Fuzzy Differential Subordination of the Atangana–Baleanu Fractional Integral

Abstract

The present paper continues the study on the relatively new concept of fuzzy differential subordination conducted in some recently published cited papers. In this article, certain fuzzy subordination results for analytical functions involving the Atangana–Baleanu fractional integral of Bessel functions are presented. Theorems giving the best dominants for some fuzzy differential subordinations are proved, and interesting corollaries are provided with the use of particular functions as fuzzy best dominants.

1. Introduction and Preliminary Results

The notion of fuzzy differential subordination was defined in two papers in 2011 [1] and 2012 [2] in order to extend the results obtained using the classical theory of differential subordination according to Mocanu and Miller that was synthesized in [3], which was a follow up to the trend among mathematicians of embedding the popular concept of the fuzzy set introduced by Lotfi A. Zadeh [4]. Fuzzy differential subordination results emerged soon after the notion was introduced, and the theory marks an important development in the past years as it can be easily seen by citing only some of the very recent papers [5,6,7]. The study performed to obtain the original results of this paper follows a line of research that combines the notion of fuzzy differential subordination with different types of operators. Such studies have recently been published [8,9,10,11], proving that the topic is of interest for researchers.

Fractional calculus has undergone tremendous development in the last few years, and has been proven to have applications in many research domains such as physics, engineering, turbulence, electric networks, biological systems with memory, computer graphics, etc. Numerous interesting approaches were taken into account and new results have emerged in an impressive number of recently published papers.

We next give some examples by citing the latest papers. A new mathematical model given by fractional-order differential equations is considered in order to analyze the complicated dynamical motion of a quarter-car suspension system with a sinusoidal road excitation force in [12]. In [13], a nonlinear mathematical model is introduced using integer and fractional-order differential equations for the investigation of a non-autonomous cardiac conduction system considering the aspects of hyperchaos analysis, optimal control, and synchronization. The dynamical behavior of a linear tri-atomic molecule is investigated in [14]. The investigation is performed firstly using a classical Lagrangian approach, which produces the classical equations of motion, and then the generalized form of the fractional Hamilton equations is formulated in the Caputo sense. A novel fractional chaotic system, including quadratic and cubic nonlinearities, is introduced and analyzed in [15], taking into account the Caputo derivative for the fractional model and an efficient nonstandard finite difference scheme in order to investigate its chaotic behavior in both the time domain and the phase plane. For a multi-term time-fractional diffusion equation, the direct and inverse problems are discussed in [16]. The formal solution, existence, uniqueness, and stability results are obtained for both of the problems.

New modifications of integral inequalities are investigated, and a refinement of the Hermite–Hadamard type integral inequality is given in [17] using a generalized fractional integral operator. An integral identity and more generalized fractional integral inequalities of the Ostrowski type with respect to certain operators are also presented in that paper. A new function called the Mittag-Leffler–confluent hypergeometric function is introduced in [18] using Riemann-Liouville fractional derivatives and integrals, and certain integral equations with several analytical implementations are examined. The study on this function is continued in [19] where new extensions on its fractional differential and integral properties are given.

A fuzzy Atangana–Baleanu fractional derivative operator is considered and studied in [20]. Since the particular topic related to fractional operators is also represented among research studies connected with the fuzzy differential subordination theory, as can be seen in [21,22], the idea of the present paper is to continue the study on the operator introduced in [23] with the use of the same Atangana–Baleanu fractional derivative. The study conducted in [23] using the classical theory of differential subordination is adapted in the present paper using the theory of fuzzy differential subordination.

In this paper, we present fuzzy subordination results in the form of two theorems and four corollaries for each of those theorems. The fuzzy subordinations obtained in the theorems are investigated in order to get their fuzzy best dominants. In order to make the appropriate choices of functions acting as fuzzy best dominants in Theorems 1 and 2, interesting and particular fuzzy subordinations are obtained in the corollaries that follow each theorem. The functions used are selected considering their univalence conditions derived from the geometric function theory.

The symmetry properties of the functions used in defining an equation or inequality could be studied to determine solutions with particular properties. Regarding the fuzzy differential subordinations, some of which are inequalities, the study of special functions could provide interesting results given their symmetry properties. Studies on the symmetry properties of different types of functions associated with the concept of quantum computing could also be investigated in a future paper.

2. Preliminaries

The study presented in this paper is performed in the general context of geometric function theory.

