Application of Fuzzy Subordinations and Superordinations for an Analytic Function Connected with $\mathfrak{q}$-Difference Operator

Abstract

This paper extends the idea of subordination from the theory of fuzzy sets to the geometry theory of analytic functions with a single complex variable. The purpose of this work is to define fuzzy subordination and illustrate its main characteristics. New fuzzy differential subordinations will be introduced with the help of this effort. We define a linear operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )$ using the concept of the $\mathfrak{q}$-calculus operators. New fuzzy differential subordinations are created by employing the previously described operator, functions from the new class, and well-known lemmas. Specific corollaries derived from the operator proved the many examples created for the fuzzy differential subordinations, as well as the theorems, and demonstrate how the new theoretical conclusions apply to the fuzzy differential superordinations provided in this research.

1. Introduction

To define fuzzy subordination, utilise the use of the fuzzy set notion initially presented by Zadeh in [1]. The fuzzy set idea that is a part of the differential subordination and superordination theories created in geometric function theory is used in this paper’s results. In geometric function theory, numerous mathematical domains have developed extensions as a result of the fuzzy set concept being used in investigations. The notion of a fuzzy set was used to investigate fuzzy subordination in geometric function theory in 2011 [2]. Since 2012 [3,4], when Miller and Mocanu’s classical theory of differential subordination [5] began to be modified by incorporating fuzzy theory elements, the theory of fuzzy differential subordination has been under development. The idea of fuzzy differential superordination was first presented in 2017 [6]. Several scholars have since examined various characteristics of differential operators, including fuzzy differential subordinations and superordinations [7,8,9,10,11,12,13].

This article presents the derivation of specific fuzzy differential subordinations and superordinations for an operator of the $\mathfrak{q}$-Ruscheweyh operator and the $\mathfrak{q}$-Cătas operator presented by Ali et al. in [14].

To derive the article’s findings, we employed the concepts and outcomes presented below:

Now suppose $\mathcal{H}=\mathcal{H}\left(\mathfrak{D}\right)$ is the class of all analytic functions in the open unit disc $\mathfrak{D}:=\{\tau \in \mathbb{C}:\left|\tau \right|<1\}$. Also, $\mathcal{H}[a,n]$ represents $\mathcal{H}$’s subclass, which includes $f\in \mathcal{H}$ provided by

$$\mathfrak{f}\left(\tau \right)=a+a_n{\tau }^n+a_{n+1}{\tau }^{n+1}+...,\tau \in \mathfrak{D},$$

we note that $\mathcal{H}[1,1]$ is the class of the function of the form

$$\mathfrak{f}\left(\tau \right)=1+a_1\tau +a_2{\tau }^2+...,\tau \in \mathfrak{D}.$$

The class A$\left(n\right)$, which is another well-known subclass of $\mathcal{H},$ is made up of $\mathfrak{f}\in \mathcal{H},$ as shown by

$$\mathfrak{f}\left(\tau \right)=\tau +\sum _{\kappa =n+1}^{\infty }{a}_{\kappa }{\tau }^{\kappa },\tau \in \mathfrak{D},$$

(1)

with $n\in \mathbb{N}=\left\{1,2,3,..\right\},$ and $\mathbf{A}=\mathbf{A}\left(1\right).$

The subclass of $\mathbf{A}$ is defined by

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

indicates the convex functions class in $\mathfrak{D}$.

The following form applies to $\mathfrak{f},\mathcal{L}\in \mathbf{A}\left(n\right)$, where $\mathfrak{f}$ is donated by (1) and $\mathcal{L}$

$$\mathcal{L}\left(\tau \right)=\tau +\sum _{\kappa =n+1}^{\infty }{b}_{\kappa }{\tau }^{\kappa },\tau \in \mathfrak{D}.$$

The definition of convolution product is: *: $A\to A$

$$(\mathfrak{f}*\mathcal{L})\left(\tau \right):=\tau +\sum _{\kappa =n+1}^{\infty }{a}_{\kappa }{b}_{\kappa }{\tau }^{\kappa },\tau \in \mathfrak{D}.$$

In particular ([15,16]) Jackson’s $\mathfrak{q}$-difference operators ${\mathfrak{d}}_{\mathfrak{q}}:$ $A\to A$ are defined by

$${\mathfrak{d}}_{\mathfrak{q}}\mathfrak{f}\left(\tau \right):=\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{f}\left(\tau \right)-\mathfrak{f}\left(\mathfrak{q}\tau \right)}{(1-\mathfrak{q})\tau }}\hfill & \left(\tau \ne 0;0<\mathfrak{q}<1\right)\hfill \\ \hfill & \\ {\mathfrak{f}}^{\prime }\left(0\right)\hfill & \left(\tau =0\right).\hfill \end{array}\right.$$

(2)

We might be able to use merely $\kappa \in \mathbb{N}$. It has already been written once

$${\mathfrak{d}}_{\mathfrak{q}}\left\{\sum _{\kappa =1}^{\infty }{a}_{\kappa }{\tau }^{\kappa }\right\}=\sum _{\kappa =1}^{\infty }{\left[\kappa \right]}_{\mathfrak{q}}{a}_{\kappa }{\tau }^{\kappa -1},$$

(3)

where

$$\begin{array}{ccc}\hfill {\left[\kappa \right]}_{\mathfrak{q}}& =& {\displaystyle \frac{1-{\mathfrak{q}}^{\kappa }}{1-\mathfrak{q}}}=1+{\displaystyle \sum _{n=1}^{\kappa -1}}{\mathfrak{q}}^n,{\displaystyle \underset{q\to 1^{-}}{lim}}{\left[\kappa \right]}_q=\kappa ,\hfill \\ \hfill {\left[\kappa \right]}_{\mathfrak{q}}!& =& \left\{\begin{array}{cc}{\displaystyle \prod _{n=1}^{\kappa }}{\left[n\right]}_{\mathfrak{q}},& \kappa \in \mathbb{N}\\ 1& \kappa =0.\end{array}\right.\hfill \end{array}$$

(4)

In [17], Aouf and Madian investigate the $\mathfrak{q}$-analogue Cătas operator ${\mathfrak{I}}_{\mathfrak{q}}^s(\nu ,\varsigma ):\mathbf{A}\to \mathbf{A}$$(s\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},\varsigma ,\nu \ge 0$, $0<\mathfrak{q}<1)$ as follows:

$${\mathfrak{I}}_{\mathfrak{q}}^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)=\tau +\sum _{\kappa =2}^{\infty }{\left({\displaystyle \frac{{\left[1+\varsigma \right]}_{\mathfrak{q}}+\nu ({\left[\kappa +\varsigma \right]}_{\mathfrak{q}}-{\left[1+\varsigma \right]}_{\mathfrak{q}})}{{\left[1+\varsigma \right]}_{\mathfrak{q}}}}\right)}^s{a}_{\kappa }{\tau }^{\kappa }.$$

Also, the $\mathfrak{q}$-Ruscheweyh operator ${\Re }_{\mathfrak{q}}^{\rho }\mathfrak{f}\left(\tau \right)$ was examined in 2014 by Aldweby and Darus [18]

$${\Re }_{\mathfrak{q}}^{\rho }\mathfrak{f}\left(\tau \right)=\tau +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{[\kappa +\rho -1]}_{\mathfrak{q}}}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}}{a}_{\kappa }{\tau }^{\kappa },(\rho \ge 0,0<\mathfrak{q}<1),$$

where ${\left[a\right]}_{\mathfrak{q}}$ and ${\left[a\right]}_{\mathfrak{q}}!$ are defined in (4).

