Applications of Fuzzy Differential Subordination to the Subclass of Analytic Functions Involving Riemann–Liouville Fractional Integral Operator

Abstract

In this research, we combine ideas from geometric function theory and fuzzy set theory. We define a new operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m:\mathcal{A}\to \mathcal{A}$ of analytic functions in the open unit disc $\Delta$ with the help of the Riemann–Liouville fractional integral operator, the linear combination of the Noor integral operator, and the generalized Sălăgean differential operator. Further, we use this newly defined operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m$ together with a fuzzy set, and we next define a new class of analytic functions denoted by $R_{ϝ}^{\zeta }(m,\alpha ,\delta ).$ Several innovative results are found using the concept of fuzzy differential subordination for the functions belonging to this newly defined class, $R_{ϝ}^{\zeta }(m,\alpha ,\delta ).$ The study includes examples that demonstrate the application of the fundamental theorems and corollaries.

1. Introduction

Fuzzy set theory is a new branch of mathematics developed by Zadeh in 1965 [1], using the idea of fuzzy sets. It has advanced rapidly, finding use in many areas of science and technology today. Many different branches of mathematics have taken up new directions with the use of the fuzzy set notion. As a generalization of the traditional idea of differential subordination given by Miller and Mocanu [2,3], it was utilized to introduce the novel notions of fuzzy subordination [4] and fuzzy differential subordination [5]. The fundamentals of differential subordination are outlined in [6,7] and the development of the concept of fuzzy differential subordination is detailed in [8]. These advances have shown the strength and adaptability of fuzzy set theory as a mathematical tool, and they have paved the way for new studies in geometric function theory and other fields of mathematics. Due to its similarities to fuzzy set theory, the same authors first put forward the classical theory of differential subordination in 1978 [2] and 1981 [3]. As was indicated before, this most likely entailed expanding the classical theory to include the idea of fuzzy differential subordination.

As well as showing potential in the mathematical modeling and analysis of actual scientific problems, the theory of real and complex-order integrals and derivatives has aided the study of geometric functions theory (GFT). There have been some remarkable discoveries made in the study of differential and integral operators recently, having applications to many areas of physics and mathematics. Two such instances include research analyzing the dynamics of dengue virus transmission [9] and a novel model of the human liver that employs Caputo–Fabrizio fractional derivatives with the exponential kernel suggested in [10]; for more information, see [11,12].

At the beginning of 2012, a new research direction was launched by combining the concept of differential subordination with the complex function domain in fuzzy set theory, which is named fuzzy differential subordination. For more details, see [5,6,13,14,15,16,17,18,19]. These articles go further into the subject of fuzzy differential subordination and how it applies to different operators. The use of the concept of fuzzy differential subordination for certain classes of operators, such as those found in geometric function theory (GFT), was likely further investigated. These developments could have led to new insights into and approaches for examining the properties of complex functions. Fuzzy differential subordination is the focus of every paper, exploring its application to various mathematical problems. Due to this, we will examine fuzzy differential subordination in this article, along with a newly developed operator and a set of analytical functions.

In the study presented in this article, fuzzy set theory and GFT interact within a single framework. This sort of study is essential for explaining the connections between distinct mathematical concepts and developing new methods for solving mathematical problems.

Here, we shall call $\mathcal{H}(\Delta )$ the set of all analytic functions on the open unit disc $\Delta =\{\tau \in \mathbb{C}:\left|\tau \right|<1\}$. Let

$$\begin{array}{ccc}\hfill \mathcal{A}& =& \left\{f\in \mathcal{H}(\Delta ):f\left(\tau \right)=\tau +{\mathrm{b}}_2{\tau }^2+\dots ,\tau \in \Delta \right\}\hfill \\ & =& \left\{f\in \mathcal{H}(\Delta ):f\left(\tau \right)=\tau +\sum _{n=2}^{\infty }{\mathrm{b}}_n{\tau }^n,\tau \in \Delta \right\}\hfill \end{array}$$

(1)

be the set of normalized analytic functions. For $\mathrm{b}\in \mathbb{C}$ and $n\in \mathbb{N},$ then

$$\mathcal{H}[\mathrm{b},n]=\left\{f\in \mathcal{H}(\Delta ):f\left(\tau \right)=\mathrm{b}+{\mathrm{b}}_n{\tau }^n+{\mathrm{b}}_{n+1}{\tau }^{n+1}+\dots \right\}.$$

The class of convex functions of order $\alpha ,$ ($0\le \alpha <1),$ denoted by $C\left(\alpha \right)$ and defined as:

$$C\left(\alpha \right)=\left\{f\in \mathcal{A}:\mathrm{Re}\left(1+\frac{\tau f^{{}^{″}}\left(\tau \right)}{f^{{}^{\prime }}\left(\tau \right)}\right)>\alpha \right\},$$

when $\alpha =0$, then the class C of convex functions is obtained.

This research is in line with recent efforts in the field of fuzzy differential subordination that develop and investigate a novel fuzzy class by introducing new operators. The current research is divided into five parts. In Section 1, we covered the fundamentals of fuzzy sets, differentiable operators, and analytic functions. To set up the conditions for the rest of the study, we shall define fuzzy differential subordination and explore several well-established ones in Section 2. In the end, we talk about a brand-new operator $R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ for analytic functions. It is based on the generalized Sălăgean differential operator, the Noor integral operator, and the Riemann–Liouville fractional integral. The class of fuzzy analytic functions is defined in Section 2 by using this operator and fuzzy differential subordination. In Section 3, we give some lemmas for our planned research on a new class of analytic functions connected with fuzzy differential subordination. Using the aforementioned lemmas, we demonstrate several helpful findings and some examples in Section 4. It is interesting to discuss the applications of the fuzzy set and system in the management sciences, modelling, optimization statistics, and probability theory. For further information, see [20,21,22] and references therein.

2. Definitions

Subsequently, we examine the definitions and supporting studies that facilitated the understanding of fuzzy differential subordination.

Definition 1

([8]). Let X be a non-empty set. A pair $\left(\mathfrak{B},F_{\mathfrak{B}}\right)$ is called a fuzzy subset of X, where

$$F_{\mathfrak{B}}:X\to [0,1]$$

and

$$\mathfrak{B}=\left\{x\in X:0<F_{\mathfrak{B}}\left(x\right)\le 1\right\}$$

are the support of the fuzzy set. The function $F_{\mathfrak{B}}:X\to [0,1]$ is called the membership function of the fuzzy subset $\left(\mathfrak{B},F_{\mathfrak{B}}\right)$ and is denoted by $\mathfrak{B}=sup\left(\mathfrak{B},F_{\mathfrak{B}}\right).$

Remark 1.

Suppose $\mathfrak{B}\subset X$, then

$$F_{\mathfrak{B}}\left(x\right)=\left\{\begin{array}{c}1,\mathrm{if}x\in \mathfrak{B},\hfill \\ 0,\mathrm{if}x\notin \mathfrak{B}.\hfill \end{array}\right.$$

The 0 shows the smallest membership of $x\in X$ to $\mathfrak{B}$, while 1 indicates the highest membership of $x\in X$ to $\mathfrak{B}$ for a fuzzy subset.