The unit disc of the complex plane is denoted by $U=\{z\in \mathbb{C}:|z|<1\}$, and the class of analytic functions in U by $\mathcal{H}\left(U\right)$. For n, a positive integer, and $a\in \mathbb{C}$, $\mathcal{H}\left[a,n\right]$ denotes the subclass of $\mathcal{H}\left(U\right)$ consisting of functions written in the form $f\left(z\right)=a+a_n{z}^n+a_{n+1}{z}^{n+1}+\dots .$, $z\in U$.

For the concept of fuzzy differential subordination to be used, the following notions are necessary:

Definition 1

([24]). A pair $(A,F_A)$, where $F_A:X\to [0,1]$ and $A=\{x\in X:0<F_A\left(x\right)\le 1\}$ is called the fuzzy subset of X. The set A is called the support of the fuzzy set $(A,F_A)$, and $F_A$ is called the membership function of the fuzzy set $(A,F_A)$. One can also denote $A=\mathrm{supp}(A,F_A)$.

Remark 1

([24]). If $A\subset X$, then $F_A\left(x\right)=\left\{\begin{array}{c}1,\mathit{if}x\in A\hfill \\ 0,\mathit{if}x\notin A\hfill \end{array}\right..$

For a fuzzy subset, the real number 0 represents the smallest membership degree of a certain $x\in X$ to A, and the real number 1 represents the biggest membership degree of a certain $x\in X$ to A.

The empty set $\varnothing \subset X$ is characterized by $F_{\varnothing }\left(x\right)=0$, $x\in X$, and the total set X is characterized by $F_X\left(x\right)=1$, $x\in X$.

Definition 2

([1]). Let $D\subset \mathbb{C}$, $z_0\in D$ be a fixed point and let the functions $f,g\in \mathcal{H}\left(D\right)$. The function f is said to be a fuzzy subordinate to g and written as $f{\prec }_{\mathcal{F}}g$ or $f\left(z\right){\prec }_{\mathcal{F}}g\left(z\right)$ if the following conditions are satisfied:

(1) $f\left(z_0\right)=g\left(z_0\right);$

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

Definition 3

([2] (Definition 2.2)). Let $\psi :{\mathbb{C}}^3\times U\to \mathbb{C}$ and h univalent in U, with $\psi \left(a,0;0\right)=h\left(0\right)=a$. If p is analytic in U, with $p\left(0\right)=a$, and satisfies the following (second-order) fuzzy differential subordination:

$$F_{\psi \left({\mathbb{C}}^3\times U\right)}\psi (p\left(z\right),z{p}^{\prime }\left(z\right),z^2{p}^{″}\left(z\right);z)\le F_{h\left(U\right)}h\left(z\right),z\in U,$$

(1)

then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, or more simply a fuzzy dominant, if $F_{p\left(U\right)}p\left(z\right)\le F_{q\left(U\right)}q\left(z\right)$, $z\in U$, for all p satisfying (1). A fuzzy dominant $\tilde{q}$ that satisfies $F_{\tilde{q}\left(U\right)}\tilde{q}\left(z\right)\le F_{q\left(U\right)}q\left(z\right)$, $z\in U$ for all fuzzy dominants q of (1) is said to be the fuzzy best dominant of (1).

The following previously obtained results are also necessary for the study:

Definition 4

([25]). Denote by Q the set of all functions f that are analytic and injective on $\overline{U}\backslash E\left(f\right)$, where $E\left(f\right)=\{\zeta \in \partial U:{\displaystyle \underset{z\to \zeta }{lim}}f\left(z\right)=\infty \},$ such that $f^{\prime }\left(\zeta \right)\ne 0$ for $\zeta \in \partial U\backslash E\left(f\right)$.

Lemma 1

([25]). Let the function q be univalent in the unit disc U, and θ and ϕ be analytic in a domain D containing $q\left(U\right)$ with $\varphi \left(w\right)\ne 0$ when $w\in q\left(U\right)$. Set $Q\left(z\right)=z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right)$ and $h\left(z\right)=\theta \left(q\left(z\right)\right)+Q\left(z\right)$. Suppose that:

1. Q is starlike univalent in U;

2. $Re\left(\frac{z{h}^{\prime }\left(z\right)}{Q\left(z\right)}\right)>0$ for $z\in U$.

