Application on Fuzzy Third-Order Subordination and Superordination Connected with Lommel Function

Abstract

This work is based on the recently introduced concepts of third-order fuzzy differential subordination and its dual, third-order fuzzy differential superordination. In order to obtain the new results that add to the development of the newly initiated lines of research, a new operator is defined here using the concept of convolution and the normalized Lommel function. The methods focusing on the basic concept of admissible function are employed. Hence, the investigation of new third-order fuzzy differential subordination results starts with the definition of the suitable class of admissible functions. The first theorems discuss third-order fuzzy differential subordinations involving the newly introduced operator. The following result shows the conditions needed such that the fuzzy best dominant can be found for a third-order fuzzy differential subordination. Next, dual results are obtained by employing the methods of third-order fuzzy differential superordination based on the same concept of an admissible function. A suitable class of admissible functions is introduced and new third-order fuzzy differential superordinations are obtained, showing how the best subordinant can be obtained under certain restrictions. As a conclusion of this study, sandwhich-type results are derived, linking the outcome of the two dual fuzzy theories.

1. Introduction

The dual ideas of third-order fuzzy differential subordination and superordination, a recently considered type of fuzzy differential subordination, are the focus of this work. Lotfi A. Zadeh first proposed the idea of the fuzzy set in 1965 [1], which was included into the differential subordination theory leading to the emergence of the fuzzy differential subordination notion in 2012 [2]. The fuzzy differential subordination theory adheres to the general differential subordination idea as investigated in [3,4]. The investigation of fuzzy differential subordination and superordination continues to deliver intriguing results in recent publications. Mittag–Leffler-type distributions are included in the research on the fuzzy differential subordination theory seen in [5,6]. Different types of operators are employed for obtaining the results seen in [7,8]. Quantum calculus operators provide the tools for the new fuzzy differential subordination results obtained in [9,10], and the dual theories of fuzzy differential subordination and superordination are used for developing sandwich-type results involving quantum calculus operators in [11]. All those results involving different types operators, including the ones involving quantum calculus, have motivated the research presented in this paper.

The context for obtaining the innovative conclusions seen in this work was given by the emergence in [12] of the idea of third-order fuzzy differential subordination. Expanding upon the notion put forth in [13], the authors of the work introduced the idea of third-order fuzzy differential subordination. The key notions to be used for the studies pertaining to this line of research for the fuzzy differential subordination theory, including the admissible functions class and basic theorems, were provided in [12], along with the main principles of third-order fuzzy subordination. As the concept of differential superordination was first proposed as a dual idea to that of differential subordination, in light of this concept, the idea of third-order fuzzy differential superordination was first presented in [14], where the dual problem of the third-order fuzzy differential superordination was studied by describing the fundamental ideas connected with the notion of third-order fuzzy differential superordination.

Since no other papers are published so far on the idea of using the two new dual fuzzy theories in order to obtain third-order fuzzy sandwich-type results, the present work adds valuable knowledge for enhancing the new lines of research recently initiated. The framework for the investigation is well known in geometric function theory.

$\mathcal{H}\left(\mathfrak{U}\right)$ is the class of analytic functions in $\mathfrak{U}=\{\zeta :\zeta \in \mathbb{C}$ and $\left|\zeta \right|<1\}$, with $\overline{\mathfrak{U}}=\{\zeta \in \mathbb{C}:\left|\zeta \right|\le 1\}$ and $\partial \mathfrak{U}=\{\zeta \in \mathbb{C}:\left|\zeta \right|=1\}$.

Certain subclasses of $\mathcal{H}\left(\mathfrak{U}\right)$ are famous and indispensable for research, such as the following:

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

where ${\mathcal{A}}_1=\mathcal{A}$, and

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

where ${\mathcal{H}}_0=\mathcal{H}[0,1]$, and ${\mathcal{H}}_1=\mathcal{H}[1,1]$ with $a\in \mathbb{C}$ and $\kappa \in \mathbb{N}=\{1,2,3,\cdots \}$.

Designate the class of convex functions as

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

Furthermore,

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

Given $\mathfrak{f},\mathfrak{g}\in \mathcal{H}\left(U\right)$, if there exists $\varpi \in \mathcal{H}\left(U\right)$, then $\mathfrak{f}$ is subordinate to $\mathfrak{g}$, denoted $\mathfrak{f}\left(\zeta \right)\prec \mathfrak{g}\left(\zeta \right)$, if $\varpi \left(0\right)=0,\zeta \in \mathfrak{U}$, and $\left|\varpi \left(\zeta \right)\right|<1$ for every $\zeta \in \mathfrak{U}$, and $\mathfrak{f}\left(\zeta \right)=\mathfrak{g}\left(\varpi \right(\zeta \left)\right)$. The following is what must be met such that $\mathfrak{f}\left(\zeta \right)\prec \mathfrak{g}\left(\zeta \right))$ we have the function $\mathfrak{g}\in S$:

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

For two functions ${\mathfrak{f}}_{\iota }\left(\zeta \right)\in \mathcal{A}(\iota =1,2)$ given by

$${\mathfrak{f}}_{\iota }\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{a}_{\kappa ,\iota }{\zeta }^{\kappa },$$

the convolution of ${\mathfrak{f}}_1\left(\zeta \right)$ and ${\mathfrak{f}}_2\left(\zeta \right)$ is defined as

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

Numerous papers refer to geometric properties of different families of special functions, particularly the generalized hypergeometric functions (see [15,16,17,18]) and the Bessel functions (see [19,20,21,22]). The Lommel functions of the first and second kind appear as specific solutions of particular second-order differential equations in the theory of Bessel functions (see, for example, [23,24,25]). We recall now the Lommel function, denoted by $L_{\delta ,\eta }\left(\zeta \right)$ and given by

$$L_{\delta ,\eta }\left(\zeta \right)={\displaystyle \frac{{\zeta }^{\delta +1}}{4}}\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^{\kappa }\Gamma \left(\frac{\delta -\eta +1}{2}\right)\Gamma \left(\frac{\delta +\eta +1}{2}\right)}{\Gamma \left(\frac{\delta -\eta +3}{2}+\kappa \right)\Gamma \left(\frac{\delta +\eta +3}{2}+\kappa \right)}}{\left({\displaystyle \frac{\zeta }{2}}\right)}^{2\kappa },$$

(1)

which is a particular solution of the inhomogeneous Bessel differential equation

$${\zeta }^2{w}^{″}\left(\zeta \right)+\zeta w^{\prime }\left(\zeta \right)+\left[{\zeta }^2-{\eta }^2\right]w\left(\zeta \right)={\zeta }^{\delta +1},$$

(2)

where $\frac{\delta -\eta +1}{2}, \frac{\delta +\eta +1}{2}\in \mathbb{C}\setminus {\mathbb{Z}}^{-}; {\mathbb{Z}}^{-}=\{-1,-2,\dots \}$, and $\Gamma$ stands for the Euler gamma function. It is clear that the function $L_{\delta ,\eta }$ is analytic for all $\zeta \in \mathbb{C}$.