Definition 2

([4]). Let fixed point ${\tau }_0\in D$ and let $D\subset \mathbb{C}.$ We take the functions $f,g\in \mathcal{H}\left(D\right)$, and f is called a fuzzy subordinate to g, written as

$$f\left(\tau \right){\prec }_{\mathcal{F}}g\left(\tau \right)\mathrm{or}f{\prec }_{\mathcal{F}}g,$$

if there exists a function $F:\mathbb{C}\to \left[0,1\right],$ such that

$$f\left({\tau }_0\right)=g\left({\tau }_0\right)$$

and

$$F_{f\left(D\right)}f\left(\tau \right)\le F_{g\left(D\right)}g\left(\tau \right),\tau \in D.$$

Remark 2.

If g is univalent, then $f{\prec }_{\mathcal{F}}g\iff f\left({\tau }_0\right)=g\left({\tau }_0\right)$ and $f\left(D\right)\subset g\left(D\right).$

Remark 3.

Such a function $F:\mathbb{C}\to \left[0,1\right]$ can be considered

$$F\left(\tau \right)=\frac{\left|\tau \right|}{1+\left|\tau \right|}\mathrm{or}F\left(\tau \right)=\frac{1}{1+\left|\tau \right|}.$$

Remark 4.

If $D=\Delta$, the conditions become $f\left(0\right)=g\left(0\right)$ and $f\left(U\right)\subset g\left(U\right)$, which is equivalent to the classical definition of subordination.

Definition 3

([5]). Let h be univalent in Δ and $\psi :{\mathbb{C}}^3\times \Delta \to \mathbb{C}$, such that $h\left(0\right)=\psi (\mathrm{b},0;0)=\mathrm{b}$. When the fuzzy differential subordination

$$F_{\psi \left({\mathbb{C}}^3\times \Delta \right)}\psi \left(\phi \left(\tau \right),\tau {\phi }^{\prime }\left(\tau \right),{\tau }^2{\phi }^{″}\left(\tau \right);\tau \right)\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta$$

(2)

is satisfied for an analytic function φ in Δ, such that $\phi \left(0\right)=b,$ then φ is called a fuzzy solution of the fuzzy differential subordination. A fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination is a univalent function q, for which $F_{\phi (\Delta )}\phi \left(\tau \right)\le F_{q(\Delta )}q\left(\tau \right),$$\tau \in \Delta$ for all φ satisfying (2). The fuzzy best dominant of (2) is a fuzzy dominant $\tilde{q}$ such that $F_{\tilde{q}(\Delta )}\tilde{q}\left(\tau \right)\le F_{q(\Delta )}q\left(\tau \right),$$\tau \in \Delta ,$ for all fuzzy dominants q of (2).

Definition 4

([13]). Let

$$\begin{array}{ccc}\hfill f\left(D\right)& =& sup\left(F_{f\left(D\right)},f\left(D\right)\right)\hfill \\ & & \\ & =& \left\{\tau \in D:0<F_{f\left(D\right)}f\left(\tau \right)\le 1\right\},\hfill \end{array}$$

where $f\left(D\right)$ is the fuzzy set $F_{f\left(D\right)}$ membership function that is connected to the function f. The membership functions of the fuzzy sets $\left(\mu f\right)\left(D\right)$ connected to the functions $\mu f$ and $f\left(D\right)$ have a one-to-one correspondence, i.e.,

$$F_{f\left(D\right)}f\left(\tau \right)=F_{\left(\mu f\right)\left(D\right)}\left(\left(\mu f\right)\left(\tau \right)\right),\tau \in D.$$

Remark 5.

Let $0<F_{f\left(D\right)}f\left(\tau \right)\le 1,$ also $0<F_{g\left(D\right)}g\left(\tau \right)\le 1$; it is obvious that

$$0<F_{(f+g)\left(D\right)}\left((f+g)\left(\tau \right)\right)\le 1,\tau \in D.$$

Definition 5.

Let $f\left(\tau \right)$ be given by (1) and $g\left(\tau \right)=z+{\displaystyle \sum _{n=2}^{\infty }}{d}_n{\tau }^n$, then the convolution of f and g is defined as:

$$\left(f\ast g\right)\left(z\right)=z+\sum _{n=2}^{\infty }{b}_n{d}_n{z}^n.$$

(3)

Now, by using the idea of fuzzy differential subordination and the linear combination of the generalized Sălăgean differential operator and the Noor integral operator together with the Riemann–Liouville fractional integral operator, we define a new class of analytic functions. So first, we give the definitions of the already-defined operators and references therein.

Definition 6

([23]). Noor studied the Noor integral operator $I^m:\mathcal{A}\to \mathcal{A}$. Let

$$f_m\left(\tau \right)=\frac{\tau }{{\left(1-\tau \right)}^{m+1}},m>-1$$

and let $f_m^{-1}\left(\tau \right)$ be defined such that

$$f_m\left(\tau \right)\ast f_m^{-1}\left(\tau \right)=\frac{\tau }{{\left(1-\tau \right)}^2}.$$

Then,

$$\begin{array}{ccc}\hfill I^{m}f\left(\tau \right)& =& f_m^{-1}\left(\tau \right)\ast f\left(\tau \right)\hfill \\ & =& {\left[\frac{\tau }{{\left(1-\tau \right)}^{m+1}}\right]}^{-1}\ast f\left(\tau \right).\hfill \end{array}$$

(4)

It follows from (4) that

$$I^{m}f\left(\tau \right)=\tau +\sum _{j=2}^{\infty }\frac{m!\mathsf{\Gamma }\left(m+1\right)}{\mathsf{\Gamma }\left(m+j\right)}{\mathrm{b}}_j{\tau }^j.$$

Remark 6.

The following identity is held for $I^{m}f\left(\tau \right)$:

$$m{I}^{m+1}f\left(\tau \right)=\left(m+1\right)I^{m+1}f\left(\tau \right)-\tau {\left(I^{m}f\left(\tau \right)\right)}^{{}^{\prime }},\tau \in \Delta$$

and

$$\begin{array}{ccc}\hfill I^{0}f\left(\tau \right)& =& \tau f^{{}^{\prime }}\left(\tau \right),\hfill \\ & & \\ \hfill I^{1}f\left(\tau \right)& =& f\left(\tau \right).\hfill \end{array}$$

The Al–Oboudi differential operator was studied in [24] and this operator is the generalization of the Salagean differential operator.

Definition 7

([24]). For $\zeta \ge 0,$$m\in {\mathbb{N}}_0,$ and $f\in \mathcal{A},$ the operator $S_{\zeta }^m:\mathcal{A}\to \mathcal{A}$ is defined by

$$\begin{array}{ccc}\hfill S_{\zeta }^{0}f\left(\tau \right)& =& f\left(\tau \right),\hfill \\ \hfill S_{\zeta }^{1}f\left(\tau \right)& =& \left(1-\zeta \right)f\left(\tau \right)+\zeta \tau f^{{}^{\prime }}\left(\tau \right)=S_{\zeta }f\left(\tau \right)\hfill \\ & & ···\hfill \\ \hfill S_{\zeta }^{m}f\left(\tau \right)& =& \left(1-\zeta \right)S^{m-1}f\left(\tau \right)+\zeta \tau {\left(S_q^{m-1}f\left(\tau \right)\right)}^{{}^{\prime }}=S_{\zeta }(S_{\zeta }^{m-1}f\left(\tau \right).\hfill \end{array}$$

After some simple calculation, we have

$$S_{\zeta }^{m}f\left(\tau \right)=\tau +\sum _{j=2}^{\infty }{\left\{\zeta \left(j-1\right)+1\right\}}^m{\mathrm{b}}_j{\tau }^j.$$

Remark 7.