If p is analytic with $p\left(0\right)=q\left(0\right)$, $p\left(U\right)\subseteq D$ and

$$F_{p\left(U\right)}\theta \left(p\left(z\right)\right)+z{p}^{\prime }\left(z\right)\varphi \left(p\left(z\right)\right)\le F_{h\left(U\right)}\theta \left(q\left(z\right)\right)+z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right),$$

then

$$F_{p\left(U\right)}p\left(z\right)\le F_{q\left(U\right)}q\left(z\right)$$

and q is the fuzzy best dominant.

The operator that will be used for the research presented in the original part of this paper was introduced in [23] and studied using the classical theory of differential subordination. In this paper, the study is extended and uses the notion of fuzzy differential subordination. The function is given below:

Definition 5

([23]). Consider $\delta ,b,c,\lambda$$\in \mathbb{C}$ with $\mathrm{Re}$ $\lambda >0$ and

$${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)=\frac{1-\lambda }{B\left(\lambda \right)}{w}_{\delta ,b,c}\left(z\right)+\frac{\lambda }{B\left(\lambda \right)}{}^{RL}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)$$

(2)

where ${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right){=}_0^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right),{}^{RL}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right){=}_0^{RL}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)$ and $B\left(\lambda \right)$ is a normalization function $B\left(0\right)=B\left(1\right)=1.$

We recall that the Riemann–Liouville fractional integral ([26]) is defined by the following relation:

$${}_c^{RL}{I}_z^{\nu }f\left(z\right)=\frac{1}{\Gamma \left(\nu \right)}{\int }_c^z{\left(z-w\right)}^{\nu -1}f\left(w\right)dw,\mathrm{Re}\left(\nu \right)>0,$$

while the extended Atangana–Baleanu integral [27] is defined for any $\nu \in \mathbb{C}$ and any $z\in D\backslash \left\{c\right\}$ by

$${}_c^{AB}{I}_z^{\nu }f\left(z\right)=\frac{1-\nu }{B\left(\nu \right)}f\left(z\right)+\frac{\nu }{B\left(\nu \right)}{}_c^{RL}{I}_z^{\nu }f\left(z\right),$$

and the generalized Bessel function of the first kind of order $\delta$ [28,29] has the following form:

$$w_{\delta ,b,c}\left(z\right)=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k{c}^k}{k!\Gamma \left(\delta +k+\frac{b+1}{2}\right)}·{\left(\frac{z}{2}\right)}^{2k+\delta },\delta ,b,c\in \mathbb{C}.$$

It follows the representation of new defined function:

$${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)=\frac{1}{B\left(\lambda \right)2^{\delta }}\sum _{k=0}^{\infty }\alpha (\delta ,b,c,k)\left(1-\lambda +\frac{\lambda \Gamma \left(2k+\delta +1\right)z^{\lambda }}{\Gamma \left(2k+\delta +\lambda +1\right)}\right)z^{2k+\delta }$$

where

$$\alpha (\delta ,b,c,k)=\frac{{\left(-1\right)}^k{c}^k}{\Gamma \left(k+1\right)\Gamma \left(\delta +k+\frac{b+1}{2}\right)4^k}.$$

3. Main Results

Theorem 1.

Let $\beta ,\gamma ,\mu ,\xi ,\alpha \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\mathrm{Re}\lambda >0$, and q be univalent in U with $q\left(z\right)\ne 0$.

Assume that $\frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}$ is univalent starlike in U. Let

$$\mathrm{Re}\left\{\frac{\alpha }{\xi }\frac{q\left(z\right)}{q^{\prime }\left(z\right)}+\frac{2\gamma }{\xi }{\left(q\left(z\right)\right)}^2+\frac{\beta }{\xi }q\left(z\right)\right\}>0$$

(3)

and

$${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z):=\xi \mu \left[\frac{z·{\left({}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)\right)}^{\prime }}{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}-1\right]+\gamma {\left[\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right]}^{2\mu }$$

(4)

$$+\beta {\left[\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right]}^{\mu }+\alpha .$$

If the following fuzzy subordination is satisfied by $q,$

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)\le F_{q\left(U\right)}\left(\xi \frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}+\gamma {\left(q\left(z\right)\right)}^2+\beta q\left(z\right)+\alpha \right),$$

(5)

then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}q\left(z\right),z\in U,z\ne 0$$

(6)

and the fuzzy best dominant is the function q.

Proof. 