Next, the normalized Lommel function ${\mathfrak{L}}_{\delta ,\eta }\left(\zeta \right)$ is considered as follows:

$${\mathfrak{L}}_{\delta ,\eta }\left(\zeta \right)=4\left(\frac{\delta -\eta +1}{2}\right)\left(\frac{\delta +\eta +1}{2}\right){\zeta }^{{\displaystyle \frac{1-\delta }{2}}}{L}_{\delta ,\eta }\left(\sqrt{\zeta }\right).$$

(3)

Using the shifted factorial ${\left(\sigma \right)}_n$ defined as

$${\left(\sigma \right)}_n={\displaystyle \frac{\Gamma \left(\sigma +n\right)}{\Gamma \left(\sigma \right)}}=\left\{\begin{array}{l}1,\left(n=0,\sigma \in \mathbb{C}\setminus \left\{0\right\}\right),\\ \sigma \left(\sigma +1\right)\dots \left(\sigma +n-1\right),\left(n\in \mathbb{N},\sigma \in \mathbb{C}\right),\end{array}\right.$$

the function ${\mathfrak{L}}_{\delta ,\eta }\left(\zeta \right)$ can be represented by the following series representation:

$${\mathfrak{L}}_{\delta ,\eta }\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{(-1)}^{\kappa -1}}{4^{\kappa -1}{\left(\frac{\delta -\eta +1}{2}+1\right)}_{\kappa -1}{\left(\frac{\delta +\eta +1}{2}+1\right)}_{\kappa -1}}}{\zeta }^{\kappa }.$$

(4)

For simplicity, let $\varrho ={\displaystyle \frac{\delta -\eta +3}{2}}$ and $\varkappa ={\displaystyle \frac{\delta +\eta +3}{2}}$. Thus, the function ${\mathfrak{L}}_{\varrho ,\varkappa }$ can be defined as follows:

$${\mathfrak{L}}_{\varrho ,\varkappa }\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{(-1)}^{\kappa -1}}{4^{\kappa -1}{\left(\varrho \right)}_{\kappa -1}{\left(\varkappa \right)}_{\kappa -1}}}{\zeta }^{\kappa }.$$

(5)

The function ${\mathfrak{L}}_{\varrho ,\varkappa }\left(\zeta \right)$ is analytic for all $\zeta \in \mathbb{C}$ and $\varrho ,\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}={\mathbb{Z}}^{-}\cup \left\{0\right\}$, and it is clear that ${\mathfrak{L}}_{\varrho ,\varkappa }\left(\zeta \right)\in \mathcal{A}$.

Using the notions presented above, the new operator used for the investigation is defined by means of a Hadamard product as follows:

$${\mathcal{L}}_{\varrho ,\varkappa }:\mathcal{A}⟶\mathcal{A},$$

$${\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right):=\left({\mathfrak{L}}_{\varrho ,\varkappa }\ast \mathfrak{f}\right)\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{(-1)}^{\kappa -1}}{4^{\kappa -1}{\left(\varrho \right)}_{\kappa -1}{\left(\varkappa \right)}_{\kappa -1}}}{a}_{\kappa }{\zeta }^{\kappa }.$$

(6)

Remark 1.

We note that by taking $\varkappa =1$ in (6), the operator ${\mathcal{L}}_{\varrho }$ is obtained, defined as follows:

$${\mathcal{L}}_{\varrho }\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{(-1)}^{\kappa -1}}{4^{\kappa -1}{\left(\varrho \right)}_{\kappa -1}(\kappa -1)!}}{a}_{\kappa }{\zeta }^{\kappa }.$$

which is related to Bessel functions of the first kind (see [21]).

The operator ${\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)$, satisfies

$$\zeta {\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right)}^{\prime }=\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1){\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),$$

(7)

and

$$\zeta {\left({\mathcal{L}}_{\varrho ,\varkappa +1}\mathfrak{f}\left(\zeta \right)\right)}^{\prime }=\varkappa {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varkappa -1){\mathcal{L}}_{\varrho ,\varkappa +1}\mathfrak{f}\left(\zeta \right).$$

(8)

2. Preliminaries

The new findings in the section that follows will be supported by the subsequent investigations.

Definition 1

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

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

and are such that $min\left|{\mathfrak{q}}^{\prime }\left(\xi \right)\right|=\rho >0$ for $\xi \in \partial \mathfrak{U}\setminus E\left(\mathfrak{q}\right)$. $E\left(\mathfrak{q}\right)$ is called the exception set, and the subclass of $\mathfrak{Q}$ when $\mathfrak{q}\left(0\right)=a$ can be denoted by $\mathfrak{Q}\left(a\right)$ and $\mathfrak{Q}\left(0\right)={\mathfrak{Q}}_0$.

From the theory of fuzzy differential subordination, we use the following:

Definition 2

([1]). A fuzzy set comprises of the pair $\left(\varsigma ,\mathcal{F}\right)$, where ς is a set, $\varsigma \ne \varphi$, and $\mathcal{F}:\varsigma \to \left[0,1\right]$ is a membership function.

Definition 3

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

The notations that are listed next are used for the investigation proposed here. Let $\mathcal{F}:\mathbb{C}\to [0,1]$. Indicate by

(i) 

$\left(\upsilon ,{\mathcal{F}}_{\upsilon }\right)=\left(\mathfrak{U},{\mathcal{F}}_{\mathfrak{U}}\right)$. Then, $\upsilon =\mathfrak{U}$, where $\upsilon =sup(\upsilon ,{\mathcal{F}}_{\upsilon })$ and $\mathfrak{U}=sup\left(\mathfrak{U},{\mathcal{F}}_{\mathfrak{U}}\right)$.

(ii) 

If $\left(\upsilon ,{\mathcal{F}}_{\upsilon }\right)\subseteq \left(\mathfrak{U},{\mathcal{F}}_{\mathfrak{U}}\right)$. Then, $I\subseteq \mathfrak{U}$ where $I=sup(I,{\mathcal{F}}_{\upsilon })$ and $\mathfrak{U}=sup\left(\mathfrak{U},{\mathcal{F}}_{\mathfrak{U}}\right)$.

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

$$\mathfrak{f}\left(\mathfrak{U}\right)=sup(\mathfrak{f}\left(\mathfrak{U}\right),{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{U}\right)})=\{\mathfrak{f}\left(\zeta \right):0<\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{U}\right)}\left(\mathfrak{f}\left(\zeta \right)\right)\right|\le 1,\zeta \in \mathfrak{U}\},$$

(9)

and

$$\mathfrak{g}\left(\mathfrak{U}\right)=sup(\mathfrak{g}\left(\mathfrak{U}\right),{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{U}\right)})=\{\mathfrak{g}\left(\zeta \right):0<\left|{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{U}\right)}\left(\mathfrak{g}\left(\zeta \right)\right)\right|\le 1,\zeta \in \mathfrak{U}\}.$$

(10)

Definition 4

([26]). Let two fuzzy subsets of η, $\left({\upsilon }_1,{\mathcal{F}}_{{\upsilon }_1}\right)$ and $\left({\upsilon }_2,{\mathcal{F}}_{{\upsilon }_2}\right)$. We state that the fuzzy subsets ${\upsilon }_1$ and ${\upsilon }_2$ are equal if ${\mathcal{F}}_{{\upsilon }_1}\left(\eta \right)={\mathcal{F}}_{{\upsilon }_2}\left(\eta \right)$, $\eta \in \varsigma$, and we denote by $\left({\upsilon }_1,{\mathcal{F}}_{{\upsilon }_1}\right)=\left({\upsilon }_2,{\mathcal{F}}_{{\upsilon }_2}\right)$. The fuzzy subset $\left({\upsilon }_1,{\mathcal{F}}_{{\upsilon }_1}\right)$ is contained in the fuzzy subset $\left({\upsilon }_2,{\mathcal{F}}_{{\upsilon }_2}\right)$ if ${\mathcal{F}}_{{\upsilon }_1}\left(\eta \right)\le {\mathcal{F}}_{{\upsilon }_2}\left(\eta \right)$, $\eta \in S$ and denote by $\left({\upsilon }_1,{\mathcal{F}}_{{\upsilon }_1}\right)\subseteq \left({\upsilon }_2,{\mathcal{F}}_{{\upsilon }_2}\right)$.