We define

$${\mathfrak{f}}_{\mathfrak{q},\nu ,\varsigma }^s\left(\tau \right)=\tau +\sum _{\kappa =2}^{\infty }{\left({\displaystyle \frac{{\left[1+\varsigma \right]}_{\mathfrak{q}}+\nu ({\left[\kappa +\varsigma \right]}_{\mathfrak{q}}-{\left[1+\varsigma \right]}_{\mathfrak{q}})}{{\left[1+\varsigma \right]}_{\mathfrak{q}}}}\right)}^s{\tau }^{\kappa }.$$

Now we define a new function ${\mathfrak{f}}_{\mathfrak{q},\nu ,\varsigma }^{s,\rho }\left(\tau \right)$ as follows:

$${\mathfrak{f}}_{\mathfrak{q},\nu ,\varsigma }^s\left(\tau \right)*{\mathfrak{f}}_{\mathfrak{q},\nu ,\varsigma }^{s,\rho }\left(\tau \right)=\tau +\sum _{\kappa =2}^{\infty }{\displaystyle \frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}}{\tau }^{\kappa }.$$

In [14], an extended multiplier operator was defined applying the operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )$ as follows:

Definition 1

([14]). For $s\in {\mathbb{N}}_0,\varsigma ,\nu ,\rho \ge 0,$ and $0<\mathfrak{q}<1$ with the operator’s assistance ${\mathfrak{f}}_{\mathfrak{q},\nu ,\varsigma }^{s,\rho }\left(\tau \right)$ we define the new linear extended multiplier by the $\mathfrak{q}$-Ruscheweyh operator and the $\mathfrak{q}$-Cătas operator, ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ):\mathbf{A}\to \mathbf{A}$ as:

$${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)={\mathfrak{f}}_{\mathfrak{q},\nu ,\varsigma }^{s,\rho }\left(\tau \right)*\mathfrak{f}\left(\tau \right).$$

(5)

For $\mathfrak{f}\in \mathbf{A}$and (5), it is instead of

$${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)=\tau +\sum _{\kappa =2}^{\infty }{\psi }_{\mathfrak{q}}^{*s}(\kappa ,\nu ,\varsigma ){\displaystyle \frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}}{a}_{\kappa }{\tau }^{\kappa },$$

(6)

where

$${\psi }_{\mathfrak{q}}^{*s}(\kappa ,\nu ,\varsigma )={\left({\displaystyle \frac{{\left[1+\varsigma \right]}_{\mathfrak{q}}}{{\left[1+\varsigma \right]}_{\mathfrak{q}}+\nu ({\left[\kappa +\varsigma \right]}_{\mathfrak{q}}-{\left[1+\varsigma \right]}_{\mathfrak{q}})}}\right)}^s.$$

Differential subordinations and $\mathfrak{q}$-calculus operators form the basis of several of the issues in geometric function theory. In 1990, Ismail et al. investigated the first applications of $\mathfrak{q}$-analogue in geometric function theory by defining the class of $\mathfrak{q}$-starlike functions [19]. A number of writers have concentrated on the $\mathfrak{q}$-analogues of the Sălăgean differential operators defined in [20] and the Ruscheweyh differential operators created in [21]. The study of differential subordinations with a particular $\mathfrak{q}$-Ruscheweyh-type derivative operator in [22] is one example.

2. Preliminaries

The below fundamentals will serve as a means of demonstrating the novel findings presented in the subsequent section.

Definition 2

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

Definition 3

([1]). A pair $\left(b,{\mathcal{F}}_b\right)$, where ${\mathcal{F}}_b:\xi \to \left[0,1\right]$ and $b=\left\{\mathrm{x}\in \xi :0<{\mathcal{F}}_b\left(\mathrm{x}\right)\le 1\right\}$ is called a fuzzy subset of ξ and ${\mathcal{F}}_b$ is said to be the membership function of the fuzzy set $\left(b,{\mathcal{F}}_b\right)$$\xi \ne \varphi ,$ An application $\mathcal{F}:\xi \to \left[0,1\right]$ is called a fuzzy subset.

Definition 4

([23]). Suppose that $\mathcal{F}:\mathbb{C}\to {\mathbb{R}}_{+}$ (${\mathbb{R}}_{+}$ is the positive real number and satisfies ${\mathcal{F}}_{\mathbb{C}}\left(\tau \right)=\left|\mathcal{F}\left(\tau \right)\right|,\tau \in \mathbb{C})$, defined as

$${\mathcal{F}}_{\mathbb{C}}\left(\mathbb{C}\right)=\{\tau :\tau \in \mathbb{C}\mathit{and}0<\left|\mathcal{F}\left(\tau \right)\right|\le 1\}=supp(\mathbb{C},{\mathcal{F}}_{\mathbb{C}}\left(\tau \right)),$$

and

$${\mathcal{F}}_{\mathbb{C}}\left(\mathbb{C}\right)=\{\tau :\tau \in \mathbb{C}\mathit{and}0<\left|\mathcal{F}\left(\tau \right)\right|\le 1\}={\mathfrak{D}}_{\mathcal{F}}(0,1).$$

It has been observed that $(\mathbb{C},{\mathcal{F}}_{\mathbb{C}}\left(\tau \right))$ corresponds to its fuzzy unit disk ${\mathfrak{D}}_{\mathcal{F}}(0,1).$

Definition 5

([2]). Let ${\tau }_0\in \mathfrak{D}$ and $\mathfrak{f},\lambda \in \mathcal{H}\left(\mathfrak{D}\right).$ $\mathfrak{f}$ be a fuzzy subordinate to λ and written as $\mathfrak{f}{\prec }_{\mathcal{F}}\lambda$ or $\mathfrak{f}\left(\tau \right){\prec }_{\mathcal{F}}\lambda \left(\tau \right)$ if every one of the following conditions is met $\mathfrak{f}\left({\tau }_0\right)=\lambda \left({\tau }_0\right)$ and

$$\left|{\mathcal{F}}_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le \left|{\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\lambda \left(\tau \right)\right)\right|,\tau \in \mathfrak{D}.$$