Consider

$$p\left(z\right):={\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu },z\in U,z\ne 0,$$

and after differentiating it, we can write

$$\frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}=\mu \left[\frac{z{{(}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right))}^{\prime }}{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}-1\right].$$

Put

$$\begin{array}{c}\theta \left(w\right):=\gamma w^2+\beta w+\alpha \\ \varphi \left(w\right):=\frac{\xi }{w}\end{array}$$

and it is evident that $\theta \left(w\right)$ is analytic in $\mathbb{C}$, $\varphi \left(w\right)$ is analytic in $\mathbb{C}\backslash \left\{0\right\}$, and $\varphi \left(w\right)\ne 0$, $w\in \mathbb{C}\backslash \left\{0\right\}$. Considering

$$Q\left(z\right)=z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right)=\xi \frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}$$

and

$$h\left(z\right):=\theta \left(q\left(z\right)\right)+Q\left(z\right)=\xi \frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}\gamma {\left(q\left(z\right)\right)}^2++\beta q\left(z\right)+\alpha ,$$

we see that Q is univalent starlike in U, and

$$\mathrm{Re}\left\{\frac{z{h}^{\prime }\left(z\right)}{Q\left(z\right)}\right\}=\mathrm{Re}\left\{\frac{\alpha }{\xi }\frac{q\left(z\right)}{q^{\prime }\left(z\right)}+\frac{2\gamma }{\xi }{\left(q\left(z\right)\right)}^2+\frac{\beta }{\xi }q\left(z\right)\right\}>0.$$

Applying Lemma 1, we then obtain the relation (6) of Theorem 1. □

For the choices $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $-1\le N<M\le 1$ and $q\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\kappa }$, $0<\kappa \le 1$ in Theorem 1, we get the corollaries.

Corollary 1.

Let $\beta ,\alpha ,\gamma ,\mu ,\xi \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\mathrm{Re}\lambda >0$, $-1\le N<M\le 1$ and

$$\mathrm{Re}\left\{\frac{\alpha }{\xi }\frac{\left(1+Mz\right)\left(1+Nz\right)}{M-N}+\frac{2\gamma }{\xi }{\left(\frac{1+Mz}{1+Nz}\right)}^2+\frac{\beta }{\xi }\frac{1+Mz}{1+Nz}\right\}>0.$$

(7)

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)\le$$

$$F_{q\left(U\right)}\left(\xi \frac{(M-N)z}{(1+Mz)(1+Nz)}+\gamma {\left(\frac{1+Mz}{1+Nz}\right)}^2+\beta \frac{1+Mz}{1+Nz}+\alpha \right)$$

(8)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\frac{1+Mz}{1+Nz}$$

(9)

and $\frac{1+Mz}{1+Nz}$ is the fuzzy best dominant.

Corollary 2.

Let $\beta ,\alpha ,\gamma ,\mu ,\xi \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\mathrm{Re}\lambda >0$, $0<\kappa \le 1$ and

$$\mathrm{Re}\left\{\frac{\alpha }{2\xi \kappa }\left(1-z^2\right)+\frac{2\gamma }{\xi }{\left(\frac{1+z}{1-z}\right)}^{2\kappa }+\frac{\beta }{\xi }{\left(\frac{1+z}{1-z}\right)}^{\kappa }\right\}>0.$$

(10)

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)\le F_{q\left(U\right)}\left(\frac{2\xi \kappa z}{1-z^2}+\gamma {\left(\frac{1+z}{1-z}\right)}^{2\kappa }+\beta {\left(\frac{1+z}{1-z}\right)}^{\kappa }+\alpha \right)$$

(11)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}{\left(\frac{1+z}{1-z}\right)}^{\kappa }$$

(12)

and ${\left(\frac{1+z}{1-z}\right)}^{\kappa }$ is the fuzzy best dominant.

For $q\left(z\right)=e^{\kappa Mz}$, $\left|\kappa M\right|<\pi$ in Theorem 1, the following result is obtained.

Corollary 3.

Let $M,\beta ,\alpha ,\gamma ,\mu ,\xi \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\left|\kappa M\right|<\pi$, $\mathrm{Re}\lambda >0$ and

$$\mathrm{Re}\left\{\frac{2\gamma }{\xi }{e}^{2\kappa Mz}+\frac{\beta }{\xi }{e}^{\kappa Mz}+\frac{\alpha }{\xi \kappa M}\right\}>0.$$

(13)

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)\le F_{q\left(U\right)}\left(\xi M\kappa z+\gamma e^{2\kappa Mz}+\beta e^{\kappa Mz}+\alpha \right)$$

(14)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\left(e^{\kappa Mz}\right)$$

(15)

and $e^{\kappa Mz}$ is the fuzzy best dominant.

Corollary 4.

