak{q}}_{\rho }\left(\tau \right)=\mathfrak{q}\left(\rho \tau \right)$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\mathfrak{q}}_{\rho }$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {{\mathfrak{q}}_{\rho }}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le \mathit{\ell }\left|{\displaystyle \frac{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}{\varkappa }}\right|,$$

(32)

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$$\left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\le {\mathcal{F}}_{\Omega }\left(\tau \right)$$

implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right),\mathit{and}\tau \in \mathfrak{D}.$$

Proof. 

Following Theorem 1, we obtain

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}{\mathfrak{q}}_{\rho }\left(\tau \right).$$

Now, the following fuzzy subordination property can be used in order to conclude the proof of Corollary 1:

$${\mathfrak{q}}_{\rho }\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right).$$

This completes the proof of Corollary 1. □

When $\Omega \ne \mathbb{C}$ is a simply connected domain, with $\Omega =\lambda \left(\mathfrak{D}\right)$ for a certain conformal mapping $\lambda \left(\tau \right)$ of $\mathfrak{D}$ onto $\Omega$, the notation ${\Psi }_{\kappa }[\lambda ,\mathfrak{q}]$ designates the class ${\Psi }_{\kappa }[\mathcal{H}\left(\mathfrak{D}\right)]$. The following two results are direct outcomes of Theorem 1 and Corollary 1.

Theorem 2.

Let $\psi \in {\Psi }_{\kappa }[\lambda ,\mathfrak{q}]$. If $\mathfrak{f}\in {\Sigma }_p$ and $\mathfrak{q}\in {\mathfrak{Q}}_1$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le \mathit{\ell }\left|{\displaystyle \frac{{\mathfrak{q}}^{\prime }\left(\xi \right)}{\varkappa }}\right|,$$

(33)

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$$\left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)$$

(34)

implies

$${\mathcal{F}}_{{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\mathfrak{D}\right)}\left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right)\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right).$$

In other words,

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\lambda \left(\tau \right),$$

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right),\mathit{and}\tau \in \mathfrak{D}.$$

Corollary 2.

Let $\Omega \subset \mathbb{C}$ and let $\mathfrak{q}\in S$ with $\mathfrak{q}\left(0\right)=1$. Let $\psi \in {\Psi }_{\kappa }[\lambda ,{\mathfrak{q}}_{\rho }]$ for certain $\rho \in (0,1)$ when ${\mathfrak{q}}_{\rho }\left(\tau \right)=\mathfrak{q}\left(\rho \tau \right)$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\mathfrak{q}}_{\rho }$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {{\mathfrak{q}}_{\rho }}^{″}\left(\xi \right)}{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le \mathit{\ell }\left|{\displaystyle \frac{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}{\varkappa }}\right|,$$

(35)

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$$\left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)$$

implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)\mathit{and}\tau \in \mathfrak{D}.$$

The fuzzy best dominant of the fuzzy differential subordination (18) or (34) is provided by our next proved result.

Theorem 3.

Let $\lambda \left(\tau \right)\in S$. Also, let $\psi :{\mathbb{C}}^4\times \mathfrak{D}⟶\mathbb{C}$ with ψ given by (27). Consider the differential equation

$$\psi \left(\mathfrak{q}\left(\tau \right),\tau {\mathfrak{q}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{q}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right);\tau \right)=\lambda \left(\tau \right)$$

(36)

having a solution $\mathfrak{q}\left(\tau \right)\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$ that satisfies (17). If $\mathfrak{f}\in {\Sigma }_p$ satisfies (32) and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \{{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),$$

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \}\le {\mathcal{F}}_{\Omega }\left(\tau \right),$$

then

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

with $\mathfrak{q}\left(\tau \right)$ as the fuzzy best dominant.

Proof. 

By applying Theorem 1, we have that $\mathfrak{q}$ is a dominant of (34) for which it is also a solution since (36) is satisfied by $\mathfrak{q}$. As a result, $\mathfrak{q}$ is the fuzzy best dominant. □

A new admissible functions class ${\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$ is introduced for developing the new results proved in the following theorem:

Definition 9.