Definition 6

([3]). Let $\mathrm{\Omega }:{\mathbb{C}}^3\times \mathfrak{D}\to \mathbb{C}$ and ζ be univalent in $\mathfrak{D}$ with Ω$(a,0,0,0)=\zeta \left(0\right)=\mathrm{a}$. If ω is analytic in $\mathfrak{D}$ with $\omega \left(0\right)=\mathrm{a}$ and satisfies the (second-order) fuzzy differential subordination,

$${\mathcal{F}}_{\mathrm{\Omega }({\mathbb{C}}^3\times \mathfrak{D})}\mathrm{\Omega }(\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right);\tau )\le \left|{\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\left(\zeta \left(\mathfrak{D}\right)\right)\right|,$$

(7)

i.e.,

$$\mathrm{\Omega }\left(\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right);\tau \right){\le }_{\mathcal{F}}\left(\zeta \left(\tau \right)\right),\tau \in \mathfrak{D},$$

then ω is a fuzzy dominant and fuzzy solution if

$$\left|{\mathcal{F}}_{\omega \left(\mathfrak{D}\right)}\omega \left(\tau \right)\right|\le \left|{\mathcal{F}}_{\chi \left(\mathfrak{D}\right)}\chi \left(\tau \right)\right|i.e.,\omega \left(\tau \right){\prec }_{\mathcal{F}}\chi \left(\tau \right)\tau \in \mathfrak{D},$$

for all ω satisfying (7). A fuzzy dominant $\tilde{\chi }$ that satisfies

$$\left|{\mathcal{F}}_{\tilde{\chi }\left(\mathfrak{D}\right)}\tilde{\chi }\left(\tau \right)\right|\le \left|{\mathcal{F}}_{\chi \left(\mathfrak{D}\right)}\chi \left(\tau \right)\right|i.e.,\tilde{\chi }\left(\tau \right){\prec }_{\mathcal{F}}\chi \left(\tau \right)\tau \in \mathfrak{D},$$

for all fuzzy dominant χ of (7) is fuzzy best dominant of (7).

Definition 7

([6]). Let $\mathrm{\Omega }:{\mathbb{C}}^3\times \mathfrak{D}\to \mathbb{C}$ and ζ be an analytic in $\mathfrak{D}$. When the univalent function ω and $\mathrm{\Omega }(\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right);\tau )$ verifies for any $\tau \in \mathfrak{D}$ the fuzzy differential superordination:

$$\left|{\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\left(\zeta \left(\mathfrak{D}\right)\right)\right|\le {\mathcal{F}}_{\mathrm{\Omega }({\mathbb{C}}^3\times \mathfrak{D})}\mathrm{\Omega }(\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{\prime \prime }\left(\tau \right);\tau ),$$

(8)

then the fuzzy differential superordination has ω as a fuzzy solution. A fuzzy subordination of the fuzzy differential superordination χ an analytic function with

$$\left|{\mathcal{F}}_{\chi \left(\mathfrak{D}\right)}\chi \left(\tau \right)\right|\le \left|{\mathcal{F}}_{\omega \left(\mathfrak{D}\right)}\omega \left(\tau \right)\right|i.e.,\chi \left(\tau \right){\prec }_{\mathcal{F}}\omega \left(\tau \right)\tau \in \mathfrak{D},$$

for all ω satisfying (8). A fuzzy subordinant $\tilde{\chi }$ that satisfies

$$\left|{\mathcal{F}}_{\chi \left(\mathfrak{D}\right)}\chi \left(\tau \right)\right|\le \left|{\mathcal{F}}_{\tilde{\chi }\left(\mathfrak{D}\right)}\tilde{\chi }\left(\tau \right)\right|i.e.,\chi \left(\tau \right){\prec }_{\mathcal{F}}\tilde{\chi }\left(\tau \right)\in \mathfrak{D},$$

for all fuzzy subordinant χ of (8) is fuzzy best subordinant of (8).

Assume that ℘ the set of analytic and injective functions on $\overline{\mathfrak{D}}\setminus E\left(\chi \right)$, with ${\chi }^{\prime }\left(\tau \right)\ne 0$ for $\tau \in \partial \mathfrak{D}\setminus E\left(\chi \right)$ and

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

Also, $\wp \left(a\right)$ is the subclass of ℘ with $\chi \left(0\right)=a.$

To derive these fuzzy inequalities, we require the lemmas listed below:

Lemma 1

([4]). Let λ be a convex function in $\mathfrak{D}$ and let the function

$$\zeta \left(\tau \right)=n\gamma \tau {\lambda }^{\prime }\left(\tau \right)+\lambda \left(\tau \right),$$

with $\tau \in \mathfrak{D},$n$\in \mathbb{N}$ and $\gamma >0.$ If the function $\eta \in \mathcal{H}\left[\lambda \right(0),n]$ and $\mathrm{\Omega }:{\mathbb{C}}^2\times \mathfrak{D}\to \mathbb{C}$

$$\mathrm{\Omega }\left(\eta \left(\tau \right),\tau {\eta }^{\prime }\left(\tau \right)\right)=\eta \left(\tau \right)+\gamma \tau {\eta }^{\prime }\left(\tau \right)$$

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

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\gamma \tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),$$

implies

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

and λ is fuzzy best $\left(\lambda \right(0),n)$-dominant.

Lemma 2

([4]). Let ζ be a convex with $\zeta \left(0\right)=a$ and set $\gamma \in {\mathbb{C}}^{*}=\mathbb{C}\setminus \left\{0\right\}$ be a complex number with $Re\left(\gamma \right)\ge 0$. If $\eta \in \mathcal{H}[a,n]$ with $\eta \left(0\right)=a$ and $\mathrm{\Omega }:{\mathbb{C}}^2\times \mathfrak{D}\to \mathbb{C},$

$$\mathrm{\Omega }\left(\eta \left(\tau \right),\tau {\eta }^{\prime }\left(\tau \right)\right)=\eta \left(\tau \right)+{\displaystyle \frac{1}{\gamma }}\tau {\eta }^{\prime }\left(\tau \right)$$

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

$${\mathcal{F}}_{\mathrm{\Omega }({\mathbb{C}}^2\times \mathfrak{D})}\left(\eta \left(\tau \right)+{\displaystyle \frac{1}{\gamma }}\tau {\eta }^{\prime }\left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

implies

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\prec {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

where

$$\lambda \left(\tau \right)={\displaystyle \frac{\gamma }{n{\tau }^{(\gamma /n)}}}{\int }_0^{\tau }\zeta \left(t\right)t^{(\gamma /n)-1}dt,\tau \in \mathfrak{D}.$$

The function λ is convex and is the fuzzy best $(a,n)$-dominant.