The following is held for $S_{\zeta }^{m}f\left(\tau \right)$:

$$S_0^{0}f\left(\tau \right)=f\left(\tau \right),S_1^{1}f\left(\tau \right)=\tau f^{{}^{\prime }}\left(\tau \right),\dots ,S_{\zeta }^{m+1}f\left(\tau \right)=\tau {\left(S_{\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }},\tau \in \Delta .$$

In the next definition, we define the operator that will be used to obtain the results of this paper.

Definition 8

([25]). Let $\alpha \ge 0,$$\zeta \ge 0$ and $m>-1.$ The operator $L_{\alpha ,\zeta }^m:\mathcal{A}\to \mathcal{A}$ is defined by

$$L_{\alpha ,\zeta }^{m}f\left(\tau \right)=\left(1-\alpha \right)S_{\zeta }^{m}f\left(\tau \right)+\alpha I^{m}f\left(\tau \right).$$

Remark 8.

$L_{\alpha ,\zeta }^m$ is a linear operator, and if $f\in \mathcal{A}$

$$f\left(\tau \right)=\tau +\sum _{j=2}^{\infty }{\mathrm{b}}_j{\tau }^j,$$

then, after some simple calculations, we have

$$L_{\alpha ,\zeta }^{m}f\left(\tau \right)=\tau +\sum _{j=2}^{\infty }\left[\left(1-\alpha \right){\left\{\zeta \left(j-1\right)+1\right\}}^m+\alpha \left(\frac{m!\mathsf{\Gamma }\left(m+1\right)}{\mathsf{\Gamma }\left(m+j\right)}\right)\right]{\mathrm{b}}_j{\tau }^j,\tau \in \Delta .$$

(5)

Remark 9.

If $\alpha =0,$ then $L_{0,\zeta }^{m}f\left(\tau \right)=S_{\zeta }^{m}f\left(\tau \right)$ and for $\alpha =1,$ then $L_{1,\zeta }^{m}f\left(\tau \right)=I^{m}f\left(\tau \right).$

Remark 10.

For $\zeta =1,$$m=0,$ then $L_{\alpha ,1}^{0}f\left(\tau \right)=\left(1-\alpha \right)S_1^{0}f\left(\tau \right)+\alpha I^{0}f\left(\tau \right)=\left(1-\alpha \right)f\left(\tau \right)+\alpha \tau f^{{}^{\prime }}\left(\tau \right).$

Remark 11.

For $m=1,$ and $\zeta =1,$ in (5), then

$$\begin{array}{ccc}\hfill L_{\alpha ,1}^{1}f\left(\tau \right)& =& \left(1-\alpha \right)S_1^{1}f\left(\tau \right)+\alpha I^{1}f\left(\tau \right)\hfill \\ & =& \left(1-\alpha \right)\tau f^{{}^{\prime }}\left(\tau \right)+\alpha f\left(\tau \right),\tau \in \Delta .\hfill \end{array}$$

We also review the definition of the Riemann–Liouville fractional integral:

Definition 9

([26]). If we apply the Riemann–Liouville fractional integral of order λ to an analytic function f, we define the following expression:

$$D_{\tau }^{-\lambda }f\left(\tau \right)=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{f\left(t\right)}{{\left(\tau -t\right)}^{1-\lambda }}dt,$$

where $\lambda >0.$

The fractional integral has been extensively exploited to construct novel operators, and these operators have given rise to intriguing subclasses of functions that have yielded practical and motivating results (see, [27,28,29,30,31,32,33]). Using Ruscheweyh and Salagean differential operators, several studies (see, for example, [34]) have generated fuzzy differential subordination. The Riemann–Liouville fractional integral in [35] is used to find the derivative of a Gaussian hypergeometric function and for studying the applications of the fractional integral operator we refer [36,37,38]. The fractional integral in [39] is used to find the derivative of a confluent hypergeometric function. In this study, we first establish linear combinations of the Al–Abodi differential operator and the Noor integral operator. Using Riemann–Liouville fractional integral and linear combinations of these two operators, we define a new operator and consider it to define a class of analytic functions of fuzzy differential subordination.

Taking the motivation from the article [25], and using the Definition 8 of Riemann–Liouville fractional integral $D_{\tau }^{-\lambda }f\left(\tau \right)$ and Definition 9 of linear combination $L_{\alpha ,\zeta }^{m}f,$ we then define a differential integral operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)$ as follows:

Definition 10.

The Riemann–Liouville fractional integral $D_{\tau }^{-\lambda }f\left(\tau \right)$ applied to the linear combination of differential operator $L_{\alpha ,\zeta }^{m}f$ is to introduce differential integral operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)$ as follows:

$$\begin{array}{ccc}& & D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\hfill \\ & =& \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{L_{\alpha ,\zeta }^{m}f\left(t\right)}{{\left(\tau -t\right)}^{1-\lambda }}dt\hfill \\ & =& \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{t}{{\left(\tau -t\right)}^{1-\lambda }}dt+\hfill \\ & & \sum _{j=2}^{\infty }\left[\left(1-\alpha \right){\left\{\zeta \left(j-1\right)+1\right\}}^m+\alpha \left(\frac{m!\mathsf{\Gamma }\left(m+1\right)}{\mathsf{\Gamma }\left(m+j\right)}\right)\right]{\mathrm{b}}_j\underset{0}{\overset{\tau }{\int }}\frac{t^j}{{\left(\tau -t\right)}^{1-\lambda }}dt,\hfill \end{array}$$

where

$$\alpha \ge 0,\lambda >0\mathrm{and}m\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\}.$$

For

$$f\left(\tau \right)=\tau +\sum _{j=2}^{\infty }{\mathrm{b}}_j{\tau }^j\in \mathcal{A}.$$

The series of the Riemann–Liouville fractional integral $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)$ is:

$$\begin{array}{ccc}\hfill D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)& =& \frac{1}{\mathsf{\Gamma }\left(2+\lambda \right)}{\tau }^{1+\lambda }+\sum _{j=2}^{\infty }\left\{\frac{\left(1-\alpha \right)\mathsf{\Gamma }\left(j+1\right){\left\{\zeta \left(j-1\right)+1\right\}}^m}{\mathsf{\Gamma }\left(j+\lambda +1\right)}+\right.\hfill \\ & & \left.+\frac{\alpha \mathsf{\Gamma }\left(j+1\right)}{\mathsf{\Gamma }\left(j+\lambda +1\right)}\left(\frac{m!\mathsf{\Gamma }\left(m+1\right)}{\mathsf{\Gamma }\left(m+j\right)}\right)\right\}{\mathrm{b}}_j{\tau }^{j+\lambda }\hfill \end{array}$$

(6)

and

$$D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\in \mathcal{H}\left[0,\lambda +1\right].$$

Remark 12.

The operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)$ discussed in this article is a more comprehensive version compared to the operator introduced in [25].

Definition 11.

Let $R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ is a fuzzy class and contains all functions $f\in \mathcal{A}$ that fulfil the fuzzy inequality

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime }>\delta ,\tau \in \Delta ,$$

(7)

where $\delta \in \left[0,1\right),$$\alpha \ge 0,$$\zeta \ge 0,$$\lambda >0$ and $m\in {\mathbb{N}}_0$.

Remark 13.

The analytic function class $R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ discussed in this study is more general in scope compared to the class introduced in [25].