Let $\Omega \subset \mathbb{C}$ and $\mathfrak{q}\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$. The functions $\psi :{\mathbb{C}}^4\times \mathfrak{D}\to \mathbb{C}$ belong to class of admissible functions ${\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$ if the admissibility condition

$$\psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)\notin \Omega ,$$

is satisfied whenever

$$\alpha =\mathfrak{q}\left(\xi \right),\beta ={\displaystyle \frac{\mathit{\ell }\xi {\mathfrak{q}}^{\prime }\left(\xi \right)+\upsilon \mathfrak{q}\left(\xi \right)}{\upsilon }},$$

$$Re\left\{\left(\upsilon +1\right)\left({\displaystyle \frac{\gamma +\alpha -2\beta }{\beta -\alpha }}\right)\right\}\ge \mathit{\ell }Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}+1\right\},$$

$$Re\left\{\left(\upsilon +1\right)\left(\upsilon +2\right)\left({\displaystyle \frac{\delta -\alpha +3\beta -3\gamma }{\beta -\alpha }}\right)\right\}\ge {\mathit{\ell }}^{2}Re\left\{{\displaystyle \frac{{\xi }^2{\mathfrak{q}}^{\prime \prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\},$$

where $\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)$, and $\tau \in \mathfrak{D}$.

Theorem 4.

Let $\psi \in {\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in {\Sigma }_p$ and $\mathfrak{q}\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le {\displaystyle \frac{\mathit{\ell }}{\upsilon }}\left|{\mathfrak{q}}^{\prime }\left(\xi \right)\right|,$$

(37)

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left(\left(\right({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau ):\tau \in \mathfrak{D}))\le {\mathcal{F}}_{\Omega }\left(\tau \right),$$

(38)

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right),\mathit{and}\tau \in \mathfrak{D},$$

implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right)(\tau \in \mathfrak{D}).$$

Proof. 

Let

$$\mathfrak{g}\left(\tau \right)={\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right).$$

(39)

By differentiating (39) and using the property of the operator ${\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )$ given in (7), we write

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)={\displaystyle \frac{\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\upsilon \mathfrak{g}\left(\tau \right)}{\upsilon }}.$$

Further, by differentiating this last equation and using again the property seen in (7), we obtain

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)={\displaystyle \frac{{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right)+2\left(\upsilon +1\right)\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\upsilon \left(\upsilon +1\right)\mathfrak{g}\left(\tau \right)}{\upsilon \left(\upsilon +1\right)}}.$$

By differentiating now this last relation and using again the property given by (7), we have

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)=\frac{{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right)+3\left(\upsilon +2\right){\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right)+3\left(\upsilon +1\right)\left(\upsilon +2\right)\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\upsilon \left(\upsilon +1\right)\left(\upsilon +2\right)\mathfrak{g}\left(\tau \right)}{\upsilon \left(\upsilon +1\right)\left(\upsilon +2\right)}.$$

(40)

Further, the transformation from ${\mathbb{C}}^4$ to $\mathbb{C}$ is expressed as

$$\alpha (t,u,v,w)=t,\beta (t,u,v,w)={\displaystyle \frac{u+\upsilon t}{\upsilon }},$$

(41)

$$\gamma (t,u,v,w)={\displaystyle \frac{v+2\left(\upsilon +1\right)u+\upsilon \left(\upsilon +1\right)t}{\upsilon \left(\upsilon +1\right)}},$$

and

$$\delta (t,u,v,w)={\displaystyle \frac{w+3\left(\upsilon +2\right)v+3\left(\upsilon +1\right)\left(\upsilon +2\right)u+\upsilon \left(\upsilon +1\right)\left(\upsilon +2\right)t}{\upsilon \left(\upsilon +1\right)\left(\upsilon +2\right)}}.$$

Let

$$\psi \left(t,u,v,w;\tau \right)=\psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)$$

$$=\psi \left(t,\frac{u+\upsilon t}{\upsilon },\frac{v+2\left(\upsilon +1\right)u+\upsilon \left(\upsilon +1\right)t}{\upsilon \left(\upsilon +1\right)},\frac{w+3\left(\upsilon +2\right)v+3\left(\upsilon +1\right)\left(\upsilon +2\right)u+\upsilon \left(\upsilon +1\right)\left(\upsilon +2\right)t}{\upsilon \left(\upsilon +1\right)\left(\upsilon +2\right)};\tau \right).$$

(42)

Using Lemma 1, (39)–(42), we have

$$\psi \left(\mathfrak{g}\left(\tau \right),\tau {\mathfrak{g}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right);\tau \right)=$$

$$\psi ({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau ).$$

(43)