Definition 5

([26]). Let ${\zeta }_0\in \mathfrak{U}$ and $\mathfrak{f},\mathfrak{g}\in \mathcal{H}\left(\mathfrak{U}\right)$. $\mathfrak{f}$ is fuzzy subordinate to $\mathfrak{g}$ and written as $\mathfrak{f}{\prec }_{\mathcal{F}}\mathfrak{g}$ or $\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{g}\left(\zeta \right)$ if each of the following is satisfied:

$$\mathfrak{f}\left({\zeta }_0\right)=\mathfrak{g}\left({\zeta }_0\right)\mathrm{and}\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{U}\right)}\left(\mathfrak{f}\left(\zeta \right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{U}\right)}\left(\mathfrak{g}\left(\zeta \right)\right)\right|,\zeta \in \mathfrak{U},$$

Proposition 1

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

$$\begin{array}{lll}\left(\mathfrak{i}\right)\mathfrak{f}\left({\zeta }_0\right)& =& \mathfrak{g}\left({\zeta }_0\right),\hfill \\ \left(\mathfrak{ii}\right)\mathfrak{f}\left(\mathfrak{U}\right)& \subseteq & \mathfrak{g}\left(\mathfrak{U}\right)\mathrm{and}\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{U}\right)}\left(\mathfrak{f}\left(\zeta \right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{U}\right)}\left(\mathfrak{g}\left(\zeta \right)\right)\right|,\zeta \in \mathfrak{U},\hfill \end{array}$$

where $\mathfrak{f}\left(\mathfrak{U}\right)$ and $\mathfrak{g}\left(\mathfrak{U}\right)$ are given by (9) and (10), respectively.

Definition 6

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

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

is satisfied, with

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

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

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

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

In particular, if we set

$$\mathfrak{q}\left(\zeta \right)=\left({\displaystyle \frac{M\zeta +\mathfrak{a}}{M+\overline{\mathfrak{a}}\zeta }}\right)M(M>0;\left|\mathfrak{a}\right|<M),$$

then

$$\mathfrak{q}\left(\mathfrak{U}\right)={\mathfrak{U}}_M=\left\{w:\left|w\right|<M\right\},\mathfrak{q}\left(0\right)=\mathfrak{a},E\left(\mathfrak{q}\left(\zeta \right)\right)=\varphi \mathrm{and}\mathfrak{q}\in \wp \left(\mathfrak{a}\right).$$

In this case, we set

$${\Psi }_{\kappa }[\Omega ,\mathfrak{q},\mathfrak{a}]={\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$$

and, in the special case when the set $\Omega ={\mathfrak{U}}_M$, the resulting class is simply denoted by ${\Psi }_{\kappa }$$[M,\mathfrak{a}]$.

Definition 7

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

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

(11)

where

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

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

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

with $\zeta \in \mathfrak{U}$, $\xi \in \partial \mathfrak{U}$ and $\mathit{\ell }\ge \kappa \ge 2$. Condition (11) is called the admissibility condition.

The definition of the concept of fuzzy dominance for the solutions of a third-order fuzzy differential subordination is given in [12] in terms of a function $\mathfrak{q}\in \mathcal{H}\left(\mathfrak{U}\right)$ that satisfies $\omega {\prec }_{\mathcal{F}}\mathfrak{q}$ whenever $\omega$ satisfies

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

i.e.,

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

(12)

with $\psi :{\mathbb{C}}^4\times \mathfrak{U}⟶\mathbb{C}$, $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ and $\lambda \in \mathcal{H}\left(\mathfrak{U}\right)$, $\omega$ being called a solution of the fuzzy differential subordination (12). For all dominants $\mathfrak{q}$ of (12), the fuzzy best dominant is a fuzzy dominant $\tilde{\mathfrak{q}}$ satisfying $\tilde{\mathfrak{q}}{\prec }_{\mathcal{F}}\mathfrak{q}$. It is known that the fuzzy best dominant is unique up to a rotation of $\mathfrak{U}$.

The dual concept of a fuzzy subordinant of a third-order fuzzy differential superordination is introduced in [14] as being a function $\mathfrak{q}\in S$ satisfying $\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}\omega \left(\zeta \right)$ or, equivalently written, as ${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right)\le {\mathcal{F}}_{\omega \left(\mathfrak{U}\right)}\left(\omega \left(\zeta \right)\right),\zeta \in \mathfrak{U}$, whenever $\omega$ satisfies

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

i.e.,

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

(13)

with $\psi :{\mathbb{C}}^4\times \mathfrak{U}⟶\mathbb{C}$, $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ and $\lambda \in \mathcal{H}\left(\mathfrak{U}\right)$, $\omega$ being called a solution for the third-order fuzzy differential superordination (13). For all fuzzy subordinants $\mathfrak{q}$ of (13), the best fuzzy subordinant is a fuzzy subordinant $\tilde{\mathfrak{q}}\in S$ satisfying $\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}\tilde{\mathfrak{q}}\left(\zeta \right)$ or, equivalently written, as ${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right)\le {\mathcal{F}}_{\tilde{\mathfrak{q}}\left(\mathfrak{U}\right)}\left(\tilde{\mathfrak{q}}\left(\zeta \right)\right),\zeta \in \mathfrak{U}$.

The next two results listed as lemmas, proved in [12] and [14], respectively, serve as tools for obtaining the new outcome of the next sections.

Lemma 1

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

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

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

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

then

$$p\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right)\mathrm{or}{\mathcal{F}}_{p\left(\mathfrak{U}\right)}p\left(\zeta \right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right).$$

Lemma 2

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

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

where $\zeta \in \mathfrak{U}$, $\xi \in \partial \mathfrak{U}\setminus E\left(\mathfrak{q}\right),\mathit{\ell }\ge \kappa \ge 2$.

If $p\left(\zeta \right)$ and $\psi (p\left(\zeta \right),\zeta p^{\prime }\left(\zeta \right),{\zeta }^2{p}^{″}\left(\zeta \right),{\zeta }^3{p}^{‴}\left(\zeta \right);\zeta )$ are univalent in $\mathfrak{U},$ then

$$\Omega \subset \left\{\psi (p\left(\zeta \right),\zeta p^{\prime }\left(\zeta \right),{\zeta }^2{p}^{″}\left(\zeta \right),{\zeta }^3{p}^{‴}\left(\zeta \right);\zeta ):\zeta \in \mathfrak{U}\right\},$$

or, equivalently, as

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

which implies that

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

In the following sections, by utilizing the third-order differential subordination results in accordance to Antonino and Miller [13] in the unit disk $\mathfrak{U}$ and the third-order fuzzy differential subordination and superordination results introduced by Oros et al. [12,14], we define certain suitable classes of admissible functions and study some third-order fuzzy differential subordination and superordination properties of univalent functions connected with the Lommel function ${\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)$ defined by (6). The results obtained using the two dual theories are linked at the end of the study by sandwich-type results.