3. Set of Lemmas

In order to demonstrate our primary findings, we shall employ the following Lemmas.

Lemma 1

([2]). Let $h\in \mathcal{A}$. If

$$\mathrm{Re}\left(\frac{\tau h^{″}\left(\tau \right)}{h^{\prime }\left(\tau \right)}+1\right)>-\frac{1}{2},\tau \in \Delta ,$$

then

$$\frac{1}{\tau }{\int }_0^{\tau }h\left(t\right)dt,\tau \in \Delta$$

is a convex function.

Lemma 2

([17]). Let $\gamma \in {\mathbb{C}}^{\ast }$, $\mathrm{Re}\gamma \ge 0$ and h be a convex function with $h\left(0\right)=b$. If $\phi \in \mathcal{H}[b,n]$ with $\phi \left(0\right)=\mathrm{b},$$\psi :{\mathbb{C}}^2\times \Delta \to \mathbb{C},$

$$\psi \left(\phi \left(\tau \right),\tau {\phi }^{\prime }\left(\tau \right);\tau \right)=\phi \left(\tau \right)+\frac{1}{\gamma }\tau {\phi }^{\prime }\left(\tau \right)$$

an analytic in Δ and

$$F_{\psi \left({\mathbb{C}}^2\times \Delta \right)}\left(\phi \left(\tau \right)+\frac{1}{\gamma }\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right),$$

(8)

then,

$$F_{\phi (\Delta )}\phi \left(\tau \right)\le F_{g(\Delta )}g\left(\tau \right)\le F_{h(\Delta )}h\left(\tau \right)$$

with the convex function

$$g\left(\tau \right)=\frac{\gamma }{n{\tau }^{\gamma /n}}{\int }_0^{\tau }h\left(t\right)t^{\gamma /n-1}dt,\tau \in \Delta$$

is the fuzzy best dominant.

Lemma 3

([17]). Let g represent a convex function in Δ and let

$$h\left(\tau \right)=g\left(\tau \right)+n\gamma \tau g^{\prime }\left(\tau \right),\tau \in \Delta ,$$

where $\gamma >0$ and $n\in {\mathbb{Z}}^{+}.$

Let

$$\phi \left(\tau \right)=g\left(0\right)+{\phi }_n{\tau }^n+{\phi }_{n+1}{\tau }^{n+1}+\dots ,\tau \in \Delta$$

be analytic in Δ and

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\gamma \tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right),$$

then the sharp result will be

$$F_{\phi (\Delta )}\phi \left(\tau \right)\le F_{g(\Delta )}g\left(\tau \right).$$

4. Main Results

Now, we establish a relation to show that the set of functions $R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ is convex.

Theorem 1.

The set $R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ is convex.

Proof. 

Let

$$f_k\left(\tau \right)=\tau +\sum _{j=2}^{\infty }{\mathrm{b}}_{jk}{\tau }^j\in R_m^F(\alpha ,\delta ,\zeta ,\lambda ),k=1,2,z\in ▵.$$

For the required result, the function

$$h\left(\tau \right)={\eta }_1{f}_1\left(\tau \right)+{\eta }_2{f}_2\left(\tau \right)$$

must belong to the class $R_m^F(\alpha ,\delta ,\zeta ,\lambda ),$ with ${\eta }_1$ and ${\eta }_2$ being non-negative such that ${\eta }_1+{\eta }_2=1$. We next show that $h\in R_m^F(\alpha ,\delta ,\zeta ,\lambda )$

$$h^{{}^{\prime }}\left(\tau \right)={\left({\mu }_1{f}_1\left(\tau \right)+{\mu }_2{f}_2\left(\tau \right)\right)}^{{}^{\prime }}\left(\tau \right)={\mu }_1{f}_1^{{}^{\prime }}\left(\tau \right)+{\mu }_2{f}_2^{{}^{\prime }}\left(\tau \right)$$

and

$$\begin{array}{ccc}\hfill {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}h\left(\tau \right)\right)}^{{}^{\prime }}& =& {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m\left({\mu }_1{f}_1\left(\tau \right)+{\mu }_2{f}_2\left(\tau \right)\right)\right)}^{{}^{\prime }}\left(\tau \right)\hfill \\ & =& {\mu }_1{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{{}^{\prime }}+{\mu }_2{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{{}^{\prime }}.\hfill \end{array}$$

From Definition 4, we obtain that

$$\begin{array}{ccc}& & F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}h\right)}^{{}^{\prime }}(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}h\left(\tau \right)\right)}^{{}^{\prime }}\hfill \\ & =& F_{({\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m\left({\mu }_1{f}_1+{\mu }_2{f}_2\right)\right)}^{{}^{\prime }}(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m\left({\mu }_1{f}_1+{\mu }_2{f}_2\right)\left(\tau \right)\right)}^{{}^{\prime }}\hfill \\ & =& F_{({\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m\left({\mu }_1{f}_1+{\mu }_2{f}_2\right)\right)}^{{}^{\prime }}(\Delta )}\left({\mu }_1{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{{}^{\prime }}+{\mu }_2{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{{}^{\prime }}\right)\hfill \\ & =& \frac{F_{({\mu }_1{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{{}^{\prime }}(\Delta )}\left({\mu }_1{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{{}^{\prime }}\right)+F_{({\mu }_2{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{{}^{\prime }}(\Delta )}\left({\mu }_2{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{{}^{\prime }}\right)}{2}\hfill \\ & =& \frac{F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{{}^{\prime }}(\Delta )}\left({\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{{}^{\prime }}\right)+F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{{}^{\prime }}(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{{}^{\prime }}}{2}.\hfill \end{array}$$

As, if $f_1,f_2\in R_m^F(\alpha ,\delta ,\zeta ,\lambda ),$ then

$$\delta <F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{\prime }\le 1,$$

as well as

$$\delta <F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{\prime }\le 1,\tau \in \Delta .$$

Therefore,

$$\delta <\frac{F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_1\left(\tau \right)\right)}^{\prime }+F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m{f}_2\left(\tau \right)\right)}^{\prime }}{2}\le 1.$$

We get

$$\delta <F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}h\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}h\left(\tau \right)\right)}^{\prime }\le 1.$$

This implies that $h\in R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ and $R_m^F(\alpha ,\delta ,\zeta ,\lambda )$ is convex. □

Theorem 2.

Let g be a convex function in Δ and let

$$h\left(\tau \right)=g\left(\tau \right)+\frac{1}{a+2}\tau g^{\prime }\left(\tau \right);$$

also $a>0,$ $\tau \in \Delta$, let $f\in R_m^F(\alpha ,\delta ,\zeta ,\lambda )$, as well as

$$G\left(\tau \right)=I_a\left(f\right)\left(\tau \right)=\frac{a+2}{{\tau }^{a+1}}{\int }_0^{\tau }{t}^{a}f\left(t\right)dt,\tau \in \Delta ;$$

then, the fuzzy differential subordination

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime }\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta .$$

(9)

This implies the sharp result

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{\prime }\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta .$$

Proof. 