Hence, (43) leads to

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi \mathfrak{g}\left(\tau \right),\tau {\mathfrak{g}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right);\tau \right)\le {\mathcal{F}}_{\Omega }\left(\tau \right).$$

Using (41) and (42), we obtain

$${\displaystyle \frac{v}{u}}+1=\left(\upsilon +1\right)\left({\displaystyle \frac{\gamma +\alpha -2\beta }{\beta -\alpha }}\right),$$

and

$${\displaystyle \frac{w}{u}}=\left(\upsilon +1\right)\left(\upsilon +2\right)\left({\displaystyle \frac{\delta -\alpha +3\beta -3\gamma }{\beta -\alpha }}\right).$$

Thus, for $\psi \in {\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$, the admissibility condition given by Definition 9 is equivalent to the one given in Definition 4 for $\psi \in {\Psi }_{\upsilon }[\Omega ,\mathfrak{q}]$. Consequently, by utilizing (37) and Lemma 1, we obtain ${\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{D}\right)}\mathfrak{g}\left(\tau \right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)$ or, equivalently, ${\mathcal{F}}_{{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\mathfrak{D}\right)}\left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)$—i.e.,

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right).$$

This completes the proof of Theorem 4. □

Example 2.

By taking $\psi =1+{\displaystyle \frac{\beta }{\alpha }},$ $\varrho =0$, and $p=\upsilon =\mathfrak{s}=1$ in Theorem 4, we obtain that

$$\begin{array}{ccc}\hfill \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)& =& 1+\left({\displaystyle \frac{\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\mathfrak{g}\left(\tau \right)}{\mathfrak{g}\left(\tau \right)}}\right)\hfill \\ & =& 1+{\displaystyle \frac{\tau {\mathfrak{f}}^{\prime }\left(\tau \right)+2\mathfrak{f}\left(\tau \right)}{\mathfrak{f}\left(\tau \right)}}\hfill \\ & =& 3+{\displaystyle \frac{\tau {\mathfrak{f}}^{\prime }\left(\tau \right)}{\mathfrak{f}\left(\tau \right)}}\hfill \end{array}$$

is analytic in $\mathfrak{D}$ and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left(3+{\displaystyle \frac{\tau {\mathfrak{f}}^{\prime }\left(\tau \right)}{\mathfrak{f}\left(\tau \right)}}\right)\le {\mathcal{F}}_{\Omega }\left(\tau \right),$$

so

$$\tau \mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right)(\tau \in \mathfrak{D}).$$

or

$${\mathcal{F}}_{\tau \mathfrak{f}\left(\mathfrak{D}\right)}\left(\tau \mathfrak{f}\left(\tau \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right).$$

When the behavior of $\mathfrak{q}$ on $\partial \mathfrak{D}$ is unknown, Theorem 4 immediately leads to the following result.

Corollary 3.

Let $\Omega \subset \mathbb{C}$ and let $\mathfrak{q}\in S$ with $\mathfrak{q}\left(0\right)=1$. Let $\psi \in {\tilde{\Psi }}_{\kappa }[\Omega ,{\mathfrak{q}}_{\rho }]$ for certain $\rho \in (0,1)$ when ${\mathfrak{q}}_{\rho }\left(\tau \right)=\mathfrak{q}\left(\rho \tau \right)$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\mathfrak{q}}_{\rho }$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {{\mathfrak{q}}_{\rho }}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le {\displaystyle \frac{\mathit{\ell }}{\upsilon }}\left|{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)\right|,$$

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi ({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),$$

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau )\le {\mathcal{F}}_{\Omega }\left(\tau \right)$$

(44)

implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)\mathit{and}\tau \in \mathfrak{D}.$$

Proof. 

Following Theorem 4, we obtain

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}{\mathfrak{q}}_{\rho }\left(\tau \right).$$

Now, the following fuzzy subordination property can be used in order to conclude the proof of Corollary 3:

$${\mathfrak{q}}_{\rho }\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right).$$

The proof is now complete. □

When $\Omega \ne \mathbb{C}$ is a simply connected domain, with $\Omega =\lambda \left(\mathfrak{D}\right)$ for a certain conformal mapping $\lambda \left(\tau \right)$ of $\mathfrak{D}$ onto $\Omega$, the notation ${\tilde{\Psi }}_{\kappa }[\lambda ,\mathfrak{q}]$ designates the class ${\tilde{\Psi }}_{\kappa }[\lambda \left(\mathfrak{D}\right),\mathfrak{q}]$. In this case, Theorem 4 and Corollary 3 provide the following immediate consequences.