3. Third-Order Fuzzy Differential Subordination Results

Throughout the study, unless otherwise indicated, we will suppose that $\varrho ,\varkappa >2$.

Certain new third-order fuzzy differential subordinations are obtained in this section. The following definition applies to the class of admissible functions for this purpose:

Definition 8.

Let $\Omega \subset \mathbb{C}$ and $\mathfrak{q}\in {\mathfrak{Q}}_0\cap {\mathcal{H}}_0$. Functions $\psi :{\mathbb{C}}^4\times \mathfrak{U}\to \mathbb{C}$ that satisfy

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

(14)

whenever

$$\alpha =\mathfrak{q}\left(\xi \right),\beta ={\displaystyle \frac{\mathit{\ell }\xi {\mathfrak{q}}^{\prime }\left(\xi \right)+(\varrho -1)\mathfrak{q}\left(\xi \right)}{\varrho }},$$

(15)

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

(16)

and

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

(17)

where $\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \partial \mathfrak{U}\setminus E\left(\mathfrak{q}\right)$ and $\zeta \in \mathfrak{U}$, form the class of admissible functions denoted as ${\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$.

Our first result is now stated and proved as Theorem 1 below.

Theorem 1.

Consider $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in \mathcal{A}$ and $\mathfrak{q}\in {\mathfrak{Q}}_0\cap {\mathcal{H}}_0$ satisfy

$$\mathrm{Re}\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{″}\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1){\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \mathit{\ell }\left|{\mathfrak{q}}^{\prime }\left(\xi \right)\right|,$$

(18)

and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi ({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta )\le {\mathcal{F}}_{\Omega }\left(\zeta \right),$$

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

then

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right)(\zeta \in \mathfrak{U}),$$

(19)

or

$${\mathcal{F}}_{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\mathfrak{U}\right)}\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right).$$

Proof. 

Define

$$\mathfrak{g}\left(\zeta \right)={\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right).$$

(20)

Utilizing (7) and (20), we have

$${\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right)+(\varrho -1)\mathfrak{g}\left(\zeta \right)}{\varrho }}.$$

(21)

Further computations show that

$${\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{{\zeta }^2{\mathfrak{g}}^{″}\left(\zeta \right)+2\left(\varrho -1\right)\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right)+(\varrho -1)(\varrho -2)\mathfrak{g}\left(\zeta \right)}{\varrho (\varrho -1)}},$$

(22)

and

$${\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{{\zeta }^3{\mathfrak{g}}^{‴}\left(\zeta \right)+3(\varrho -1){\zeta }^2{\mathfrak{g}}^{″}\left(\zeta \right)+3(\varrho -1)(\varrho -2)\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right)+(\varrho -1)(\varrho -2)(\varrho -3)\mathfrak{g}\left(\zeta \right)}{\varrho (\varrho -1)(\varrho -2)}}.$$

(23)

Now, we specify how ${\mathbb{C}}^4$ transforms to $\mathbb{C}$ by

$$\alpha (t,u,\upsilon ,w)=t,\beta (t,u,\upsilon ,w)={\displaystyle \frac{u+(\varrho -1)t}{\varrho }},$$

(24)

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

(25)

and

$$\begin{array}{ccc}\hfill \delta (t,u,\upsilon ,w;\zeta )& =& {\displaystyle \frac{w+3(\varrho -1)\upsilon +3(\varrho -1)(\varrho -2)u+(\varrho -1)(\varrho -2)(\varrho -3)t}{\varrho (\varrho -1)(\varrho -2)}}.\hfill \end{array}$$

(26)

Let

$$\begin{array}{ccc}\hfill \psi \left(t,u,\upsilon ,w;\zeta \right)& =& \psi \left(\alpha ,\beta ,\gamma ,\delta ;\zeta \right)\hfill \\ & =& \psi \left(t,{\displaystyle \frac{u+(\varrho -1)t}{\varrho }},{\displaystyle \frac{\upsilon +2\left(\varrho -1\right)u+(\varrho -1)(\varrho -2)t}{\varrho (\varrho -1)}},\right.\hfill \\ & & \left.{\displaystyle \frac{w+3(\varrho -1)\upsilon +3(\varrho -1)(\varrho -2)u+(\varrho -1)(\varrho -2)(\varrho -3)t}{\varrho (\varrho -1)(\varrho -2)}};\zeta \right)\hfill \end{array}$$

(27)

By applying Lemma 1, conditions (20)–(27), we write

$$\begin{array}{l}\psi \left(\mathfrak{g}\left(\zeta \right),\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right),{\zeta }^2{\mathfrak{g}}^{″}\left(\zeta \right),{\zeta }^3{\mathfrak{g}}^{‴}\left(\zeta \right);\zeta \right)\\ =\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right).\end{array}$$

(28)

Hence, (19) leads to

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

(29)

Additionally, applying (24)–(26), it is simple to obtain

$${\displaystyle \frac{\upsilon }{u}}+1={\displaystyle \frac{\varrho (\varrho -1)\gamma -(\varrho -1)(\varrho -2)\alpha }{\varrho \beta -(\varrho -1)\alpha }}-(2\varrho -3),$$

(30)

and

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

(31)

Accordingly, the $\psi \in {\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ admissibility condition in Definition 8 corresponds to the $\psi \in {\Psi }_2[\Omega ,\mathfrak{q}]$ admissibility condition in Definition 6. Thus, by applying (18) and Lemma 1, we write ${\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{U}\right)}\mathfrak{g}\left(\zeta \right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right)$ or, equivalently, ${\mathcal{F}}_{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\mathfrak{U}\right)}\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right)$,

i.e.,

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right),$$

The proof is completed. □

Example 1.

By taking $\psi ={\displaystyle \frac{\beta }{\alpha }}-1$, in Theorem 1 we obtain

$$\begin{array}{ccc}\hfill \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)& =& {\displaystyle \frac{u+(\varrho -1)t}{\varrho t}}-1\hfill \\ & =& {\displaystyle \frac{u-t}{\varrho t}}\hfill \\ & =& {\displaystyle \frac{\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1)t-t}{\nearrow t}}\hfill \\ & =& {\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}}-1\hfill \end{array}$$

which is analytic in $\mathfrak{U}$ and

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

then

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right)(\zeta \in \mathfrak{U}),$$

(32)

or

$${\mathcal{F}}_{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\mathfrak{U}\right)}\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right).$$

When $\mathfrak{q}$ behaves in an unknown manner on $\partial \mathfrak{U}$, the following result is generated by Theorem 1.

Corollary 1.