By using $G\left(\tau \right)$, we get

$${\tau }^{a+1}G\left(\tau \right)=(a+2){\int }_0^{\tau }{t}^{a}f\left(t\right)dt.$$

(10)

Differentiating Equation (10) with respect to $\tau$, we have

$$(a+1)G\left(\tau \right)+\tau G^{\prime }\left(\tau \right)=(a+2)f\left(\tau \right);$$

also,

$$\begin{array}{ccc}& & (a+1)D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)+\tau {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{\prime }\hfill \\ & =& (a+2)D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right),\tau \in \Delta .\hfill \end{array}$$

(11)

Differentiating (11), we have

$${\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{\prime }+\frac{1}{a+2}\tau {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{″}={\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime },\tau \in \Delta ,$$

and the inequality (9) may be formulated to express the fuzzy differential subordination:

$$\begin{array}{ccc}& & F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G(\Delta )}\left({\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{\prime }+\frac{1}{a+2}\tau {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{″}\right)\hfill \\ & \le & F_{g(\Delta )}\left(g\left(\tau \right)+\frac{1}{a+2}\tau g^{\prime }\left(\tau \right)\right).\hfill \end{array}$$

(12)

Let

$$\phi \left(\tau \right)={\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{\prime },\tau \in \Delta$$

(13)

and $\phi \in \mathcal{H}\left[1,n\right]$. Putting (13) in (12), we get

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\frac{1}{a+2}\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{g(\Delta )}\left(g\left(\tau \right)+\frac{1}{a+2}\tau g^{\prime }\left(\tau \right)\right),\tau \in \Delta .$$

By Lemma 3, we have

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{\prime }\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta ,$$

where g is the best dominant. □

In the following theorem, we proved the inclusion result for the class $R_m^F(\alpha ,\delta ,\zeta ,\lambda ).$

Theorem 3.

Suppose that

$$h\left(\tau \right)=\frac{1+(2\delta -1)\tau }{1+\tau },\delta \in [0,1),$$

as $a>0$. Let $\lambda >0,$ and $m>-1$

$$G\left(\tau \right)=I_a\left(f\right)\left(\tau \right)=\frac{a+2}{{\tau }^{a+1}}{\int }_0^{\tau }{t}^{a}f\left(t\right)dt,\tau \in \Delta ,$$

then

$$G\left(\tau \right)\left[R_m^F(\alpha ,\delta ,\zeta ,\lambda )\right]\subset R_m^F(\alpha ,{\delta }^{\ast },\zeta ,\lambda ),$$

(14)

where

$${\delta }^{\ast }=2\delta -1+\frac{2(a+2)(1-\delta )}{n}{\int }_0^1\frac{t^{a+1}}{1+t}dt.$$

Proof. 

By repeating the procedures used to prove Theorem 2, while keeping in mind the presumption of Theorem 3 and that

$$h\left(\tau \right)=\frac{1+(2\delta -1)\tau }{1+\tau }$$

is a convex function, we obtain

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\frac{1}{a+2}\tau {\phi }^{{}^{\prime }}\left(\tau \right)\right)\le f_{h(\Delta )}h\left(\tau \right),$$

where $\phi \left(\tau \right)$ is given by (13). Using the Lemma 2, we get

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\right)}^{{}^{\prime }}(\Delta )}{\left({}_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G}\right)}^{{}^{\prime }}\le F_{g(\Delta )}g\left(\tau \right)\le F_{h(\Delta )}h\left(\tau \right)$$

and

$$\begin{array}{ccc}\hfill g\left(\tau \right)& =& \frac{a+2}{n{\tau }^{a+2}}{\int }_0^{\tau }{t}^{\frac{2+a}{n}-1}\frac{1+(2\delta -1)}{1+t}dt\hfill \\ & =& \left(2\delta -1\right)+\frac{2(a+2)(1-\delta )}{n{\tau }^{\frac{2+a}{n}}}{\int }_0^1\frac{t^{\frac{2+a}{n}-1}}{1+t}dt.\hfill \end{array}$$

Since $g(\Delta )$ is symmetric with respect to the real axis and g is convex, therefore we may write

$$\begin{array}{ccc}& & F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}G\left(\tau \right)\right)}^{{}^{\prime }}\hfill \\ & \ge & \underset{\left|\tau \right|=1}{min}{F}_{g(\Delta )}g\left(\tau \right)=F_{g(\Delta )}g\left(1\right)\hfill \end{array}$$

(15)

and

$${\delta }^{\ast }=g\left(1\right)=2\delta -1+\frac{2(a+2)(1-\delta )}{n}{\int }_0^1\frac{t^{\frac{2+a}{n}-1}}{1+t}dt.$$

Inclusion (14) follows obviously from (15). □

Theorem 4.

Let g be a convex function and $g\left(0\right)=1;$ define

$$h\left(\tau \right)=g\left(\tau \right)+\tau g^{\prime }\left(\tau \right),\tau \in \Delta .$$

Let $f\in \mathcal{A}$ satisfy

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime }\le F_{h(\Delta )}h\left(\tau \right).$$

(16)

Let $m\in {\mathbb{N}}_0,$ then we get the sharp fuzzy differential subordination

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\le F_{g(\Delta )}g\left(\tau \right).$$

Proof. 

Consider $\phi \left(\tau \right)=\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau },$ we can write $\tau \phi \left(\tau \right)=D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right),$ for $\tau \in \Delta$. One differentiates the obtained expression

$${\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}=\phi \left(\tau \right)+\tau {\phi }^{{}^{\prime }}\left(\tau \right).$$

The inequality (16) can be written as follows:

$$\begin{array}{ccc}\hfill F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\tau {\phi }^{{}^{\prime }}\left(\tau \right)\right)& \le & F_{h(\Delta )}h\left(\tau \right)\hfill \\ & =& F_{g(\Delta )}\left(g\left(\tau \right)+\tau g^{{}^{\prime }}\left(\tau \right)\right).\hfill \end{array}$$

Using the Lemma 3, we have

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\right)}^{{}^{\prime }}(\Delta )}\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta .$$

□

Example 1.

Consider

$$g\left(\tau \right)=\frac{-2\tau }{1+\tau }$$

a convex function in Δ, and we obtain that $g\left(0\right)=0$ and $g^{{}^{\prime }}\left(\tau \right)=\frac{-2}{{\left(1+\tau \right)}^2}.$ Define

$$h\left(\tau \right)=g\left(\tau \right)+\tau g^{{}^{\prime }}\left(\tau \right)=\frac{-2{\tau }^2-4\tau }{{\left(1+\tau \right)}^2}.$$

Take $\alpha =2,$$\zeta =1,$$m=0,$$f\left(\tau \right)=\tau +{\tau }^2$ and, after a short computation, we obtain

$$\begin{array}{ccc}\hfill L_{2,1}^{0}f\left(\tau \right)& =& -S_1^{0}f\left(\tau \right)+2I^{0}f\left(\tau \right)\hfill \\ & =& -f\left(\tau \right)+2\tau f^{\prime }\left(\tau \right)\hfill \\ & =& -\tau -{\tau }^2+2\tau +4{\tau }^2\hfill \\ & =& \tau +3{\tau }^2.\hfill \end{array}$$

Then,

$${\left(L_{2,1}^{0}f\left(\tau \right)\right)}^{\prime }=1+6\tau$$

and

$$\frac{L_{2,1}^{0}f\left(\tau \right)}{\tau }=1+3\tau .$$

Now,

$$\begin{array}{ccc}\hfill D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)& =& \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{L_{2,1}^{0}f\left(\tau \right)}{{\left(\tau -t\right)}^{1-\lambda }}dt=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{\tau +3{\tau }^2}{{\left(\tau -t\right)}^{1-\lambda }}dt\hfill \\ & =& \frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +2}\hfill \end{array}$$

and, differentiating it,

$${\left(D_{\tau }^{-\lambda }{L}_{2,1}^{1}f\left(\tau \right)\right)}^{{}^{\prime }}=\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}.$$

Applying Theorem 4, we get the following fuzzy differential subordination:

$$\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}{\prec }_{\mathcal{F}}\frac{-2{\tau }^2-4\tau }{{\left(1+\tau \right)}^2},\tau \in ▵.$$

Induce the following fuzzy differential subordination:

$$\frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +1}{\prec }_{\mathcal{F}}\frac{-2\tau }{1+\tau },\tau \in ▵.$$

Theorem 5.

