**2. Preliminaries**

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

The unit disc of the complex plane is denoted by *U* = {*z* ∈ C : |*z*| < 1} and the class of analytic functions in *U* by H(*U*). For *n* a positive integer and *a* ∈ C, H[*<sup>a</sup>*, *n*] denotes the subclass of H(*U*) consisting of functions written in the form *f*(*z*) = *a* + *anz<sup>n</sup>* + *an*+1*zn*+<sup>1</sup> + . . . ., *z* ∈ *U*.

A function with beautiful applications in defining operators is the fractional integral of order *λ* given as:

**Definition 1** ([37])**.** *The fractional integral of order λ (λ* > 0*) is defined for a function f by*

$$D\_z^{-\lambda}f(z) = \frac{1}{\Gamma(\lambda)} \int\_0^z \frac{f(t)}{(z-t)^{1-\lambda}}dt\lambda$$

*where f is an analytic function in a simply connected region of the z-plane containing the origin, and the multiplicity of* (*z* − *t*)*<sup>λ</sup>*−<sup>1</sup> *is removed by requiring* log(*z* − *t*) *to be real, when* (*z* − *t*) > 0.

The definitions of the notions used in the present investigation are next recalled. Confluent (or Kummer) hypergeometric function is defined as:

**Definition 2** ([10] p. 5)**.** *Let a and c be complex numbers with c* = 0, −1, −2, . . . *and consider*

$$\Phi(a,c;z) = \,\_1F\_1(a,c;z) = 1 + \frac{a}{c}\frac{z}{1!} + \frac{a(a+1)}{c(c+1)}\frac{z^2}{2!} + \dots, z \in \mathcal{U}.\tag{1}$$

*This function is called confluent (Kummer) hypergeometric function, is analytic in* C *and satisfies Kummer's differential equation*

$$zw''(z) + (c-z)w'(z) - aw(z) = 0.$$

The operator introduced in [36] using the fractional integral of confluent hypergeometric function is given in the following definition:

**Definition 3** ([36])**.** *Let a and c be complex numbers with c* = 0, −1, −2, ... *and λ* > 0. *We define the fractional integral of confluent hypergeometric function*

$$D\_z^{-\lambda} \phi(a, c; z) = \frac{1}{\Gamma(\lambda)} \int\_0^z \frac{\phi(a, c; t)}{\left(z - t\right)^{1 - \lambda}} dt = \tag{2}$$

$$\frac{1}{\Gamma(\lambda)} \frac{\Gamma(c)}{\Gamma(a)} \sum\_{k=0}^\infty \frac{\Gamma(a + k)}{\Gamma(c + k)\Gamma(k + 1)} \int\_0^z \frac{t^k}{\left(z - t\right)^{1 - \lambda}} dt.$$

**Remark 1** ([36])**.** *The fractional integral of confluent hypergeometric function can be written*

$$D\_z^{-\lambda} \phi(a, c; z) = \frac{\Gamma(c)}{\Gamma(a)} \sum\_{k=0}^{\infty} \frac{\Gamma(a+k)}{\Gamma(c+k)\Gamma(\lambda+k+1)} z^{k+\lambda},\tag{3}$$

.

*after a simple calculation. Evidently D*−*<sup>λ</sup> z φ*(*<sup>a</sup>*, *c*; *z*) ∈ H[0, *<sup>λ</sup>*].

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

**Definition 4** ([38])**.** *A pair* (*<sup>A</sup>*, *FA*)*, where FA* : *X* → [0, 1] *and A* = {*x* ∈ *X* : 0 < *FA*(*x*) ≤ 1} *is called the fuzzy subset of X. The set A is called the support of the fuzzy set* (*<sup>A</sup>*, *FA*) *and FA is called the membership function of the fuzzy set* (*<sup>A</sup>*, *FA*)*. One can also denote A* = supp(*<sup>A</sup>*, *FA*)*.*

**Remark 2** ([38])**.** *If A* ⊂ *X, then FA*(*x*) = 1*, if x* ∈ *A* 0*, if x* ∈/ *A*

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

*The empty set* ∅ ⊂ *X is characterized by <sup>F</sup>*∅(*x*) = 0*, x* ∈ *X, and the total set X is characterized by FX*(*x*) = 1*, x* ∈ *X.*

**Definition 5** ([4])**.** *Let D* ⊂ C*, z*0 ∈ *D be a fixed point and let the functions f* , *g* ∈ H(*D*)*. The function f is said to be fuzzy subordinate to g and write f* ≺F *g or f*(*z*) ≺F *g*(*z*)*, if are satisfied the conditions:*

*(1) f*(*<sup>z</sup>*0) = *g*(*<sup>z</sup>*0), *(2) Ff*(*D*) *f*(*z*) ≤ *Fg*(*D*)*g*(*z*)*, z* ∈ *D*.