Lemma 3

([6]). Let ζ be a convex with $\zeta \left(0\right)=a,$ and set $\gamma \in {\mathbb{C}}^{*},$ with $Re\left(\gamma \right)\ge 0$. When $\eta \in \wp \cap \mathcal{H}[a,n],$$\eta \left(\tau \right)+{\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{\gamma }}$ verifies for any $\tau \in \mathfrak{D}$ the fuzzy differential superordination

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\eta \left(\tau \right)+{\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{\gamma }}\right),$$

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

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right),\tau \in \mathfrak{D},$$

is satisfied for any $\tau \in \mathfrak{D}$ by the convex function $\lambda \left(\tau \right)={\displaystyle \frac{\gamma }{n{\tau }^{(\gamma /n)}}}{\int }_0^{\tau }\zeta \left(t\right)t^{(\gamma /n)-1}dt,$where λ is the fuzzy best subordinant.

Lemma 4

([6]). Let a convex λ in $\mathfrak{D},$ and

$$\zeta \left(\tau \right)=\lambda \left(\tau \right)+{\displaystyle \frac{\tau {\lambda }^{\prime }\left(\tau \right)}{\gamma }},\tau \in \mathfrak{D},$$

with $\gamma \in {\mathbb{C}}^{*},$$Re\left(\gamma \right)\ge 0$. If $\eta \in \wp \cap \mathcal{H}[a,n],$$\eta \left(\tau \right)+{\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{\gamma }}$ verifies for any $\tau \in \mathfrak{D}$ the fuzzy differential superordination

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\lambda \left(\tau \right)+{\displaystyle \frac{\tau {\lambda }^{\prime }\left(\tau \right)}{\gamma }}\right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\eta \left(\tau \right)+{\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{\gamma }}\right),\tau \in \mathfrak{D},$$

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

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right),\tau \in \mathfrak{D},$$

is satisfied for any $\tau \in \mathfrak{D}$ by the convex function $\lambda \left(\tau \right)={\displaystyle \frac{\gamma }{n{\tau }^{(\gamma /n)}}}{\int }_0^{\tau }\zeta \left(t\right)t^{(\gamma /n)-1}dt,$which is the fuzzy best subordinant.

For i,u,e and e(e$\notin {\mathbb{Z}}_0^{-}=\{0,-1,-2,...\})$ let

$${}_{2}F_1(\mathrm{i},\mathrm{u};\mathrm{e};\tau )=1+{\displaystyle \frac{\mathrm{iu}}{\mathrm{e}}}.{\displaystyle \frac{\tau }{1!}}+{\displaystyle \frac{\mathrm{i}(i+1)\mathrm{u}(\mathrm{u}+1)}{\mathrm{e}(\mathrm{e}+1)}}.{\displaystyle \frac{{\tau }^2}{2!}}+....$$

Lemma 5

([24]). For i,u and e(e$\notin {\mathbb{Z}}_0^{-})$, complex parameters

$${\int }_0^{\mathfrak{1}}{t}^{u-1}{(1-t)}^{e-u-1}{(1-\tau t)}^{-i}dt={\displaystyle \frac{\mathrm{\Gamma }\left(u\right)\mathrm{\Gamma }(e-i)}{\mathrm{\Gamma }\left(e\right)}}{}_{2}F_1(i,u;e;\tau )(Re\left(e\right)>Re\left(u\right)>0);$$

$${}_{2}F_1(i,u;e;\tau )={}_{2}F_1(u,i;e;\tau );$$

$${}_{2}F_1(i,u;e;\tau )={(1-\tau )}^{-i}{}_{2}F_1(i,e-u;e;{\displaystyle \frac{\tau }{\tau -1}});$$

$${}_{2}F_1(1,1;2;{\displaystyle \frac{i\tau }{i\tau +1}})={\displaystyle \frac{(1+i\tau )ln(1+i\tau )}{i\tau }}$$

$${}_{2}F_1(1,1;3;{\displaystyle \frac{i\tau }{i\tau +1}})={\displaystyle \frac{2(1+i\tau )}{i\tau }}\left(1-{\displaystyle \frac{ln(1+i\tau )}{i\tau }}\right).$$

By using the operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ),$ we study the fuzzy differential subordination results in Section 3. Additionally, examples are given to illustrate possible applications of the findings. The best subordinants are also identified for fuzzy differential superordinations concerning the operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )$, which are examined in Section 4. Examples are also provided to highlight the significance of the findings.

3. Fuzzy Differential Subordination Results

We construct the class $\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;\upsilon ),$ and establish the fuzzy differential subordination for the function that belongs to this class by utilising the operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)$.

Definition 8.

Let $\upsilon \in [0,1).$ The class $\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;\upsilon )$ includes the function $\mathfrak{f}\in \mathbf{A}$ with

$${\mathcal{F}}_{{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\right)}^{\prime }\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }>\upsilon ,\tau \in \mathfrak{D}.$$

(9)

We see that for $\upsilon =0,$ the class $\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;\upsilon )$ reduces to $\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ).$

Theorem 1.

Let λ be a convex function in $\mathfrak{D}$, and $d>0$ and let

$$\zeta \left(\tau \right)=\lambda \left(\tau \right)+{\displaystyle \frac{\tau {\lambda }^{\prime }\left(\tau \right)}{d+2}},\tau \in \mathfrak{D}.$$

(10)

For $\mathfrak{f}\in \mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;\upsilon ),$ consider

$${\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)={\displaystyle \frac{d+2}{{\tau }^{d+1}}}{\int }_0^{\tau }{t}^d\mathfrak{f}\left(t\right)dt,\tau \in \mathfrak{D},$$

(11)

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),$$

(12)

implies

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),$$

λ is the fuzzy best dominant for $\tau \in \mathfrak{D}.$

Proof.  

We can write (11) as follows:

$${\tau }^{d+1}{\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)=(d+2){\int }_0^{\tau }{t}^d\mathfrak{f}\left(t\right)dt,\tau \in \mathfrak{D},$$

and differentiating it, we obtain

$$\tau {\left({\mathcal{L}}_d\left(\mathfrak{f}\right)\right)}^{\prime }\left(\tau \right)+(d+1){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)=(d+2)\mathfrak{f}\left(\tau \right),$$

and

$$\tau {\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }+(d+1){\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)=(d+2){\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right),\tau \in \mathfrak{D}.$$

Differentiating the last relation, we obtain

$${\displaystyle \frac{\tau {\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{″}}{d+2}}+{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }={\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D}.$$

Applying the final relation, the fuzzy differential subordination (12) will be

$$\begin{array}{ccc}& & {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{″}}{d+2}}+{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }\right)\hfill \\ & \le & {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\lambda }^{\prime }\left(\tau \right)}{d+2}}+\lambda \left(\tau \right)\right).\hfill \end{array}$$

(13)

This means

$$\eta \left(\tau \right)={\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }\in \mathcal{H}[1,1],$$

(14)

the fuzzy differential subordination (13) have the next type:

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{d+2}}+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\lambda }^{\prime }\left(\tau \right)}{d+2}}+\lambda \left(\tau \right)\right).$$

Through Lemma 1, we find

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),$$

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\tau \right)}^{\prime }\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)$$

where $\lambda$ is the fuzzy best dominant. □

Theorem 2.