Suppose that h is a holomorphic function such that $\mathrm{Re}\left(1+\frac{\tau h^{{}^{″}}\left(\tau \right)}{h^{{}^{\prime }}\left(\tau \right)}\right)>\frac{-1}{2}$, and $h\left(0\right)=1$. Let $f\in \mathcal{A}$, and the fuzzy differential subordination holds

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m)f\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime }\le F_{h(\Delta )}h\left(\tau \right);$$

(17)

then,

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\le F_{q(\Delta )}q\left(\tau \right),$$

where

$$q\left(\tau \right)=\frac{1}{\tau }{\int }_0^{\tau }h\left(t\right)dt$$

is the best fuzzy dominant and a convex set.

Proof. 

Consider

$$\mathrm{Re}\left(1+\frac{\tau h^{{}^{″}}\left(\tau \right)}{h^{{}^{\prime }}\left(\tau \right)}\right)>\frac{-1}{2},\tau \in \Delta .$$

By Lemma 1, we get that

$$q\left(\tau \right)=\frac{1}{\tau }{\int }_0^{\tau }h\left(t\right)dt$$

is a convex function and verifies the differential equation associated with the fuzzy differential subordination (17)

$$q\left(\tau \right)+\tau q^{\prime }\left(\tau \right)=h\left(\tau \right).$$

Therefore, it is the fuzzy best dominant and, by differentiating, we obtain

$${\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime }=\phi \left(\tau \right)+\tau {\phi }^{\prime }\left(\tau \right),\tau \in \Delta .$$

Then, we get

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta .$$

By Lemma 3, we have

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\le F_{q(\Delta )}q\left(\tau \right),\tau \in \Delta .$$

□

Corollary 1.

Suppose that

$$h\left(\tau \right)=\frac{1+(2\delta -1)\tau }{1+\tau }$$

is a convex function in $\Delta ,$ for $0\le \delta <1$. Let $m\in {\mathbb{N}}_0,$$\zeta \ge 0,$$\alpha \ge 0,$$f\in \mathrm{A}$ and verify the fuzzy differential subordination

$$F_{{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\right)}^{\prime }(\Delta )}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{\prime }\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta ;$$

(18)

then,

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\le F_{q(\Delta )}q\left(\tau \right),\tau \in \Delta$$

and

$$q\left(\tau \right)=2\delta -1+2(1-\delta )\frac{log\left(\tau +1\right)}{\tau },\tau \in \Delta .$$

The function $q\left(\tau \right)$ is convex and fuzzy best dominant.

Proof. 

Let

$$h\left(\tau \right)=\frac{1+(2\delta -1)\tau }{1+\tau }.$$

Also,

$$h\left(0\right)=1\mathrm{and}{h}^{\prime }\left(\tau \right)=\frac{-2(1-\delta )}{{(1+\tau )}^2},$$

as well as

$$h^{″}\left(\tau \right)=\frac{4(1-\delta )}{{(1+\tau )}^3},$$

also

$$\begin{array}{ccc}& & \mathrm{Re}\left(\frac{\tau h^{″}\left(\tau \right)}{h^{\prime }\left(\tau \right)}+1\right)\hfill \\ & =& \mathrm{Re}\left(\frac{1-\tau }{1+\tau }\right)\hfill \\ & =& \mathrm{Re}\left(\frac{1-\varphi cos\theta -i\varphi sin\theta }{1+\varphi cos\theta +i\varphi sin\theta }\right)\hfill \\ & =& \frac{1-{\varphi }^2}{1+2\varphi cos\theta +{\varphi }^2}>0>-\frac{1}{2}.\hfill \end{array}$$

Using the same steps of the proof of Theorem 5, and considering

$$\phi \left(\tau \right)=\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau },$$

the fuzzy differential subordination (18) becomes

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left(\phi \left(\tau \right)+\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta .$$

According to the Lemma 2 for $\gamma =1$,

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\le F_{q(\Delta )}q\left(\tau \right),$$

where

$$q\left(\tau \right)=\frac{1}{\tau }{\int }_0^{\tau }h\left(t\right)dt,\tau \in \Delta$$

and

$$\begin{array}{ccc}\hfill q\left(\tau \right)& =& \frac{1}{\tau }\left({\int }_0^{\tau }\frac{1+\left(2\delta -1\right)t}{1+t}dt,\tau \in \Delta \right)\hfill \\ & =& 2\delta -1+\frac{2(1-\delta )}{\tau }{\int }_0^{\tau }\frac{1}{1+t}dt,\tau \in \Delta ,\hfill \\ & =& 2\delta -1+\frac{2(1-\delta )}{\tau }\frac{log\left(\tau +1\right)}{\tau }.\hfill \end{array}$$

□

Example 2.

Suppose

$$h\left(\tau \right)=\frac{-2\tau }{1+\tau }$$

with

$$h\left(0\right)=1,h^{\prime }\left(\tau \right)=\frac{-2}{{(1+\tau )}^2},$$

also

$$h^{″}\left(\tau \right)=\frac{4}{{(1+\tau )}^3}.$$

As well as

$$\begin{array}{ccc}& & \mathrm{Re}\left(\frac{\tau h^{″}\left(\tau \right)}{h^{\prime }\left(\tau \right)}+1\right)\hfill \\ & =& \mathrm{Re}\left(\frac{1-\tau }{1+\tau }\right)\hfill \\ & =& \mathrm{Re}\left(\frac{1-\varphi cos\theta -i\varphi sin\theta }{1+\varphi cos\theta +i\varphi sin\theta }\right)\hfill \\ & =& \frac{1-{\varphi }^2}{1+2\varphi cos\theta +{\varphi }^2}>0>-\frac{1}{2},\hfill \end{array}$$

the function h is convex in Δ.

Suppose

$$f\left(\tau \right)=\tau +{\tau }^2,\tau \in \Delta .$$

For $\zeta =1,$$\alpha =2$, and $m=0,$ we get

$$D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)=\left(1-\alpha \right)S_{\zeta }^{m}f\left(\tau \right)+\alpha I^{m}f\left(\tau \right).$$

$$\begin{array}{ccc}\hfill L_{2,1}^{0}f\left(\tau \right)& =& -S_1^{0}f\left(\tau \right)+2I^{0}f\left(\tau \right)\hfill \\ & =& -f\left(\tau \right)+2\tau f^{\prime }\left(\tau \right)\hfill \\ & =& -\tau -{\tau }^2+2\tau +4{\tau }^2\hfill \\ & =& \tau +3{\tau }^2.\hfill \end{array}$$

Then,

$${\left(L_{2,1}^{0}f\left(\tau \right)\right)}^{\prime }=1+6\tau$$

and

$$\begin{array}{ccc}\hfill D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)& =& \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{L_{2,1}^{0}f\left(\tau \right)}{{\left(\tau -t\right)}^{1-\lambda }}dt=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{\tau +3{\tau }^2}{{\left(\tau -t\right)}^{1-\lambda }}dt\hfill \\ & =& \frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +2}.\hfill \end{array}$$

Differentiating it,

$${\left(D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)\right)}^{{}^{\prime }}=\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}.$$

Additionally, we get

$$q\left(\tau \right)=\frac{1}{\tau }{\int }_0^{\tau }\frac{-2t}{1+t}dt=\frac{2ln(1+\tau )}{\tau }-2.$$

By Theorem 5, we get the following fuzzy differential subordination:

$$\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}{\prec }_{\mathcal{F}}\frac{-2\tau }{1+\tau },\tau \in ▵,$$

and induce the following fuzzy differential subordination:

$$\frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +2}{\prec }_{\mathcal{F}}\frac{2ln(1+\tau )}{\tau }-2,\tau \in ▵.$$

Theorem 6.