Consider $\Omega \subset \mathbb{C}$ and take $\mathfrak{q}\in S$ with $\mathfrak{q}\left(0\right)=1$. Consider $\psi \in {\Psi }_{\kappa }[\Omega ,{\mathfrak{q}}_{\rho }]$, for certain $\rho \in (0,1)$ when ${\mathfrak{q}}_{\rho }\left(\zeta \right)=\mathfrak{q}\left(\rho \zeta \right)$. If $\mathfrak{f}\in \mathcal{A}$ and ${\mathfrak{q}}_{\rho }$ satisfy

$$\mathrm{Re}\left\{{\displaystyle \frac{\xi {\mathfrak{q}}_{\rho }^{″}\left(\xi \right)}{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1){\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \mathit{\ell }\left|{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)\right|,$$

(33)

$$(\xi \in \partial \mathfrak{U}\setminus E\left({\mathfrak{q}}_{\rho }\right);\zeta \in \mathfrak{U}),$$

and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi ({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta )\le {\mathcal{F}}_{\Omega }\left(\zeta \right),$$

then

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right).$$

Proof. 

Following Theorem 1, we obtain

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}{\mathfrak{q}}_{\rho }\left(\zeta \right).$$

The assertion of Corollary 1 is obtained from

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

□

If $\Omega \ne \mathbb{C}$ is a simply connected domain, then $\Omega =\lambda \left(\mathfrak{U}\right)$ for a well-chosen conformal mapping $\lambda \left(\zeta \right)$ of $\mathfrak{U}$ onto $\Omega$. In this situation, the class ${\Psi }_{\kappa }[\mathcal{H}\left(\mathfrak{U}\right),\mathfrak{q}]$ is written as ${\Psi }_{\kappa }[\lambda ,\mathfrak{q}]$. The next listed results are direct outcomes of Theorem 1 and Corollary 1.

Theorem 2.

Suppose that $\psi \in {\Psi }_{\kappa }[\lambda ,\mathfrak{q}]$. If $\mathfrak{f}\in \mathcal{A}$ and $\mathfrak{q}\in {\mathfrak{Q}}_0$ satisfy Condition (18), then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi ({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta )\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right),$$

(34)

implies that

$${\mathcal{F}}_{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\mathfrak{U}\right)}\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right)\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right),$$

i.e.,

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\lambda \left(\zeta \right)(\zeta \in \mathfrak{U}).$$

Corollary 2.

Consider $\Omega \subset \mathbb{C}$ and take $\mathfrak{q}\in S$ with $\mathfrak{q}\left(0\right)=1$. Consider $\psi \in {\Psi }_{\kappa }[\lambda ,{\mathfrak{q}}_{\rho }]$ for certain $\rho \in (0,1)$ when ${\mathfrak{q}}_{\rho }\left(\zeta \right)=\mathfrak{q}\left(\rho \zeta \right)$. If $\mathfrak{f}\in \mathcal{A}$ and ${\mathfrak{q}}_{\rho }$ satisfy

$$\mathrm{Re}\left\{{\displaystyle \frac{\xi {\mathfrak{q}}_{\rho }^{″}\left(\xi \right)}{{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1){\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \mathit{\ell }\left|{\mathfrak{q}}_{\rho }^{\prime }\left(\xi \right)\right|,$$

(35)

$$(\xi \in \partial \mathfrak{U}\setminus E\left({\mathfrak{q}}_{\rho }\right);\zeta \in \mathfrak{U}),$$

and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi ({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta )\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right),$$

then

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right).$$

The fuzzy best dominant of the fuzzy differential subordination (19) or (34) is obtained by our next theorem.

Theorem 3.

Suppose that $\lambda \left(\zeta \right)$ be univalent in $\mathfrak{U}$. Also, let $\psi :{\mathbb{C}}^4\times \mathfrak{U}⟶\mathbb{C}$ and ψ be given by (27). Let

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

(36)

have a solution $\mathfrak{q}\left(\zeta \right)\in {\mathfrak{Q}}_0\cap {\mathcal{H}}_0$, which satisfies the conditions in (18). If $\mathfrak{f}\in \mathcal{A}$ satisfies Condition (33) and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \}\le {\mathcal{F}}_{\Omega }\left(\zeta \right)$$

is analytic in $\mathfrak{U}$, then

$${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right),$$

and $\mathfrak{q}\left(\zeta \right)$ is the fuzzy best dominant.

Proof. 

We determine that $\mathfrak{q}$ is a dominant of (34) by using Theorem 1. $\mathfrak{q}$ is also a solution of (34) since it satisfies (36). As a result, all dominants will dominate $\mathfrak{q}$. As a result, the fuzzy best dominant is $\mathfrak{q}$. □

In the particular case when $\mathfrak{q}\left(\zeta \right)=M\zeta (M>0)$ and, in view of Definition 8, the class ${\Psi }_{\kappa }[\Omega ,\mathfrak{q}]$ of admissible functions, which we denote by ${\Psi }_{\kappa }[\Omega ,M]$, is described below.

Definition 9.

Let Ω be a set in $\mathbb{C}$ and $M>0$. The class ${\Psi }_{\kappa }$$[\Omega ,M]$ of admissible functions consists of the functions $\phi :{\mathbb{C}}^4\times \mathfrak{U}\to \mathbb{C}$ such that

$$\begin{array}{l}\psi (M{e}^{i\theta },{\displaystyle \frac{(\mathit{\ell }+(\varrho -1))M{e}^{i\theta }}{\varrho }},{\displaystyle \frac{L+\left[2\left(\varrho -1\right)\mathit{\ell }+(\varrho -1)(\varrho -2)\right]M{e}^{i\theta }}{\varrho (\varrho -1)}},\\ {\displaystyle \frac{N+3(\varrho -1)L+\left[3(\varrho -1)(\varrho -2)\mathit{\ell }+(\varrho -1)(\varrho -2)(\varrho -3)\right]M{e}^{i\theta }}{\varrho (\varrho -1)(\varrho -2)}})\notin \Omega \end{array}$$

whenever $\zeta \in \mathfrak{U}$, $\theta \in \mathbb{R}$ and

$$Re\left(L{e}^{-i\theta }\right)\geqq (\mathit{\ell }-1)\mathit{\ell }M\mathrm{and}Re\left(N{e}^{-i\theta }\right)\geqq 0(\forall \theta \in \mathbb{R};\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}).$$

Corollary 3.

Let $\psi \in {\Psi }_{\kappa }[\Omega ,M]$. If $\mathfrak{f}\in \mathcal{A}$, then

$$\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1){\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \mathit{\ell }M.$$

If

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi ({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta )\le {\mathcal{F}}_{\Omega }\left(\zeta \right),$$

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

then

$$\left|{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|{\prec }_{\mathcal{F}}M(\zeta \in \mathfrak{U}),$$

or

$${\mathcal{F}}_{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\mathfrak{U}\right)}\left|\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right)\right|\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}M,$$

where $\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},M>0$ and $\zeta \in \mathfrak{U}$.

Then, as follows, we define a new admissible class ${\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$:

Definition 10.

Suppose that Ω is a set in $\mathbb{C}$ and $\mathfrak{q}\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$. The class of admissible functions ${\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$ consists of function $\psi :{\mathbb{C}}^4\times \mathfrak{U}\to \mathbb{C}$ which satisfies

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

whenever

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

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

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

where $\varrho \in \mathbb{C}\setminus {\mathbb{Z}}^{-},\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \partial \mathfrak{U}\setminus E\left(\mathfrak{q}\right)$ and $\zeta \in \mathfrak{U}$.