Let $\zeta \left(\tau \right)={\displaystyle \frac{1-(2\upsilon -1)\tau }{1-\tau }},\upsilon \in [0,1).$ For ${\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)$ given by (11), with $d>0,$ then,

$${\mathcal{L}}_d\left[\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;\upsilon )\right]\subset \mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;{\upsilon }^{*}),$$

(15)

where

$${\upsilon }^{*}=2(1-\upsilon )-{(\upsilon -1)}_2{F}_1(1,1,d+3;{\displaystyle \frac{1}{2}}).$$

(16)

Proof.  

Using the identical procedures as the Theorem 1, then

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{d+2}}+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),$$

holds, with $\eta$ defined by (14).

Through Lemma 2, we find

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),$$

similar to

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\tau \right)}^{\prime }\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\prec {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),$$

where

$$\begin{array}{ccc}\hfill \lambda \left(\tau \right)& =& {\displaystyle \frac{d+2}{{\tau }^{d+2}}}{\int }_0^{\tau }{t}^{d+1}{\displaystyle \frac{1-(2\upsilon -1)t}{1-t}}dt\hfill \\ & =& (2\upsilon -1)-{\displaystyle \frac{2(d+2)(\upsilon -1)}{{\tau }^{d+2}}}{\int }_0^{\tau }{\displaystyle \frac{t^{d+1}}{1-t}}dt.\hfill \end{array}$$

By using Lemma 5, we find

$$\lambda \left(\tau \right)=(2\upsilon -1)-2(\upsilon -1){{(1-\tau )}^{-1}}_2{F}_1(1,1,d+3;{\displaystyle \frac{\tau }{\tau -1}}).$$

Given that $\lambda$ is a convex function and $\lambda \left(\mathfrak{D}\right)$ is symmetric around the real axis, we have

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\tau \right)}^{\prime }& \ge & \underset{\left|\tau \right|=1}{min}{\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\lambda \left(\tau \right)\right)={\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda (-1)={\upsilon }^{*}\hfill \\ & =& (2\upsilon -1)-{(\upsilon -1)}_2{F}_1(1,1,d+3;{\displaystyle \frac{1}{2}}).\hfill \end{array}$$

□

If we put $\upsilon =0,$ in Theorem 2 we find

Corollary 1.

Let

$${\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)={\displaystyle \frac{d+2}{{\tau }^{d+1}}}{\int }_0^{\tau }{t}^d\mathfrak{f}\left(t\right)dt,d>0,$$

then,

$${\mathcal{L}}_d\left[\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\right]\subset \mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;{\upsilon }^{*}),$$

by using Lemma 5

$${\upsilon }^{*}=-1+{}_{2}F_1(1,1,d+3;{\displaystyle \frac{1}{2}}).$$

Example 1.

If $d=0$ in Corollary 1, we find

$${\mathcal{L}}_0\left(\mathfrak{f}\right)\left(\tau \right)={\displaystyle \frac{2}{\tau }}{\int }_0^{\tau }\mathfrak{f}\left(t\right)dt,$$

then,

$${\mathcal{L}}_0\left[\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\right]\subset \mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;{\upsilon }^{*}),$$

where

$$\begin{array}{ccc}\hfill {\upsilon }^{*}& =& -1+{}_{2}F_1(1,1,3;{\displaystyle \frac{1}{2}})\hfill \\ & =& 3-4ln2\hfill \end{array}$$

Theorem 3.

Let λ be convex with $\lambda \left(0\right)=1$, we define

$$\zeta \left(\tau \right)=\tau {\lambda }^{\prime }\left(\tau \right)+\lambda \left(\tau \right),\tau \in \mathfrak{D}.$$

If $\mathfrak{f}\in \mathbf{A}$ verifies

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

(17)

then the fuzzy differential subordination

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),\tau \in \mathfrak{D},$$

(18)

holds.

Proof.  

Considering

$$\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}={\displaystyle \frac{\tau +{\sum }_{\kappa =2}^{\infty }{\psi }_{\mathfrak{q}}^{*s}(\kappa ,\nu ,\varsigma )\frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}{a}_{\kappa }{\tau }^{\kappa }}{\tau }}=1+{\eta }_1\tau +{\eta }_2{\tau }^2+....,\tau \in \mathfrak{D},$$

clearly $\eta \in \mathcal{H}[1,1],$ we will write

$$\tau \eta \left(\tau \right)={\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right),$$

and differentiating it, we obtain

$${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right).$$

The fuzzy differential subordination (17) takes the form

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)={\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\tau {\lambda }^{\prime }\left(\tau \right)+\lambda \left(\tau \right)\right).$$

(19)

Lemma 1 permits us to have ${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\lambda \left(\tau \right)\right);$ then, (18) holds.

The reality that $\lambda$ is the fuzzy best dominant. □

Theorem 4.

Assume that ζ is convex and $\zeta \left(0\right)=1$, if $\mathfrak{f}\in \mathbf{A}$ verifies

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

(20)

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),\tau \in \mathfrak{D},$$

for the convex function $\lambda \left(\tau \right)=(2\upsilon -1)+{\displaystyle \frac{2(\upsilon -1)}{\tau }}ln(1-\tau ),$ being the fuzzy best dominant.

Proof.  

Let

$$\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}=1+\sum _{\kappa =2}^{\infty }{\psi }_{\mathfrak{q}}^{*s}(\kappa ,\nu ,\varsigma ){\displaystyle \frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}}{a}_{\kappa }{\tau }^{\kappa -1}\in \mathcal{H}[1,1],\tau \in \mathfrak{D}.$$

Differentiating it, we obtain

$${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right),$$

and the fuzzy differential subordination (20) obtains

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right).$$

Lemma 2 enables us to obtain

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)\prec {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt,$$

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),$$

and

$$\lambda \left(\tau \right)=(2\upsilon -1)+{\displaystyle \frac{2(\upsilon -1)}{\tau }}ln(1-\tau ),$$

is a convex function that achieves the fuzzy differential subordination differential (20),

$$\tau {\lambda }^{\prime }\left(\tau \right)+\lambda \left(\tau \right)=\zeta \left(\tau \right).$$

Consequently, it is the fuzzy best dominant. □

If we put $\upsilon =0,$ in Theorem 5, we have

Corollary 2.

Considering ζ convex with $\zeta \left(0\right)=1$, if $\mathfrak{f}\in \mathbf{A}$ verifies

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

(21)

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),\tau \in \mathfrak{D},$$

where

$$\lambda \left(\tau \right)=-1-{\displaystyle \frac{2}{\tau }}ln(1-\tau ),\tau \in \mathfrak{D},$$

is convex and it is the fuzzy best dominant.