Let $h\left(\tau \right)=g\left(\tau \right)+\tau g^{\prime }\left(\tau \right),$$\tau \in \Delta$, and function g is a convex in Δ with $g\left(0\right)=1$. Let $f\in \mathcal{A}$ satisfy

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}{\left(\frac{\tau D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m+1}f\left(\tau \right)}{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}\right)}^{\prime }\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta .$$

(19)

As $\alpha \ge 0,$$m>-1$ and $n\in \mathbb{N}$, then we get sharp fuzzy differential subordination,

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left(\frac{\tau D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m+1}f\left(\tau \right)}{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}\right)\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta .$$

Proof. 

Consider

$$\phi \left(\tau \right)=\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m+1}f\left(\tau \right)}{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)};$$

differentiating it, we get

$$\phi \left(\tau \right)+\tau {\phi }^{\prime }\left(\tau \right)={\left(\frac{\tau L_{\alpha ,\zeta }^{m+1}f\left(\tau \right)}{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}\right)}^{\prime }.$$

Relation (19) becomes

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right)=F_{g(\Delta )}\left(g\left(\tau \right)+\tau g^{\prime }\left(\tau \right)\right),\tau \in \Delta .$$

According to Lemma 3, we have

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m+1}f\left(\tau \right)}{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right))}\right)\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta .$$

□

Theorem 7.

Define a function $h\left(\tau \right)=g\left(\tau \right)+\gamma \tau g^{\prime }\left(\tau \right),$$\tau \in \Delta$, and function g is a convex in Δ with $g\left(0\right)=1$, $\alpha \ge 0,$$\zeta \ge 0,$$\gamma >0,$$\beta >0,$$m>-1$, when $f\in \mathcal{A}$ and the fuzzy differential subordination

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left({\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}\right)\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta$$

(20)

hold, then we get sharp fuzzy differential subordination

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}{\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta }\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta .$$

Proof. 

Consider

$$\phi \left(\tau \right)={\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta };$$

differentiating it, we get

$$\begin{array}{ccc}\hfill \tau {\phi }^{\prime }\left(\tau \right)& =& \beta \left({\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}-\beta {\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta }\right)\hfill \\ & =& \beta \left({\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}-\beta \phi \left(\tau \right)\right)\hfill \\ & & \phi \left(\tau \right)+\frac{1}{\beta }\tau {\phi }^{\prime }\left(\tau \right)={\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}.\hfill \end{array}$$

Inequality (20) becomes

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\frac{1}{\beta }\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right)=F_{g(\Delta )}\left(g\left(\tau \right)+\gamma \tau g^{\prime }\left(\tau \right)\right),\tau \in \Delta .$$

According to the Lemma 3, we have

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}{\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta }\le F_{g(\Delta )}g\left(\tau \right),\tau \in \Delta .$$

□

Example 3.

Consider that

$$g\left(\tau \right)=\frac{-2\tau }{1+\tau }$$

is a convex function in Δ and $g\left(0\right)=0$, and $g^{{}^{\prime }}\left(\tau \right)=\frac{-2}{{\left(1+\tau \right)}^2}.$ Define

$$h\left(\tau \right)=g\left(\tau \right)+\tau g^{{}^{\prime }}\left(\tau \right)=\frac{-2{\tau }^2-4\tau }{{\left(1+\tau \right)}^2}.$$

Take $\alpha =2,$$\zeta =1,$$m=0,$$f\left(\tau \right)=\tau +{\tau }^2$ and, after a short computation, we obtain

$$\begin{array}{ccc}\hfill L_{2,1}^{0}f\left(\tau \right)& =& -S_1^{0}f\left(\tau \right)+2I^{0}f\left(\tau \right)\hfill \\ & =& -f\left(\tau \right)+2\tau f^{\prime }\left(\tau \right)\hfill \\ & =& -\tau -{\tau }^2+2\tau +4{\tau }^2\hfill \\ & =& \tau +3{\tau }^2.\hfill \end{array}$$

Then,

$${\left(L_{2,1}^{0}f\left(\tau \right)\right)}^{\prime }=1+6\tau$$

and

$$\frac{L_{2,1}^{0}f\left(\tau \right)}{\tau }=1+3\tau .$$

Now,

$$\begin{array}{ccc}\hfill D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)& =& \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{L_{2,1}^{0}f\left(\tau \right)}{{\left(\tau -t\right)}^{1-\lambda }}dt=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{\tau +3{\tau }^2}{{\left(\tau -t\right)}^{1-\lambda }}dt\hfill \\ & =& \frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +2};\hfill \end{array}$$

differentiating it, we have

$${\left(D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)\right)}^{{}^{\prime }}=\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}.$$

Applying Theorem 7, we get the following fuzzy differential subordination:

$${\left(\frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +1}\right)}^{\beta -1}\left(\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}\right){\prec }_{\mathcal{F}}\frac{-2{\tau }^2-4\tau }{{\left(1+\tau \right)}^2},\tau \in ▵,$$

induce the following fuzzy differential subordination

$${\left(\frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +1}\right)}^{\beta }{\prec }_{\mathcal{F}}\frac{-2\tau }{1+\tau },\tau \in ▵.$$

Theorem 8.

Suppose that h is a holomorphic function such that $\mathrm{Re}\left(1+\frac{\tau h^{{}^{″}}\left(\tau \right)}{h^{{}^{\prime }}\left(\tau \right)}\right)>\frac{-1}{2}$, and $h\left(0\right)=1$, $\alpha \ge 0,$$\zeta \ge 0,$$\gamma >0,\beta >0,$$m>-1,$ when $f\in \mathcal{A}$ and the fuzzy differential subordination

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left({\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}\right)\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta$$

(21)

hold, we get sharp fuzzy differential subordination,

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}{\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta }\le F_{q(\Delta )}q\left(\tau \right),\tau \in \Delta ,$$

where the fuzzy best dominant $q\left(\tau \right)=\frac{1}{\tau }\underset{0}{\overset{\tau }{\int }}h\left(t\right)dt$ is convex.

Proof. 