Theorem 5.

Let $\psi \in {\tilde{\Psi }}_{\kappa }[\lambda ,\mathfrak{q}]$. If $\mathfrak{f}\in {\Sigma }_p$ and $\mathfrak{q}\in {\mathfrak{Q}}_1$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le {\displaystyle \frac{\mathit{\ell }}{\upsilon }}\left|{\mathfrak{q}}^{\prime }\left(\xi \right)\right|,$$

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi ({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),$$

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau )\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)$$

(45)

implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right),\mathit{and}\tau \in \mathfrak{D}.$$

Corollary 4.

Let $\Omega \subset \mathbb{C}$ and let $\mathfrak{q}\in S$ with $\mathfrak{q}\left(0\right)=1$. Let $\psi \in {\tilde{\Psi }}_{\kappa }[\lambda ,{\mathfrak{q}}_{\rho }]$ for certain $\rho \in (0,1)$ when ${\mathfrak{q}}_{\rho }\left(\tau \right)=\mathfrak{q}\left(\rho \tau \right)$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\mathfrak{q}}_{\rho }$ satisfy

$$Re\left\{{\displaystyle \frac{\xi {{\mathfrak{q}}_{\rho }}^{\prime \prime }\left(\xi \right)}{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le {\displaystyle \frac{\mathit{\ell }}{\upsilon }}\left|{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)\right|,$$

then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi ({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),$$

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau )\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right),$$

(46)

implies

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

$$\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},\xi \in \partial \mathfrak{D}\setminus E\left(\mathfrak{q}\right)\mathit{and}\tau \in \mathfrak{D}.$$

The fuzzy best dominant of the fuzzy differential subordination given in (38) or (46) is provided by our next proved result.

Theorem 6.

Let $\lambda \left(\tau \right)\in S$ and $\psi :{\mathbb{C}}^4\times \mathfrak{D}⟶\mathbb{C}$ with ψ given by (42). Consider the differential equation

$$\psi (\mathfrak{q}\left(\tau \right),\tau {\mathfrak{q}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{q}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right);\tau )=\lambda \left(\tau \right)$$

(47)

having a solution $\mathfrak{q}\left(\tau \right)\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$, satisfying (37). If $\mathfrak{f}\in {\Sigma }_p$ satisfies Condition (44) and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \{{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),$$

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \}\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right),$$

then

$${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\tau \right),$$

with $\mathfrak{q}\left(\tau \right)$ as the fuzzy best dominant.

Proof. 

By applying the outcome of Theorem 4, we have that $\mathfrak{q}$ is a dominant of (45) for which it is also a solution since (47) is satisfied by $\mathfrak{q}$. As a result, $\mathfrak{q}$ is the fuzzy best dominant. □

3.2. Third-Order Fuzzy Differential Superordination Advancements

This section presents new results regarding third-order fuzzy differential superordination. The following definition of the class of admissible functions is used for this purpose:

Definition 10.

Let $\Omega \subset \mathbb{C}$, $\mathfrak{q}\in {\mathcal{H}}_1,{\mathfrak{q}}^{\prime }\left(\tau \right)\ne 0$, and $\vartheta \in \mathbb{N}\setminus \left\{1\right\}$. The functions $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{D}}\to \mathbb{C}$ belong to the class of admissible functions ${{\Psi }_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$ if the admissibility condition

$$\psi \left(\alpha ,\beta ,\gamma ,\delta ;\xi \right)\in \Omega$$

(48)

is satisfied whenever

$$\alpha =\mathfrak{q}\left(\tau \right),\beta ={\displaystyle \frac{\tau {\mathfrak{q}}^{\prime }\left(\tau \right)+\vartheta \varkappa \mathfrak{q}\left(\tau \right)}{\vartheta \varkappa }},$$

(49)

$$Re\left\{{\displaystyle \frac{\varkappa \left(\gamma +\alpha -2\beta \right)}{\beta -\alpha }}\right\}\le {\displaystyle \frac{1}{\vartheta }}Re\left\{1+{\displaystyle \frac{\tau {\mathfrak{q}}^{\prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

(50)

$$Re\left\{{\displaystyle \frac{1}{\beta -\alpha }}\left[{\varkappa }^2\left(\delta -\alpha \right)+3\varkappa \left(\varkappa +1\right)\left(2\beta -\alpha -\gamma \right)\right]-3{\varkappa }^2+2\right\}\le$$

$${\displaystyle \frac{1}{{\vartheta }^2}}Re\left\{{\displaystyle \frac{{\tau }^2{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

(51)

where $\vartheta \ge 2$, $\xi \in \partial \mathfrak{D}$, and $\tau \in \mathfrak{D}$.