Proof.  

By Theorem 4 putting $\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }},$ the fuzzy differential subordination (21) has the following shape:

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right).$$

Lemma 2, enables us to obtain

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)\prec {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt,$$

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),$$

and

$$\lambda \left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }{\displaystyle \frac{1+t}{1-t}}dt=-1-{\displaystyle \frac{2}{\tau }}ln(1-\tau ),$$

is the fuzzy best dominant. □

Example 2.$\left(i\right)$ From Corollary 2 if

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

we obtain

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\ge \underset{\left|\tau \right|=1}{min}{\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)={\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda (-1)=-1+2ln2.$$

$\left(ii\right)$ Let $\zeta \left(\tau \right)={\displaystyle \frac{1+\tau }{1-\tau }}$ be convex in $\mathfrak{D}$ with $\zeta \left(0\right)=1$ and $\mathfrak{f}\left(\tau \right)={\tau }^2+\tau ,$$\tau \in \mathfrak{D}.$ For $s=0,$ and $\rho =2,$ we obtain ${\mathfrak{I}}_2^0(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)={\left[3\right]}_{\mathfrak{q}}{\tau }^2+\tau$ and ${\left({\mathfrak{I}}_2^0(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=2{\left[3\right]}_{\mathfrak{q}}\tau +1$ and ${\displaystyle \frac{{\mathfrak{I}}_2^0(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}={\left[3\right]}_{\mathfrak{q}}\tau +1.$ We conclude that $\lambda \left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }{\displaystyle \frac{1+t}{1-t}}dt=-1-{\displaystyle \frac{2}{\tau }}ln(1-\tau ).$ Applying Theorem 4 yields

$${\mathcal{F}}_{\left(\mathfrak{D}\right)}\left(2{\left[3\right]}_{\mathfrak{q}}\tau +1\right)\le {\mathcal{F}}_{\left(\mathfrak{D}\right)}\left({\displaystyle \frac{1+\tau }{1-\tau }}\right),\tau \in \mathfrak{D},$$

implies

$${\mathcal{F}}_{\left(\mathfrak{D}\right)}\left({\left[3\right]}_{\mathfrak{q}}\tau +1\right)\le {\mathcal{F}}_{\left(\mathfrak{D}\right)}\left(-1-{\displaystyle \frac{2}{\tau }}ln(1-\tau )\right),\tau \in \mathfrak{D},$$

Theorem 5.

Let λ be convex with $\lambda \left(0\right)=1$; we define $\zeta \left(\tau \right)=\tau {\lambda }^{\prime }\left(\tau \right)+\lambda \left(\tau \right),\tau \in \mathfrak{D}.$ If $\mathfrak{f}\in \mathbf{A}$ verifies

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right),\tau \in \mathfrak{D},$$

(22)

then

$${\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}\left({\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right),\tau \in \mathfrak{D},$$

(23)

holds.

Proof.  

For

$$\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}={\displaystyle \frac{\tau +{\sum }_{\kappa =2}^{\infty }{\psi }_{\mathfrak{q}}^{*s+1}(\kappa ,\nu ,\varsigma )\frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}{a}_{\kappa }{\tau }^{\kappa }}{\tau +{\sum }_{\kappa =2}^{\infty }{\psi }_{\mathfrak{q}}^{*s}(\kappa ,\nu ,\varsigma )\frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}{a}_{\kappa }{\tau }^{\kappa }}}.$$

Differentiating it, we obtain

$${\eta }^{\prime }\left(\tau \right)={\displaystyle \frac{{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}-\eta \left(\tau \right){\displaystyle \frac{{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},$$

then

$$\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)={\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }.$$

The fuzzy differential subordination (22), using the above notation, takes

$${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)={\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\tau {\lambda }^{\prime }\left(\tau \right)+\lambda \left(\tau \right)\right).$$

Lemma 1 permits us to have ${\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\eta \left(\tau \right)\right)\le {\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\left(\lambda \left(\tau \right)\right);$ then, (23) holds.

The reality is that $\lambda$ is the fuzzy best dominant. □

4. Fuzzy Differential Superordination Results

In this section, we provide the best subordinant for each fuzzy differential superordination that is being studied.

Theorem 6.

Assume that $\mathfrak{f}\in \mathbf{A}$, ζ is convex in $\mathfrak{D}$ such that $\zeta \left(0\right)=1$, and ${\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)$ defined in (11). We let ${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }$ be a univalent in $\mathfrak{D},$${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }\in \wp \cap \mathcal{H}[1,1].$ If

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\prec {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

(24)

holds, then the fuzzy superordination

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\prec {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

with $\lambda \left(\tau \right)={\displaystyle \frac{d+2}{{\tau }^{d+2}}}{\int }_0^{\tau }{t}^{d+1}\zeta \left(t\right)dt$ the fuzzy best subordinant, which is convex.

Proof.  

Differentiating (14), then $\tau {\mathcal{L}}_d{\left(\mathfrak{f}\right)}^{\prime }\left(\tau \right)+(d+1){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)=(d+2)\mathfrak{f}\left(\tau \right)$ can be stated as

$$\tau {\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }+(d+1){\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)=(d+2){\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right),$$

which after differentiating it again, has the form

$${\displaystyle \frac{\tau {\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{″}}{(d+2)}}+{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }={\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }.$$

Using the final relation, (24) can be expressed the fuzzy differential superordination

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{″}}{(d+2)}}+{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }\right).$$

(25)

Define

$$\eta \left(\tau \right)={\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

(26)

the fuzzy differential superordination (25), we obtain

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{(d+2)}}+\eta \left(\tau \right)\right),\tau \in \mathfrak{D}.$$

Using Lemma 3, we deduce ${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)$; similarly,

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }$$

by the fuzzy best subordinant $\lambda \left(\tau \right)={\displaystyle \frac{d+2}{{\tau }^{d+2}}}{\int }_0^{\tau }{t}^{d+1}\zeta \left(t\right)dt$ convex function. □

Theorem 7.

Let $\mathfrak{f}\in \mathbf{A}$, ${\mathcal{L}}_d\left(\mathfrak{f}\right)\tau ={\displaystyle \frac{d+2}{{\tau }^{d+1}}}{\int }_0^{\tau }{t}^d\mathfrak{f}\left(t\right)dt$ and $\zeta \left(\tau \right)={\displaystyle \frac{1-(2\upsilon -1)\tau }{1-\tau }}$ where $Red>-2,\upsilon \in [0,1).$ Suppose that ${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }$ is a univalent in $\mathfrak{D},$${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime }\in \wp \cap \mathcal{H}[1,1]$ and

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

(27)

then

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

is satisfied for the convex function $\lambda \left(\tau \right)=(2\upsilon -1)-2(\upsilon -1){{(1-\tau )}^{-1}}_2{F}_1(1,1,d+3;{\displaystyle \frac{\tau }{\tau -1}})$ as the fuzzy best subordinant.