Let $\beta ,\gamma ,\mu ,\xi ,\kappa ,\alpha \in \mathbb{C},$$\kappa \ne 0,$$\mu \ne 0,$$\xi \ne 0,$$\mathrm{Re}\lambda >0,-1\le N<M\le 1$ and

$$\mathrm{Re}\left\{\frac{\alpha (1+Nz)}{\xi \kappa \left(M-N\right)}+\frac{2\gamma }{\xi }{(1+Nz)}^{\frac{2\kappa (M-N)}{N}}+\frac{\beta }{\xi }{(1+Mz)}^{\frac{\kappa (M-N)}{M}}\right\}>0.$$

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)\le$$

$$F_{q\left(U\right)}\left(\xi \frac{z\kappa \left(M-N\right)}{1+Nz}+\gamma {(1+Nz)}^{\frac{2\kappa (M-N)}{N}}+\beta {(1+Nz)}^{\frac{\kappa (M-N)}{N}}+\alpha \right),$$

(16)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\gamma ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\left({(1+Nz)}^{\frac{\kappa (M-N)}{N}}\right)$$

(17)

and ${(1+Nz)}^{\frac{\kappa (M-N)}{M}}$ is the fuzzy best dominant.

The function $q\left(z\right)={(1+Nz)}^{\frac{\kappa (M-N)}{N}}$ is univalent if and only if either

$$\left|\frac{\kappa (M-N)}{N}-1\right|\le 1\mathrm{or}\left|\frac{\kappa (M-N)}{N}+1\right|\le 1.$$

Theorem 2.

Let the function q be univalent in the unit disc U, with $q\left(z\right)\ne 0$ and $\beta ,\alpha ,\mu ,\xi \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\mathrm{Re}\lambda >0$.

Suppose that $\frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}$ is univalent starlike in U. Consider the inequality

$$\mathrm{Re}\left\{\frac{z{q}^{″}\left(z\right)}{q^{\prime }\left(z\right)}+\frac{\beta }{\xi }+1\right\}>0$$

(18)

and

$${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z):=\xi \mu {\left({}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)\right)}^{\prime }{\left[\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right]}^{\mu -1}+$$

(19)

$$+\left(\beta -\mu \xi \right){\left[\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right]}^{\mu }+\alpha .$$

If q satisfies the following fuzzy subordination:

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)\le F_{q\left(U\right)}\left(\xi z{q}^{\prime }\left(z\right)+\beta q\left(z\right)+\alpha \right),$$

(20)

then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}q\left(z\right),z\in U,z\ne 0$$

(21)

and the fuzzy best dominant is the function q.

Proof. 

Considering

$$p\left(z\right):={\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu },z\in U,z\ne 0.$$

By differentiating it, we get the following:

$$p^{\prime }\left(z\right)=\mu {\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu -1}\left[\frac{{\left({}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)\right)}^{\prime }}{z}-\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z^2}\right].$$

(22)

Put $\theta \left(w\right):=\beta w+\alpha$ and $\varphi \left(w\right):=\xi$ and it is evident that $\theta \left(w\right)$ is analytic in $\mathbb{C}$, $\varphi \left(w\right)$ is analytic in $\mathbb{C}\backslash \left\{0\right\}$, and $\varphi \left(w\right)\ne 0$, $w\in \mathbb{C}\backslash \left\{0\right\}$. Considering

$$Q\left(z\right)=z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right)=\xi z{q}^{\prime }\left(z\right)$$

and

$$h\left(z\right):=\theta \left(q\left(z\right)\right)+Q\left(z\right)=\xi z{q}^{\prime }\left(z\right)+\beta q\left(z\right)+\alpha ,$$

we get that $Q\left(z\right)$ is univalent starlike in U and

$$\mathrm{Re}\left\{\frac{z{h}^{\prime }\left(z\right)}{Q\left(z\right)}\right\}=\mathrm{Re}\left\{\frac{z{q}^{″}\left(z\right)}{q^{\prime }\left(z\right)}+\frac{\beta }{\xi }+1\right\}>0.$$

Applying Lemma 1 yields the assertion (21) of Theorem 2. □

Theorem 2 gives the following corollaries:

Corollary 5.