Theorem 4.

Consider $\psi \in {\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in \mathcal{A}$ and $\mathfrak{q}\in {\mathfrak{Q}}_1$ satisfy

$$\mathrm{Re}\left\{{\displaystyle \frac{\xi {\mathfrak{q}}^{″}\left(\xi \right)}{{\mathfrak{q}}^{\prime }\left(\xi \right)}}\right\}\ge 0,\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-\varrho {\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \mathit{\ell }\left|\zeta {\mathfrak{q}}^{\prime }\left(\xi \right)\right|,$$

(37)

and

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\left(\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right):\zeta \in \mathfrak{U}\right)\le {\mathcal{F}}_{\Omega }\left(\zeta \right),$$

(38)

then

$${\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}{\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right)(\zeta \in \mathfrak{U}).$$

Proof. 

Let

$$\mathfrak{g}\left(\zeta \right)={\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}.$$

(39)

Utilizing (7) and (39), we have

$${\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}={\displaystyle \frac{\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right)+\varrho \mathfrak{g}\left(\zeta \right)}{\varrho }}.$$

Further computations show that

$${\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}={\displaystyle \frac{{\zeta }^2{\mathfrak{g}}^{″}\left(\zeta \right)+2\varrho \zeta {\mathfrak{g}}^{\prime }\left(\zeta \right)+\varrho (\varrho -1)\mathfrak{g}\left(\zeta \right)}{\varrho (\varrho -1)}},$$

and

$${\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}={\displaystyle \frac{{\zeta }^3{\mathfrak{g}}^{‴}\left(\zeta \right)+3\varrho {\zeta }^2{\mathfrak{g}}^{″}\left(\zeta \right)+3\varrho (\varrho -1)\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right)+\varrho (\varrho -1)(\varrho -2)\mathfrak{g}\left(\zeta \right)}{\varrho (\varrho -1)(\varrho -2)}}.$$

(40)

We now specify how ${\mathbb{C}}^4$ transforms to $\mathbb{C}$ by

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

(41)

$$\gamma (t,u,\upsilon ,w)={\displaystyle \frac{\upsilon +2\varrho u+\varrho (\varrho -1)t}{\varrho (\varrho -1)}},$$

and

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

Let

$$\begin{gathered} \psi \left(t,u,v,w;\zeta \right)=\psi \left(\alpha ,\beta ,\gamma ,\delta ;\zeta \right) \\ =\psi \left(t,{\displaystyle \frac{u+\varrho t}{\varrho }},{\displaystyle \frac{\upsilon +2\varrho u+\varrho (\varrho -1)t}{\varrho (\varrho -1)}},{\displaystyle \frac{w+3v\upsilon +3\varrho (\varrho -1)u+\varrho (\varrho -1)(\varrho -2)t}{\varrho (\varrho -1)(\varrho -2)}};\zeta \right) \end{gathered}$$

(42)

By applying Lemma 1, conditions (39)–(42), we get

$$\begin{gathered} \psi \left(\mathfrak{g}\left(\zeta \right),\zeta {\mathfrak{g}}^{\prime }\left(\zeta \right),{\zeta }^2{\mathfrak{g}}^{″}\left(\zeta \right),{\zeta }^3{\mathfrak{g}}^{‴}\left(\zeta \right);\zeta \right) \\ =\left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right). \end{gathered}$$

(43)

Hence, (43) leads to

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

Using (41) and (42), we have

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

and

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

Accordingly, the $\psi \in {\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$ admissibility condition in Definition 10 corresponds to the $\psi \in {\Psi }_2[\Omega ,\mathfrak{q}]$ admissibility condition in Definition 6. Thus, by applying Lemma 1 and (37), we write ${\mathcal{F}}_{\mathfrak{g}\left(\mathfrak{U}\right)}\mathfrak{g}\left(\zeta \right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right)$ or, equivalently, ${\mathcal{F}}_{{\displaystyle \frac{1}{\zeta }}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\mathfrak{U}\right)}\left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}\right)\le {\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{U}\right)}\mathfrak{q}\left(\zeta \right)$,

i.e.,

$${\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}{\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right),$$

The proof is completed. □

If $\Omega \ne \mathbb{C}$ is a simply connected domain, then $\Omega =\lambda \left(\mathfrak{U}\right)$ for a well-chosen conformal mapping $\lambda \left(\zeta \right)$ of $\mathfrak{U}$ onto $\Omega$. In this case, the class ${\tilde{\Psi }}_{\kappa }[\lambda \left(\mathfrak{U}\right),\mathfrak{q}]$ is written as ${\tilde{\Psi }}_{\kappa }[\lambda ,\mathfrak{q}]$. The following result is an immediate consequence of Theorem 4.

Corollary 4.

Assume that $\psi \in {\tilde{\Psi }}_{\kappa }[\lambda ,\mathfrak{q}]$. If $\mathfrak{f}\in \mathcal{A}$ and $\mathfrak{q}\in {\mathfrak{Q}}_1$ satisfies Condition (37), then

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\left(\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right):\zeta \in \mathfrak{U}\right)\le {\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right),$$

implies that

$${\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}{\prec }_{\mathcal{F}}\mathfrak{q}\left(\zeta \right)(\zeta \in \mathfrak{U}).$$

In the particular case when $\mathfrak{q}\left(\zeta \right)=1+M\zeta (M>0)$ and, in view of Definition 8, the class ${\tilde{\Psi }}_{\kappa }[\Omega ,\mathfrak{q}]$ of admissible functions, which we denote by ${\tilde{\Psi }}_{\kappa }[\Omega ,M]$, is described below.

Definition 11.

Let Ω be a set in $\mathbb{C}$ and $M>0$. The class ${\tilde{\Psi }}_{\kappa }[\Omega ,M]$ of admissible functions consists of the functions $\phi :{\mathbb{C}}^4\times \mathfrak{U}\to \mathbb{C}$ such that

$$\begin{gathered} \psi \left(1+M{e}^{i\theta },1+{\displaystyle \frac{(\mathit{\ell }+\varrho )M{e}^{i\theta }}{\varrho }},1+{\displaystyle \frac{L+\left[2\varrho \mathit{\ell }+\varrho (\varrho -1)\right]M{e}^{i\theta }}{\varrho (\varrho -1)}},\right. \\ \left.1+{\displaystyle \frac{N+3\varrho L+\left[3\varrho (\varrho -1)\mathit{\ell }+\varrho (\varrho -1)(\varrho -2)\right]M{e}^{i\theta }}{\varrho (\varrho -1)(\varrho -2)}}\right)\notin \Omega \end{gathered}$$

whenever $\zeta \in \mathfrak{U}$, $\theta \in \mathbb{R}$ and

$$Re\left(L{e}^{-i\theta }\right)\geqq (\mathit{\ell }-1)\mathit{\ell }M\mathrm{and}Re\left(N{e}^{-i\theta }\right)\geqq 0(\forall \theta \in \mathbb{R};\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\}).$$

Corollary 5.