Proof.  

Assume that $\eta \left(\tau \right)={\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime },$ the fuzzy differential superordination (27) becomes

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left({\displaystyle \frac{\tau {\eta }^{\prime }\left(\tau \right)}{d+2}}+\eta \left(\tau \right)\right).$$

Through the use of Lemma 4, we obtain ${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\prec {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right),$ with

$${\mathcal{F}}_{\lambda \left(\mathfrak{D}\right)}\lambda \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\tau \right)\right)}^{\prime },$$

and

$$\begin{array}{ccc}\hfill \lambda \left(\tau \right)& =& {\displaystyle \frac{d+2}{{\tau }^{d+2}}}{\int }_0^{\tau }{\displaystyle \frac{1-(2\upsilon -1)t}{1-t}}{t}^{d+1}dt\hfill \\ & =& (2\upsilon -1)-2(\upsilon -1){{(1-\tau )}^{-1}}_2{F}_1(1,1,d+3;{\displaystyle \frac{\tau }{\tau -1}})\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma ){\mathcal{L}}_d\left(\mathfrak{f}\right)\tau \right)}^{\prime },\hfill \end{array}$$

is convex and the fuzzy best subordinant. □

Example 3.

Let $\zeta \left(\tau \right)={\displaystyle \frac{1+\tau }{1-\tau }}$ be convex in $\mathfrak{D}$ with $\zeta \left(0\right)=1$ and $\mathfrak{f}\left(\tau \right)={\tau }^2+\tau ,$$\tau \in \mathfrak{D}.$ For $s=0,$$\rho =2$ and $d=2$ we obtain

$${\mathcal{L}}_2\left(\mathfrak{f}\right)\tau ={\displaystyle \frac{4}{{\tau }^3}}{\int }_0^{\tau }{t}^2(t^2+t)dt={\displaystyle \frac{4}{5}}{\tau }^2+\tau$$

and

$$\begin{array}{ccc}\hfill {\mathfrak{I}}_2^0(\nu ,\varsigma ){\mathcal{L}}_2\left(\mathfrak{f}\right)\tau & =& {\displaystyle \frac{4}{5}}{\left[3\right]}_{\mathfrak{q}}{\tau }^2+\tau \hfill \\ \hfill {\left({\mathfrak{I}}_2^0(\nu ,\varsigma ){\mathcal{L}}_2\left(\mathfrak{f}\right)\tau \right)}^{\prime }& =& {\displaystyle \frac{8}{5}}{\left[3\right]}_{\mathfrak{q}}\tau +1\in \wp \cap \mathcal{H}[1,1].\hfill \end{array}$$

We deduce $\lambda \left(\tau \right)={\displaystyle \frac{4}{{\tau }^4}}{\int }_0^{\tau }{t}^3{\displaystyle \frac{1+\tau }{1-\tau }}dt=-{\displaystyle \frac{8ln(1-\tau )}{{\tau }^4}}-{\displaystyle \frac{8}{{\tau }^3}}-{\displaystyle \frac{4}{{\tau }^2}}-{\displaystyle \frac{8}{3\tau }}-1.$Applying Theorem 7, we obtain

$${\mathcal{F}}_{\left(\mathfrak{D}\right)}\left({\displaystyle \frac{1+\tau }{1-\tau }}\right)\le {\mathcal{F}}_{\left(\mathfrak{D}\right)}\left(2{\left[3\right]}_{\mathfrak{q}}\tau +1\right),\tau \in \mathfrak{D},$$

induce

$${\mathcal{F}}_{\left(\mathfrak{D}\right)}\left(-{\displaystyle \frac{8ln(1-\tau )}{{\tau }^4}}-{\displaystyle \frac{8}{{\tau }^3}}-{\displaystyle \frac{4}{{\tau }^2}}-{\displaystyle \frac{8}{3\tau }}-1\right)\le {\mathcal{F}}_{\left(\mathfrak{D}\right)}\left({\displaystyle \frac{8}{5}}{\left[3\right]}_{\mathfrak{q}}\tau +1\right),\tau \in \mathfrak{D}.$$

Theorem 8.

Assume that $\mathfrak{f}\in \mathbf{A}$ and ζ be convex with $\zeta \left(0\right)=1$. Considering ${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }$ is a univalent and ${\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\in \wp \cap \mathcal{H}[1,1],$ if the fuzzy differential superordination

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

(28)

holds, then the fuzzy differential superordination

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }},\tau \in \mathfrak{D},$$

is satisfied for the convex function ${\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt$ the fuzzy best subordinant.

Proof.  

Denoting

$$\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}={\displaystyle \frac{\tau +{\sum }_{\kappa =2}^{\infty }{\psi }_{\mathfrak{q}}^{*s}(\kappa ,\nu ,\varsigma )\frac{{[\kappa +\rho -1]}_{\mathfrak{q}}!}{{\left[\rho \right]}_{\mathfrak{q}}!{[\kappa -1]}_{\mathfrak{q}}!}{a}_{\kappa }{\tau }^{\kappa }}{\tau }}\in \mathcal{H}[1,1].$$

We are able to write ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)=\tau \eta \left(\tau \right)$ and differentiating it, we have

$${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right).$$

Using the form, the fuzzy differential superordination (28) becomes

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right).$$

By Lemma 3, we obtain

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}for{\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt,$$

stated as

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}for{\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt,$$

convex and the fuzzy best subordinant. □

Theorem 9.

Let $\zeta \left(\tau \right)={\displaystyle \frac{1-(2\upsilon -1)\tau }{1-\tau }}$ with$\upsilon \in [0,1).$ For $\mathfrak{f}\in \mathbf{A}$, let ${\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }$ be a univalent and ${\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\in \wp \cap \mathcal{H}[1,1].$ If the fuzzy differential superordination

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime },\tau \in \mathfrak{D},$$

(29)

holds, then

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }},\tau \in \mathfrak{D},$$

is satisfied by the fuzzy best subordinant

$${\lambda }_1\left(\tau \right)=(2\upsilon -1)+{\displaystyle \frac{2(\upsilon -1)}{\tau }}ln(1-\tau ),$$

convex function for $\tau \in \mathfrak{D}.$

Proof.  

At the presentation of the proof of Theorem 8 at $\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }},$ the fuzzy superordination (29) assumes the shape

$$\zeta \left(\tau \right)={\displaystyle \frac{1-(2\upsilon -1)\tau }{1-\tau }}\le \tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right).$$

Using Lemma 3, we obtain ${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right),$ by

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)& \le & {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}\hfill \\ \hfill {\lambda }_1\left(\tau \right)& =& {\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }{\displaystyle \frac{1-(2\upsilon -1)t}{1-t}}dt\hfill \\ & =& (2\upsilon -1)+{\displaystyle \frac{2(\upsilon -1)}{\tau }}ln(1-\tau )\le {\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }},\hfill \end{array}$$

${\lambda }_1$ is convex and the fuzzy best subordinant. □

Example 4.