Theorem 7.

Let $\psi \in {{\Psi }_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1$ satisfies

$$Re\left\{{\displaystyle \frac{\tau {\mathfrak{q}}^{\prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le \vartheta \left|{\displaystyle \frac{{\mathfrak{q}}^{\prime }\left(\tau \right)}{\varkappa }}\right|$$

(52)

and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)$$

is univalent in $\mathfrak{D}$, then

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\Omega }\left(\tau \right)& \le & {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right.,\hfill \\ & & \left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau :\tau \in \mathfrak{D}\right)\hfill \end{array}$$

(53)

implies

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

(54)

Proof. 

Let $\mathfrak{g}\left(\tau \right)$ be given by (20) and $\psi$ be given by (27). Since $\psi \in {{\Psi }_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}],$ (28) and (53) yield

$${\mathcal{F}}_{\Omega }\left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi \left(\mathfrak{g}\left(\tau \right),\tau {\mathfrak{g}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right);\tau \right):\tau \in \mathfrak{D}\right).$$

(55)

For $\psi \in {{\Psi }_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$, we conclude from (27) that the admissible condition seen in Definition 10 is equivalent to the one expressed in Definition 8 for $\psi$. Thus, by utilizing Lemma 2 and the conditions in (52), we establish that

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{g}\left(\tau \right),$$

(56)

or, equivalently,

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

(57)

Hence, the proof is completed. □

Example 3.

By taking $\psi =2\alpha -\beta ,$ $\varrho =-1$, and $p=\upsilon =\mathfrak{s}=1$ in Theorem 7, we obtain

If

$$\begin{array}{ccc}\hfill \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)& =& 2\mathfrak{g}\left(\tau \right)-{\displaystyle \frac{\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\varkappa \mathfrak{g}\left(\tau \right)}{\varkappa }}\hfill \\ & =& \mathfrak{g}\left(\tau \right)-{\displaystyle \frac{\tau {\mathfrak{g}}^{\prime }\left(\tau \right)}{\varkappa }}\hfill \\ & =& \tau \mathfrak{f}\left(\tau \right)-{\displaystyle \frac{\tau {\left(\tau \mathfrak{f}\left(\tau \right)\right)}^{\prime }}{\varkappa }}\hfill \end{array}$$

is analytic in $\mathfrak{U}$ and

$${\mathcal{F}}_{\Omega }\left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left(\tau \mathfrak{f}\left(\tau \right)-{\displaystyle \frac{{\tau }^2{\mathfrak{f}}^{\prime }\left(\tau \right)}{\varkappa }}-{\displaystyle \frac{\tau \mathfrak{f}\left(\tau \right)}{\varkappa }}\right),$$

then

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}\tau \mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

or

$${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)\le {\mathcal{F}}_{\tau \mathfrak{f}\left(\mathfrak{D}\right)}\left(\tau \mathfrak{f}\left(\tau \right)\right).$$

When $\Omega \ne \mathbb{C}$ is a simply connected domain with $\Omega =\lambda \left(\mathfrak{D}\right)$ for a certain conformal mapping $\lambda \left(\tau \right)$ of $\mathfrak{D}$ onto $\Omega$, the notation ${{\Psi }_{\kappa }}^{\prime }[\lambda ,\mathfrak{q}]$ designates the class ${{\Psi }_{\kappa }}^{\prime }[\lambda \left(\mathfrak{D}\right),\mathfrak{q}]$. Using the same techniques as in the previous section, Theorem 7 yields the following result.

Corollary 5.

Let $\psi \in {{\Psi }_{\kappa }}^{\prime }[\lambda ,\mathfrak{q}]$ and let $\lambda \in \mathcal{H}\left(\mathfrak{D}\right)$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1$ satisfy the conditions in (52) and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)$$

is univalent in $\mathfrak{D}$, then

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)& \le & {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.\hfill \\ & & \left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\hfill \end{array}$$

(58)

implies

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

(59)

For a correctly chosen $\psi$, it is established by the following theorem that the best subordinant for (58) exists.