Let $\psi \in {\tilde{\Psi }}_{\kappa }[\Omega ,M]$. If $\mathfrak{f}\in \mathcal{A}$ satisfies

$$\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-\varrho {\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \mathit{\ell }M$$

If

$${\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\left(\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right):\zeta \in \mathfrak{U}\right)\le {\mathcal{F}}_{\Omega }\left(\zeta \right),$$

then

$${\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}{\prec }_{\mathcal{F}}1+M\zeta (\zeta \in \mathfrak{U}).$$

where $\mathit{\ell }\in \mathbb{N}\setminus \left\{1\right\},M>0$ and $\zeta \in \mathfrak{U}$.

In the special case when

$$\Omega =\mathfrak{q}\left(\mathfrak{U}\right)=\left\{w:\left|w-1\right|<M(M>0)\right\},$$

the class ${\tilde{\Psi }}_{\kappa }[\Omega ,M]$ is simply denoted by ${\tilde{\Psi }}_{\kappa }\left[M\right]$.

4. Third-Order Fuzzy Differential Superordination Results

We derive certain third-order fuzzy differential superordinations in this section. The following definition applies to the class of admissible functions for this purpose:

Definition 12.

Consider $\Omega \subset \mathbb{C}$, $\mathfrak{q}\in {\mathcal{H}}_0$ with ${\mathfrak{q}}^{\prime }\left(\zeta \right)\ne 0$ and $\vartheta \in \mathbb{N}\setminus \left\{1\right\}$. Functions $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{U}}\to \mathbb{C}$ that satisfy

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

(44)

whenever

$$\alpha =\mathfrak{q}\left(\zeta \right),\beta ={\displaystyle \frac{\zeta {\mathfrak{q}}^{\prime }\left(\zeta \right)+\vartheta (\varrho -1)\mathfrak{q}\left(\zeta \right)}{\vartheta \varrho }},$$

(45)

$$\mathrm{Re}\left\{{\displaystyle \frac{\varrho (\varrho -1)\gamma -(\varrho -1)(\varrho -2)\alpha }{\varrho \beta -(\varrho -1)\alpha }}-(2\varrho -3)\right\}\le {\displaystyle \frac{1}{\vartheta }}\mathrm{Re}\left\{1+{\displaystyle \frac{\zeta {\mathfrak{q}}^{″}\left(\zeta \right)}{{\mathfrak{q}}^{\prime }\left(\zeta \right)}}\right\},$$

(46)

$$\mathrm{Re}\left\{{\displaystyle \frac{\varrho (\varrho -1)\left[(\varrho -2)\delta +3(1-\varrho )\gamma +2(\varrho -2)\alpha \right]}{\varrho \beta -(\varrho -1)\alpha }}+3\varrho (\varrho -1)\right\}\le {\displaystyle \frac{1}{{\vartheta }^2}}\mathrm{Re}\left\{{\displaystyle \frac{{\zeta }^2{\mathfrak{q}}^{‴}\left(\zeta \right)}{{\mathfrak{q}}^{\prime }\left(\zeta \right)}}\right\},$$

(47)

where $\vartheta \ge 2$, $\xi \in \partial \mathfrak{U}$ and $\zeta \in \mathfrak{U}$, form the class of admissible functions denoted by ${\Psi }_{\kappa }^{\prime }[\Omega ,\mathfrak{q}]$.

Theorem 5.

Let $\psi \in {\Psi }_{\kappa }^{\prime }[\Omega ,\mathfrak{q}]$. If $\mathfrak{f}\in \mathcal{A}$ and ${\mathcal{L}}_{\varrho +1,\varkappa }$$\mathfrak{f}\left(\zeta \right)\in {\mathfrak{Q}}_0$ satisfy

$$\mathrm{Re}\left\{{\displaystyle \frac{\zeta {\mathfrak{q}}^{″}\left(\zeta \right)}{{\mathfrak{q}}^{\prime }\left(\zeta \right)}}\right\}\ge 0,\left|\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1){\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right|\le \vartheta \left|{\mathfrak{q}}^{\prime }\left(\xi \right)\right|,$$

(48)

and

$$\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right)$$

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

$${\mathcal{F}}_{\Omega }\left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\left(\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right):\zeta \in \mathfrak{U}\right),$$

implies

$$\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)(\zeta \in \mathfrak{U}).$$

(49)

Proof. 

Let $\mathfrak{g}\left(\zeta \right)$ be given by (20) and $\psi$ be given by (27). Since $\psi \in {\Psi }_{\kappa }^{\prime }[\Omega ,\mathfrak{q}]$, (28) and (49) yield

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

(50)

From (27), we deduce that the $\psi \in {\Psi }_{\kappa }^{\prime }[\Omega ,\mathfrak{q}]$ admissibility condition in Definition 12 corresponds to the admissibility condition for $\psi$ in Definition 8. Therefore, by applying the terms in (48) and Lemma 2, we write

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

(51)

or, equivalently,

$$\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)(\zeta \in \mathfrak{U}).$$

(52)

The proof is now complete. □

Example 2.

$\psi =2\alpha -\beta$. In Theorem 4, we obtain

$$\begin{array}{ccc}\hfill \psi \left(\alpha ,\beta ,\gamma ,\delta ;\tau \right)& =& 2t-{\displaystyle \frac{u+(\varrho -1)t}{\varrho t}}\hfill \\ & =& 2t-{\displaystyle \frac{\varrho {\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)-(\varrho -1)t+(\varrho -1)t}{\varrho }}\hfill \\ & =& 2t-{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\hfill \\ & =& 2{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)-{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\hfill \end{array}$$

which is analytic in $\mathfrak{U}$ and

$${\mathcal{F}}_{\Omega }\left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \left(2{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)-{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\right);$$

thus,

$$\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)(\zeta \in \mathfrak{U}),$$

(53)

or

$${\mathcal{F}}_{\mathfrak{q}\left(\mathfrak{D}\right)}\mathfrak{q}\left(\zeta \right)\le {\mathcal{F}}_{\zeta \mathfrak{f}\left(\mathfrak{D}\right)}\left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\right).$$

If $\Omega \ne \mathbb{C}$ is a simply connected domain and $\Omega =\lambda \left(\mathfrak{U}\right)$ for a well-chosen mapping $\lambda \left(\zeta \right)$ of $\mathfrak{U}$ onto $\Omega$, then ${\Psi }_{\kappa }^{\prime }[\lambda \left(\mathfrak{U}\right),\mathfrak{q}]={\Psi }_{\kappa }^{\prime }[\lambda ,\mathfrak{q}]$. The following theorem is derived from Theorem 5 using procedures similar to those in the previous section.

Theorem 6.

Consider $\psi \in {\Psi }_{\kappa }^{\prime }[\lambda ,\mathfrak{q}]$, and λ to be analytic in $\mathfrak{U}$. If $\mathfrak{f}\in \mathcal{A}$ and ${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\in {\mathfrak{Q}}_0$ satisfy the conditions in (48) and

$$\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right)$$

(54)

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

$${\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right),$$

implies

$$\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)(\zeta \in \mathfrak{U}).$$

(55)

For a well-chosen $\psi$, the next result establishes the existence of the best fuzzy subordinant of (55).

Theorem 7.