Consider

$$\phi \left(\tau \right)={\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta };$$

differentiating it, we get

$$\begin{array}{ccc}\hfill \tau {\phi }^{\prime }\left(\tau \right)& =& \beta \left({\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}-\beta {\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta }\right)\hfill \\ & =& \beta \left({\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}-\beta \phi \left(\tau \right)\right),\hfill \\ & & \phi \left(\tau \right)+\frac{1}{\beta }\tau {\phi }^{\prime }\left(\tau \right)={\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta -1}{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}.\hfill \end{array}$$

Inequality (21) becomes

$$F_{\phi (\Delta )}\left(\phi \left(\tau \right)+\frac{1}{\beta }\tau {\phi }^{\prime }\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right)=F_{g(\Delta )}\left(g\left(\tau \right)+\gamma \tau g^{\prime }\left(\tau \right)\right),\tau \in ▵.$$

According to Lemma 3, we get

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}{\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau }\right)}^{\beta }\le F_{q(\Delta )}q\left(\tau \right),\tau \in \Delta .$$

Taking into account that

$$\mathrm{Re}\left(1+\frac{\tau h^{{}^{″}}\left(\tau \right)}{h^{{}^{\prime }}\left(\tau \right)}\right)>\frac{-1}{2}.$$

Applying the Lemma 1, we obtain that

$$q\left(\tau \right)=\frac{1}{\tau }\underset{0}{\overset{\tau }{\int }}h\left(t\right)dt,$$

where $q\left(\tau \right)$ is a convex function and it is a solution of the differential equation of the fuzzy differential subordination (21) and $q\left(\tau \right)+\tau q^{\prime }\left(\tau \right)=h\left(\tau \right),$ so $q\left(\tau \right)$ is the fuzzy best dominant. □

Example 4.

Suppose that

$$h\left(\tau \right)=\frac{-2\tau }{1+\tau },$$

with

$$h\left(0\right)=1,h^{\prime }\left(\tau \right)=\frac{-2}{{(1+\tau )}^2};$$

also,

$$h^{″}\left(\tau \right)=\frac{4}{{(1+\tau )}^3}.$$

Additionally,

$$\begin{array}{ccc}& & \mathrm{Re}\left(\frac{\tau h^{″}\left(\tau \right)}{h^{\prime }\left(\tau \right)}+1\right)\hfill \\ & =& \mathrm{Re}\left(\frac{1-\tau }{1+\tau }\right)\hfill \\ & =& \mathrm{Re}\left(\frac{1-\varphi cos\theta -i\varphi sin\theta }{1+\varphi cos\theta +i\varphi sin\theta }\right)\hfill \\ & =& \frac{1-{\varphi }^2}{1+2\varphi cos\theta +{\varphi }^2}>0>-\frac{1}{2}.\hfill \end{array}$$

Hence, the function h is convex in Δ.

Suppose that

$$f\left(\tau \right)=\tau +{\tau }^2,\tau \in \Delta .$$

For $\zeta =1,$$\alpha =2$, and $m=0,$ we get

$$\begin{array}{ccc}\hfill L_{2,1}^{0}f\left(\tau \right)& =& -S_1^{0}f\left(\tau \right)+2I^{0}f\left(\tau \right)\hfill \\ & =& -f\left(\tau \right)+2\tau f^{\prime }\left(\tau \right)\hfill \\ & =& -\tau -{\tau }^2+2\tau +4{\tau }^2\hfill \\ & =& \tau +3{\tau }^2.\hfill \end{array}$$

Then,

$${\left(L_{2,1}^{0}f\left(\tau \right)\right)}^{\prime }=1+6\tau$$

and

$$\begin{array}{ccc}\hfill D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)& =& \frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{L_{2,1}^{0}f\left(\tau \right)}{{\left(\tau -t\right)}^{1-\lambda }}dt=\frac{1}{\mathsf{\Gamma }\left(\lambda \right)}\underset{0}{\overset{\tau }{\int }}\frac{\tau +3{\tau }^2}{{\left(\tau -t\right)}^{1-\lambda }}dt\hfill \\ & =& \frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +2}.\hfill \end{array}$$

Differentiating it,

$${\left(D_{\tau }^{-\lambda }{L}_{2,1}^{0}f\left(\tau \right)\right)}^{{}^{\prime }}=\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}.$$

Additionally, we get

$$q\left(\tau \right)=\frac{1}{\tau }{\int }_0^{\tau }\frac{-2t}{1+t}dt=\frac{2ln(1+\tau )}{\tau }-2.$$

By Theorem 8, we get the following fuzzy differential subordination:

$${\left(\frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +1}\right)}^{\beta -1}\left(\frac{1}{\mathsf{\Gamma }\left(\lambda +1\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda +1}\right){\prec }_{\mathcal{F}}\frac{-2\tau }{1+\tau },\tau \in ▵;$$

induce the following fuzzy differential subordination:

$$\frac{1}{\mathsf{\Gamma }\left(\lambda +2\right)}{\tau }^{\lambda }+\frac{6}{\mathsf{\Gamma }\left(\lambda +3\right)}{\tau }^{\lambda +1}{\prec }_{\mathcal{F}}\frac{2ln(1+\tau )}{\tau }-2,\tau \in ▵.$$

Theorem 9.

Considering a convex function g with the property $g\left(0\right)=\frac{1}{\lambda +1}$ and defining

$$h\left(\tau \right)=g\left(\tau \right)+\tau g^{{}^{\prime }}\left(\tau \right),$$

when $f\in \mathcal{A}$ and the fuzzy differential subordination

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left(1-\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right){\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{″}}}{{\left[{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}\right]}^2}\right)\le F_{h(\Delta )}h\left(\tau \right),\tau \in \Delta$$

hold, then we obtain the sharp result,

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}}\right)\le F_{g(\Delta )}g\left(\tau \right).$$

Proof. 

Differentiating

$$p\left(\tau \right)=\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}},$$

we obtain

$$\tau p^{{}^{\prime }}\left(\tau \right)+p\left(\tau \right)=1-\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right){\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{″}}}{{\left[{\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}\right]}^2},\tau \in ▵.$$

Using this notation, the fuzzy differential subordination can be written as:

$$F_{p(\Delta )}\left(\tau p^{{}^{\prime }}\left(\tau \right)+p\left(\tau \right)\right)\le F_{h(\Delta )}h\left(\tau \right)=F_{g(\Delta )}\left(\tau g^{{}^{\prime }}\left(\tau \right)+g\left(\tau \right)\right).$$

According to the Lemma 3, we obtain the sharp result

$$F_{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f(\Delta )}\left(\frac{D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)}{\tau {\left(D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^{m}f\left(\tau \right)\right)}^{{}^{\prime }}}\right)\le F_{g(\Delta )}g\left(\tau \right).$$

□

5. Conclusions

In this research, we combined ideas from geometric function theory and fuzzy set theory. As the first step of our research, we developed an interesting operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m:\mathcal{A}\to \mathcal{A}$ of analytic functions in an open unit disc $\Delta$ by combining the help of the Riemann–Liouville fractional integral operator. Then, we considered the operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m$, and defined a special fuzzy class of analytic functions represented by $R_F^{\zeta }(m,\alpha ,\delta )$. Using the idea of fuzzy differential subordination, we found many new findings that are relevant to this class. Following our discussion of the main theorems and corollaries, we provided many examples related to these results.

The subclass provided in Definition 11 and the operator specified in Definition 10 may be used in further studies, and other studies involving neighborhood analysis, convexity, close-to-convexity, coefficient estimates, closure theorems, and distortion theorems may all be investigated. The dual theory of fuzzy differential superordination may provide similar results using the operator $D_{\tau }^{-\lambda }{L}_{\alpha ,\zeta }^m$ and the class $R_F^{\zeta }(m,\alpha ,\delta )$, and these may be related to the results given here for sandwich-type theorems.