Theorem 8.

Let $\lambda \in \mathcal{H}\left(\mathfrak{D}\right)$ and let $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{D}}⟶\mathbb{C}$ as given by (27). Consider the differential equation

$$\psi \left(\mathfrak{q}\left(\tau \right),\tau {\mathfrak{q}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{q}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right);\tau \right)=\lambda \left(\tau \right)$$

having a solution $\mathfrak{q}\left(\tau \right)\in Q_1$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1$ satisfy the condition (52) and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)$$

is univalent in $\mathfrak{D}$, then

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)& \le & {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.\hfill \\ & & \left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\hfill \end{array}$$

(60)

implies

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}),$$

with $\mathfrak{q}$ as the fuzzy best subordinant.

Proof. 

The proof of Theorem 3, which is similar to the proof of Theorem 8, is not given in detail. □

A new class of admissible functions, ${{\tilde{\Psi }}_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$, is introduced for obtaining new fuzzy superordination results.

Definition 11.

Let $\Omega \subset \mathbb{C}$, $\mathfrak{q}\in {\mathcal{H}}_1$ with ${\mathfrak{q}}^{\prime }\left(\tau \right)\ne 0$ and $\vartheta \in \mathbb{N}\setminus \left\{1\right\}$. Functions $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{D}}\to \mathbb{C}$ belong to the class of admissible functions ${{\tilde{\Psi }}_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$ if the admissibility condition

$$\psi \left(\alpha ,\beta ,\gamma ,\delta ;\xi \right)\in \Omega ,$$

is satisfied, whenever

$$\alpha =\mathfrak{q}\left(\tau \right),\beta ={\displaystyle \frac{\tau {\mathfrak{q}}^{\prime }\left(\tau \right)+\vartheta \upsilon \mathfrak{q}\left(\tau \right)}{\vartheta \upsilon }},$$

$$Re\left\{\left(\upsilon +1\right)\left({\displaystyle \frac{\gamma +\alpha -2\beta }{\beta -\alpha }}\right)\right\}\le {\displaystyle \frac{1}{\vartheta }}Re\left\{1+{\displaystyle \frac{\tau {\mathfrak{q}}^{\prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

$$Re\left\{\left(\upsilon +1\right)\left(\upsilon +2\right)\left({\displaystyle \frac{\delta -\alpha +3\beta -3\gamma }{\beta -\alpha }}\right)\right\}\le {\displaystyle \frac{1}{{\vartheta }^2}}Re\left\{{\displaystyle \frac{{\tau }^2{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\},$$

where $\vartheta \ge 2$, $\xi \in \partial \mathfrak{D}$, and $\tau \in \mathfrak{D}$.

Theorem 9.

Let $\psi \in {{\tilde{\Psi }}_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1$ satisfy

$$Re\left\{{\displaystyle \frac{\tau {\mathfrak{q}}^{″}\left(\tau \right)}{{\mathfrak{q}}^{\prime }\left(\tau \right)}}\right\}\ge 0\mathit{and}\left|{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)-{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\right|\le {\displaystyle \frac{\vartheta }{\upsilon }}\left|{\mathfrak{q}}^{\prime }\left(\tau \right)\right|,$$

(61)

and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)$$

is univalent in $\mathfrak{D}$, then

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\Omega }\left(\tau \right)& \le & {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left(({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.\hfill \\ & & \left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau ):\tau \in \mathfrak{D}\right)\hfill \end{array}$$

(62)

implies

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

Proof. 

Let $\mathfrak{g}\left(\tau \right)$ be given by (39) and $\psi$ be given by (42). Since $\psi \in {{\tilde{\Psi }}_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}],$ (43) and (62) yield

$${\mathcal{F}}_{\Omega }\left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left\{\psi \left(\mathfrak{g}\left(\tau \right),\tau {\mathfrak{g}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{g}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{g}}^{\prime \prime \prime }\left(\tau \right);\tau \right):\tau \in \mathfrak{D}\right\}.$$

For $\psi \in {{\tilde{\Psi }}_{\kappa }}^{\prime }[\Omega ,\mathfrak{q}]$, we conclude from (42) that the admissible condition seen in Definition 2 is equivalent to the one expressed in Definition 8 for $\psi$. Thus, utilizing Lemma 2 and the conditions in (61), we establish that

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}\mathfrak{g}\left(\tau \right),$$

or, equivalently,

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

Hence, the proof is concluded. □

When $\Omega \ne \mathbb{C}$ is a simply connected domain with $\Omega =\lambda \left(\mathfrak{D}\right)$ for a certain conformal mapping $\lambda \left(\tau \right)$ of $\mathfrak{D}$ onto $\Omega$, the notation ${{\tilde{\Psi }}_{\kappa }}^{\prime }[\lambda ,\mathfrak{q}]$ designates the class ${{\tilde{\Psi }}_{\kappa }}^{\prime }[\lambda \left(\mathfrak{D}\right),\mathfrak{q}]$. Using the same techniques as in the previous section, Theorem 9 yields the following result

Corollary 6.