Consider λ to be analytic in $\mathfrak{U}$, $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{U}}⟶\mathbb{C}$ and ψ to be given by (27). Let the differential equation

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

have a solution $\mathfrak{q}\left(\zeta \right)\in {\mathfrak{Q}}_0$. If $\mathfrak{f}\in \mathcal{A}$ and ${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\in {\mathfrak{Q}}_0$ satisfy Condition (48) and

$$\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right)$$

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

$${\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right),$$

implies

$$\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)(\zeta \in \mathfrak{U}).$$

and $\mathfrak{q}$ is the best fuzzy subordinant.

Proof. 

The proof of Theorem 7 is similar to that of Theorem 3 so we omitted it. □

Next, we consider a new definition for the class of admissible functions, ${\tilde{\Psi }}_{\kappa }^{\prime }[\Omega ,\mathfrak{q}]$.

Definition 13.

Consider $\Omega \subset \mathbb{C}$, $\mathfrak{q}\in {\mathcal{H}}_1$ with ${\mathfrak{q}}^{\prime }\left(\zeta \right)\ne 0$ and $\vartheta \in \mathbb{N}\setminus \left\{1\right\}$. Functions $\psi :{\mathbb{C}}^4\times \overline{\mathfrak{U}}\to \mathbb{C}$ that satisfy

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

whenever

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

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

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

where $\vartheta \ge 2$, $\xi \in \partial \mathfrak{U}$ and $\zeta \in \mathfrak{U}$, form the class of admissible functions denoted by ${\tilde{\Psi }}_{\kappa }^{\prime }[\Omega ,\mathfrak{q}]$.

If $\Omega \ne \mathbb{C}$ is a simply connected domain and $\Omega =\lambda \left(\mathfrak{U}\right)$ for a well-chosen conformal mapping $\lambda \left(\zeta \right)$ of $\mathfrak{U}$ onto $\Omega$, then the class ${\tilde{\Psi }}_{\kappa }^{\prime }[\lambda \left(\mathfrak{U}\right),\mathfrak{q}]={\tilde{\Psi }}_{\kappa }^{\prime }[\lambda ,\mathfrak{q}]$. The following theorem is derived from Theorem 7 using procedures similar to those in the previous section.

Theorem 8.

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

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

have a solution $\mathfrak{q}\left(\zeta \right)\in {\mathfrak{Q}}_1$. If $f\in A$ and ${\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}\in {\mathfrak{Q}}_1$ satisfy the conditions in (37) and

$$\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right)$$

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

$${\mathcal{F}}_{\lambda (\Omega )}\lambda \left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right),$$

implies

$$\mathfrak{q}\left(\zeta \right){\prec }_{\mathcal{F}}{\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }}(\zeta \in \mathfrak{U}),$$

and $\mathfrak{q}$ is the best fuzzy subordinant.

Proof. 

We left out Theorem 8 since its proof is comparable to that of Theorem 4. □

5. Sandwich-Type Results

Two sandwich-type outcomes are shown in this section. Combining Theorems 1 and 5 yields the sandwich-type result that follows:

Theorem 9.

Let the functions ${\lambda }_1$ and ${\mathfrak{q}}_1$ be analytic functions in $\mathfrak{U}$. Also, let ${\lambda }_2$ be univalent in $\mathfrak{U}$, ${\mathfrak{q}}_2\in {\mathfrak{Q}}_0$ with ${\mathfrak{q}}_1\left(0\right)={\mathfrak{q}}_2\left(0\right)=1$ and $\psi \in {\Psi }_{\kappa }[{\lambda }_2,{\mathfrak{q}}_2]\cap {\Psi }_{\kappa }^{\prime }[{\lambda }_1,{\mathfrak{q}}_1]$. If $\mathfrak{f}\in \mathcal{A}$ and ${\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)\in {\mathfrak{Q}}_0\cap {\mathcal{H}}_0$ and

$$\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right)\in S,$$

with Conditions (18) and (48) being satisfied, then

$${\mathcal{F}}_{{\lambda }_1(\Omega )}{\lambda }_1\left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \left({\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right),{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right);\zeta \right)\le {\mathcal{F}}_{{\lambda }_2(\Omega )}{\lambda }_2\left(\zeta \right),$$

implies that

$${\lambda }_1\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}{\lambda }_2\left(\zeta \right).$$

Similarly, combining Theorems 4 and 8, we obtain

Theorem 10.

Let the functions ${\tilde{\lambda }}_1,{\tilde{\mathfrak{q}}}_1\in \mathcal{H}\left(U\right)$. Also, let ${\tilde{\lambda }}_2\in S$, ${\tilde{\mathfrak{q}}}_2\in {\mathfrak{Q}}_1$ with ${\tilde{\mathfrak{q}}}_1\left(0\right)={\tilde{\mathfrak{q}}}_2\left(0\right)=1$ 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 $\mathfrak{f}\in \mathcal{A}$ and ${\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)\in {\mathfrak{Q}}_1\cap {\mathcal{H}}_1$ and

$$\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right)\in S,$$

with conditions (18) and (48) being satisfied, then

$${\mathcal{F}}_{{\tilde{\lambda }}_1(\Omega )}{\tilde{\lambda }}_1\left(\zeta \right)\le {\mathcal{F}}_{\psi ({\mathbb{C}}^4\times \mathfrak{U})}\psi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho +1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -1,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }},{\displaystyle \frac{{\mathcal{L}}_{\varrho -2,\varkappa }\mathfrak{f}\left(\zeta \right)}{\zeta }};\zeta \right)\le {\mathcal{F}}_{{\tilde{\lambda }}_2(\Omega )}{\tilde{\lambda }}_2\left(\zeta \right),$$

implies that

$${\tilde{\lambda }}_1\left(\zeta \right){\prec }_{\mathcal{F}}{\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right){\prec }_{\mathcal{F}}{\tilde{\lambda }}_2\left(\zeta \right).$$

Remark 2.

By taking $\varkappa =1$ in (6), we obtain the operator ${\mathcal{L}}_{\varrho }$ defined as follows:

$${\mathcal{L}}_{\varrho }\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{(-1)}^{\kappa -1}}{4^{\kappa -1}{\left(\varrho \right)}_{\kappa -1}(\kappa -1)!}}{a}_{\kappa }{\zeta }^{\kappa }.$$

which is related to Bessel functions. If this function is used, new results in third-order fuzzy differential subordination can be developed.

6. Conclusions

The theory of differential subordination and superordination advances as a result of the new findings from the inquiry presented in this paper. The basic concepts needed for this study, the Lommel function ${\mathcal{L}}_{\varrho ,\varkappa }\mathfrak{f}\left(\zeta \right)$, and the motivation for the topic’s inquiry are all included in the introduction in Section 1. Section 2 reports the research’s preliminary known results, whereas Section 3 presents the primary findings. The fundamental theorems of the new third-order fuzzy differential subordination theory were established and validated. The findings in Section 4 of this paper contribute to the field of third-order fuzzy differential superordination presenting dual results to those reported in Section 3. Using the previously established new outcome in Section 3 and Section 4, sandwich-type results are stated in Section 5.

The present endeavor provides important knowledge to improve the newly started lines of research because there are currently no published papers on the idea of using the two new dual theories to develop third-order fuzzy sandwich-type outcomes. Also, new results that would enhance our knowledge regarding the topic of third-order fuzzy differential subordination theory could be further developed in view of the statement of Remark 2.