Consider $\beta ,\alpha ,\mu ,\xi \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\mathrm{Re}\lambda >0$, $-1\le N<M\le 1$ and

$$\mathrm{Re}\left\{1+\frac{\beta }{\xi }-\frac{2Nz}{1+Nz}\right\}>0.$$

(23)

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)\le F_{q\left(U\right)}\left(\xi \frac{(M-N)z}{{(1+Nz)}^2}+\beta \frac{1+Mz}{1+Nz}+\alpha ,\right)$$

(24)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\left(\frac{1+Mz}{1+Nz}\right)$$

(25)

and $\frac{1+Mz}{1+Nz}$ is the fuzzy best dominant.

Corollary 6.

Consider $\beta ,\alpha ,\mu ,\xi \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\mathrm{Re}\lambda >0$, $0<\kappa \le 1$ and

$$\mathrm{Re}\left\{1+\frac{\beta }{\xi }+\frac{2z\left(\kappa +z\right)}{1-z^2}\right\}>0.$$

(26)

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)\le F_{q\left(U\right)}\left(\xi z\frac{{\left(1+z\right)}^{\kappa -1}}{{\left(1-z\right)}^{\kappa +1}}+\beta {\left(\frac{1+z}{1-z}\right)}^{\kappa }+\alpha \right)$$

(27)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(U\right)}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{\kappa }\right)$$

(28)

and ${\left(\frac{1+z}{1-z}\right)}^{\kappa }$ is the fuzzy best dominant.

Corollary 7.

Let $M,\beta ,\alpha ,\mu ,\xi ,\kappa \in \mathbb{C}$, $\xi \ne 0$, $\mu \ne 0$, $\left|\kappa M\right|<\pi$, $\mathrm{Re}\lambda >0$ and

$$\mathrm{Re}\left(1+\frac{\beta }{\xi }+z\kappa M\right)>0.$$

(29)

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)\le F_{q\left(U\right)}\left(e^{\kappa Mz}\left(\beta +\xi M\kappa z\right)+\alpha \right),$$

(30)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\left(e^{\kappa M}\right)$$

(31)

and $e^{\kappa Mz}$ is the fuzzy best dominant.

Corollary 8.

Let $\beta ,\alpha ,\mu ,\xi ,\kappa \in \mathbb{C},$$\mu \ne 0,$$\xi \ne 0,$$\kappa \ne 0,$$\mathrm{Re}\lambda >0,$$-1\le N<M\le 1$ and

$$\mathrm{Re}\left\{\frac{\beta }{\xi }\frac{z\left[\kappa \left(M-N\right)-N\right]}{(1+Nz)}+1\right\}>0.$$

If

$$F_{{\psi }_{\lambda }^{\delta ,b,c}\left(U\right)}{\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)\le$$

$$F_{q\left(U\right)}\left(\xi z\kappa \left(M-N\right){(1+Nz)}^{\frac{\kappa (M-N)-N}{N}}+\beta {(1+Nz)}^{\frac{\kappa (M-N)}{M}}+\alpha \right),$$

(32)

with ${\psi }_{\lambda }^{\delta ,b,c}(\mu ,\alpha ,\beta ,\xi ;z)$ introduced in (4), then

$$F_{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}}{\left(\frac{{}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)}{z}\right)}^{\mu }\le F_{q\left(U\right)}\left({(1+Nz)}^{\frac{\kappa (M-N)}{N}}\right)$$

(33)

and ${(1+Nz)}^{\frac{\kappa (M-N)}{N}}$ is the fuzzy best dominant.

4. Conclusions

In this paper, the original results are obtained using the theory of fuzzy differential subordination on the function ${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)$ given in Definition 5, introduced with the help of the Atangana–Baleanu fractional derivative. Fuzzy subordinations involving the function ${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)$ are investigated in the two theorems that formulate the essence of the study. The importance of this study comes from the new approach to the function ${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)$, which is investigated here using the notion of fuzzy differential subordination. This sort of investigation is at its beginning since the notion itself is rather new in the literature. Interesting corollaries emerge when known univalent functions with particular properties are used as fuzzy best dominants in the proved theorems. As suggestions for future research, we propose investigations on this function using the dual notion of fuzzy differential superordination introduced in 2017 [30]. Due to such a study, sandwich-type results related to the study performed here are made possible. The prospect of introducing fuzzy classes of analytic functions using the function ${}^{AB}{I}_z^{\lambda }{w}_{\delta ,b,c}\left(z\right)$ must also be investigated. This approach is familiar in the classical theory of differential subordination and could provide interesting results using the theory of fuzzy differential subordination. Since the theories involving the notion of the fuzzy set have offered many applications in real life, it is hoped that interdisciplinary studies will be implemented considering this new function and the fuzzy differential subordinations obtained in this paper.