Let $\psi \in {{\tilde{\Psi }}_{\kappa }}^{\prime }[\lambda ,\mathfrak{q}]$ and $\lambda \in \mathcal{H}\left(\mathfrak{D}\right)$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1$ satisfy the conditions in (61) and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)$$

is univalent in $\mathfrak{D}$, then

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)& \le & {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.\hfill \\ & & \left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\hfill \end{array}$$

(63)

implies

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

For a suitable choice of $\psi$, it is established by the following theorem that the fuzzy best subordinant for (63) exists.

Proposition 2.

Let $\lambda \in \mathcal{H}\left(\mathfrak{D}\right)$ and let $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{D}}⟶\mathbb{C}$ as given by (42). Consider the differential equation

$$\psi \left(\mathfrak{q}\left(\tau \right),\tau {\mathfrak{q}}^{\prime }\left(\tau \right),{\tau }^2{\mathfrak{q}}^{\prime \prime }\left(\tau \right),{\tau }^3{\mathfrak{q}}^{\prime \prime \prime }\left(\tau \right);\tau \right)=\lambda \left(\tau \right),$$

having a solution $\mathfrak{q}\left(\tau \right)\in {\mathfrak{Q}}_1$. If $\mathfrak{f}\in {\Sigma }_p$ and ${\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1$ satisfy the conditions in (61) and the function

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)$$

is univalent in $\mathfrak{D}$, then

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\tau \right)& \le & {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.\hfill \\ & & \left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\hfill \end{array}$$

implies

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D})$$

with $\mathfrak{q}$ as the fuzzy best subordinant.

Proof. 

As it is similar to the proof of Theorem 6, the proof of Theorem 2 is not given in detail. □

Example 4.

By taking $\psi =2\alpha -\beta ,$ $\varrho =0$, and $p=\upsilon =\mathfrak{s}=1$ in Theorem 8, we obtain

If

$$\begin{array}{ccc}\hfill \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)& =& 2\mathfrak{g}\left(\tau \right)-\tau {\mathfrak{g}}^{\prime }\left(\tau \right)+\mathfrak{g}\left(\tau \right)\hfill \\ & =& \mathfrak{g}\left(\tau \right)-\tau {\mathfrak{g}}^{\prime }\left(\tau \right)\hfill \\ & =& -{\tau }^2{\mathfrak{f}}^{\prime }\left(\tau \right)\hfill \end{array}$$

is analytic in $\mathfrak{U}$ and

$${\mathcal{F}}_{\Omega }\left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\psi \left(-{\tau }^2{\mathfrak{f}}^{\prime }\left(\tau \right)\right),$$

then

$$\mathfrak{q}\left(\tau \right){\prec }_{\mathcal{F}}\tau \mathfrak{f}\left(\tau \right)(\tau \in \mathfrak{D}).$$

or

$${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\tau \right)\le {\mathcal{F}}_{\tau \mathfrak{f}\left(\mathfrak{D}\right)}\left(\tau \mathfrak{f}\left(\tau \right)\right).$$

3.3. Sandwich-Type Advancements

Combining Theorems 2 and 5, we write

Proposition 3.

Let ${\lambda }_1,{\mathfrak{q}}_1\in \mathcal{H}\left(\mathfrak{D}\right)$. Also, let ${\lambda }_2\in S,{\mathfrak{q}}_2\in {\mathfrak{Q}}_1$ satisfying ${\mathfrak{q}}_1\left(0\right)={\mathfrak{q}}_2\left(0\right)=1$, and $\psi \in {\Psi }_{\kappa }[{\lambda }_2,{\mathfrak{q}}_2]\cap {{\Psi }_{\kappa }}^{\prime }[{\lambda }_1,{\mathfrak{q}}_1]$. If $\mathfrak{f}\in {\Sigma }_p,{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$ and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\in S,$$

with conditions (17) and (52) satisfied, then

$${\mathcal{F}}_{{\lambda }_1(\Omega )}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi {\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$$\left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho -2}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\le {\mathcal{F}}_{{\lambda }_2(\Omega )}{\lambda }_2\left(\tau \right)$$

implies that

$${\lambda }_1\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho +1}(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}{\lambda }_2\left(\tau \right).$$

The following sandwich-type result can also be obtained by combining Theorems 5 and 6:

Proposition 4.