Let $\zeta \left(\tau \right)={\displaystyle \frac{1+\tau }{1-\tau }}$ be convex in $\mathfrak{D}$ with $\zeta \left(0\right)=1$ and $\mathfrak{f}\left(\tau \right)={\tau }^2+\tau ,$$\tau \in \mathfrak{D}$. For $s=0,$ and $\rho =2,$ we obtain ${\mathfrak{I}}_2^0(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)={\left[3\right]}_{\mathfrak{q}}{\tau }^2+\tau$ and ${\left({\mathfrak{I}}_2^0(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=2{\left[3\right]}_{\mathfrak{q}}\tau +1$univalent in $\mathfrak{D}$ and ${\displaystyle \frac{{\mathfrak{I}}_2^0(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{\tau }}={\left[3\right]}_{\mathfrak{q}}\tau +1\in \wp \cap \mathcal{H}[1,1].$ Since ${\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }{\displaystyle \frac{1+t}{1-t}}dt=-1-{\displaystyle \frac{2}{\tau }}ln(1-\tau ).$ Applying Theorem 9, we deduce

$${\mathcal{F}}_{\left(\mathfrak{D}\right)}\left({\displaystyle \frac{1+\tau }{1-\tau }}\right)\le {\mathcal{F}}_{\left(\mathfrak{D}\right)}\left(2{\left[3\right]}_{\mathfrak{q}}\tau +1\right),\tau \in \mathfrak{D},$$

implies

$${\mathcal{F}}_{\left(\mathfrak{D}\right)}\left(-{\displaystyle \frac{2ln(1-\tau )}{\tau }}-1\right)\le {\mathcal{F}}_{\left(\mathfrak{D}\right)}\left({\left[3\right]}_{\mathfrak{q}}\tau +1\right),\tau \in \mathfrak{D}.$$

Theorem 10.

Let ζ be convex, with $\zeta \left(0\right)=1,$ for $\mathfrak{f}\in \mathbf{A},$ and let ${\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }$ be univalent in $\mathfrak{D}$ and ${\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\in \wp \cap \mathcal{H}[1,1].$ If

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime },\tau \in \mathfrak{D},$$

(30)

holds, then

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},\tau \in \mathfrak{D},$$

where the convex ${\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt$ is the fuzzy best subordinant.

Proof.  

Suppose

$$\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},$$

after differentiating it, we can write

$${\eta }^{\prime }\left(\tau \right)={\displaystyle \frac{{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}-\eta \left(\tau \right){\displaystyle \frac{{\left({\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},$$

in the form $\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)={\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }.$

In these conditions, the fuzzy differential superordination (30) becomes ${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right).$ Utilising Lemma 3, we acquire ${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right)$, written as

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},$$

with the convex ${\lambda }_1\left(\tau \right)={\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }\zeta \left(t\right)dt$, the fuzzy best subordinant. □

Theorem 11.

Assume that $\zeta \left(\tau \right)={\displaystyle \frac{1-(2\upsilon -1)\tau }{1-\tau }}$ with$\upsilon \in [0,1).$ At $\mathfrak{f}\in \mathbf{A}$ Assume that ${\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }$is univalent and ${\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\in \wp \cap \mathcal{H}[1,1].$ If

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\left({\displaystyle \frac{\tau {\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime },\tau \in \mathfrak{D},$$

(31)

holds, then

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},\tau \in \mathfrak{D},$$

and the fuzzy best suordinant is the convex function

$${\lambda }_1\left(\tau \right)=(2\upsilon -1)+{\displaystyle \frac{2(\upsilon -1)}{\tau }}ln(1-\tau ),\tau \in \mathfrak{D}.$$

Proof.  

By using $\eta \left(\tau \right)={\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},$ the fuzzy differential superordination (31) takes the form

$${\mathcal{F}}_{\zeta \left(\mathfrak{D}\right)}\zeta \left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\left(\tau {\eta }^{\prime }\left(\tau \right)+\eta \left(\tau \right)\right).$$

Utilising Lemma 3, we obtain ${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{\eta \left(\mathfrak{D}\right)}\eta \left(\tau \right),$ with

$${\mathcal{F}}_{{\lambda }_1\left(\mathfrak{D}\right)}{\lambda }_1\left(\tau \right)\le {\mathcal{F}}_{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\mathfrak{D}\right)}{\displaystyle \frac{{\mathfrak{I}}_{\mathfrak{q},\rho }^{s+1}(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}{{\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )\mathfrak{f}\left(\tau \right)}},$$

and

$$\begin{array}{ccc}\hfill {\lambda }_1\left(\tau \right)& =& {\displaystyle \frac{1}{\tau }}{\int }_0^{\tau }{\displaystyle \frac{1-(2\upsilon -1)t}{1-t}}dt\hfill \\ & =& (2\upsilon -1)+{\displaystyle \frac{2(\upsilon -1)}{\tau }}ln(1-\tau ),\hfill \end{array}$$

$\lambda$ is convex and the fuzzy best subordinant. □

5. Conclusions

The innovative findings demonstrated in this work, as stated in Definition 8, are associated with a new class of analytic functions $\mathcal{F}{\aleph }_{\mathfrak{q},\rho }^s(\nu ,\varsigma ;\upsilon ),$ presented in Definition 2. Applying the notion of the $\mathfrak{q}$-difference operator, we build the $\mathfrak{q}$-analogue multiplier-Ruscheweyh operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )$ to introduce a subclass of univalent functions. These subclasses are then further investigated using techniques from fuzzy differential subordination theory in Section 3. In Section 4, we develop fuzzy differential fuzzy superordinations for the $\mathfrak{q}$-analogue multiplier-Ruscheweyh operator ${\mathfrak{I}}_{\mathfrak{q},\rho }^s(\nu ,\varsigma )$.

The results discussed in this work further advance the topic of introducing and investigating new classes of analytic functions with the help of quantum calculus operators. Motivated by the encouraging outcomes of integrating components of quantum calculus into the research, we obtained some practical examples in Section 3 and Section 4.

Fuzzy differential subordinates are a powerful tool in mathematical analysis that allows us to extend the concept of derivatives to fuzzy functions. They have important applications in fields such as fuzzy control systems, where uncertainty and imprecision are prevalent. By incorporating fuzzy derivatives into our mathematical models, we can better understand and manipulate complex systems, leading to more effective and adaptive control strategies. Given that the second-order fuzzy differential subordinations found here could be extended to third-order fuzzy differential subordinations in light of the very recent findings by [25,26], the novel result in this paper will stimulate further research in the field of geometric function theory.