Let ${\tilde{\lambda }}_1,{\tilde{\mathfrak{q}}}_1\in \mathcal{H}\left(\mathfrak{D}\right)$. Consider ${\tilde{\lambda }}_2\in S$, ${\tilde{\mathfrak{q}}}_2\in Q_1$ satisfying ${\tilde{\mathfrak{q}}}_1\left(0\right)={\tilde{\mathfrak{q}}}_2\left(0\right)=1$, and $\psi \in {\tilde{\Psi }}_{\kappa }[{\tilde{\lambda }}_2,{\tilde{\mathfrak{q}}}_2]\cap {{\tilde{\Psi }}_{\kappa }}^{\prime }[{\tilde{\lambda }}_1,{\tilde{\mathfrak{q}}}_1]$. If $\mathfrak{f}\in {\Sigma }_p,{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right)\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$ and

$$\psi \left({\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right),$$

$\psi \in S$ with (44) and (61) satisfied, then

$${\mathcal{F}}_{{\tilde{\lambda }}_1(\Omega )}{\tilde{\lambda }}_1\left(\tau \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{D})}\left(\psi {\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +1,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),\right.$$

$$\left.{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +2,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right),{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon +3,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right);\tau \right)\le {\mathcal{F}}_{{\tilde{\lambda }}_2(\Omega )}{\tilde{\lambda }}_2\left(\tau \right)$$

implies that

$${\tilde{\lambda }}_1\left(\tau \right){\prec }_{\mathcal{F}}{\tau }^p{\mathfrak{k}}_{\varkappa ,p}^{\varrho }(\upsilon ,\mathfrak{s};\tau )\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}{\tilde{\lambda }}_2\left(\tau \right).$$

4. Discussion

The results of the present investigation enhance the recently started research on third-order differential subordination and superordination theories. The emergence of fresh insights for these relatively new lines of study is what makes this work innovative. The first part of the investigation provides means to determine the third-order fuzzy dominant of several third-order fuzzy differential subordinations involving the operator given in relation (5) applied to meromorphic multivalent functions. Then, when the third-order fuzzy differential subordination admits of the third-order fuzzy best dominant, it is made clear how to find it. In the following part of the investigation, using the operator given in relation (5) applied to meromorphic multivalent functions, dual results are obtained regarding means for obtaining the third-order fuzzy subordinant of certain third-order fuzzy differential superordinations and the third-order fuzzy best subordinant, respectively. The investigation is concluded by stating sandwich-type results that connect the outcomes of the two dual theories. The approach concerns the key notion of the class of admissible functions of those theories. Research context is presented in Section 1, while the basic notions and previously established results necessary for the development of the new outcomes are contained in Section 2. Section 3 reveals the original results gathered in three subsections—third-order fuzzy differential subordination results, third-order fuzzy differential superordination results, and sandwich-type theorems connecting those results, respectively. The third-order fuzzy differential subordination outputs of this study consist of taking into consideration two classes of admissible functions, each used for proving theorems and associated corollaries that enrich the knowledge of this recently identified field of research. The fuzzy best dominant is also established for the investigated third-order fuzzy differential subordinations. The dual third-order fuzzy differential superordination outputs follow a similar pattern by taking into consideration two classes of admissible functions and using them for the establishment of theorems and associated corollaries. The fuzzy best subordinant is also established for the investigated third-order fuzzy differential superordinations. The study is concluded by stating two sandwich-type theorems that connect the outcomes of the two dual theories.

5. Conclusions

The results could inspire the use of the new operator given by Relation (5) for introducing new fuzzy classes of analytic functions as seen in the recent publications [26,27]. Also, the pattern applied here for this operator is likely to be followed using other operators for which second-order fuzzy differential subordinations and superordinations have been previously obtained, and it could be extended to fit the new context of higher-order fuzzy differential subordinations and superordinations. Such results can be cited as [28,29,30].