**Definition 6** ([8] Definition 2.2)**.** *Let ψ* : C<sup>3</sup> × *U* → C *and h univalent in U, with ψ*(*<sup>a</sup>*, 0; 0) = *h*(0) = *a. If p is analytic in U, with p*(0) = *a and satisfies the (second-order) fuzzy differential subordination*

$$F\_{\psi(\mathbb{C}^3 \times \mathcal{U})} \psi(p(z), zp'(z), z^2 p''(z); z) \le F\_{\hbar(\mathcal{U})} \hbar(z), \quad z \in \mathcal{U},\tag{4}$$

*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 Fp*(*U*) *p*(*z*) ≤ *Fq*(*U*)*q*(*z*)*, z* ∈ *U, for all p satisfying (4). A fuzzy dominant q that satisfies Fq*(*U*)*q*˜(*z*) ≤ *Fq*(*U*)*q*(*z*)*, z* ∈ *U, for all fuzzy dominants q of (4) is said to be the fuzzy best dominant of (4).*

**Definition 7** ([11])**.** *Let ϕ* : C<sup>3</sup> × *U* → C *and let h be analytic in U. If p and ϕ*(*p*(*z*), *zp*(*z*), *z*2 *p*(*z*); *z*) *are univalent in U and satisfy the (second-order) fuzzy differential superordination*

$$F\_{\hbar(\mathsf{U})}h(z) \le F\_{q(\mathsf{C}^3 \times \mathsf{U})}q(p(z), zp'(z), z^2p''(z); z), \quad z \in \mathsf{U},\tag{5}$$

*i.e.,*

$$h(z) \prec\_{\mathcal{F}} \varphi(p(z), zp'(z), z^2p''(z); z), \quad z \in \mathcal{U},$$

*then p is called a fuzzy solution of the fuzzy differential superordination. An analytic function q is called fuzzy subordinant of the fuzzy differential superordination, or more simply a fuzzy subordination if*

$$F\_{\mathfrak{q}(\mathcal{U})}q(z) \le F\_{\mathfrak{p}(\mathcal{U})}p(z), \ z \in \mathcal{U}\_{\prime}$$

*for all p satisfying (5). A univalent fuzzy subordination q that satisfies Fq*(*U*)*<sup>q</sup>* ≤ *Fq*(*U*)*q for all fuzzy subordinate q of (5) is said to be the fuzzy best subordinate of (5). Please note that the fuzzy best subordinant is unique to a relation of U.*

The purpose of this paper is to obtain several fuzzy differential subordination and superordination results, using the following known results.

**Definition 8** ([8])**.** *Denote by Q the set of all functions f that are analytic and injective on <sup>U</sup>*\*E*(*f*)*, where E*(*f*) = {*ζ* ∈ *∂U* : lim *z*<sup>→</sup>*ζf*(*z*) = <sup>∞</sup>}, *and are such that f* (*ζ*) = 0 *for ζ* ∈ *∂U*\*E*(*f*)*.*

**Lemma 1** ([8])**.** *Let the function q be univalent in the unit disc U and θ and φ be analytic in a domain D containing q*(*U*) *with φ*(*w*) = 0 *when w* ∈ *q*(*U*)*. Set Q*(*z*) = *zq*(*z*)*φ*(*q*(*z*)) *and h*(*z*) = *<sup>θ</sup>*(*q*(*z*)) + *Q*(*z*)*. Suppose that*

*1. Q is starlike univalent in U and* 

*2. Rezh*(*z*) *Q*(*z*)> 0 *for z* ∈ *U.*

*If p is analytic with p*(0) = *q*(0)*, p*(*U*) ⊆ *D and*

$$F\_{p(\mathcal{U})}\theta(p(z)) + zp'(z)\phi(p(z)) \le F\_{h(\mathcal{U})}\theta(q(z)) + zq'(z)\phi(q(z)),$$

*then*

$$F\_{p(\mathcal{U})}p(z) \le F\_{q(\mathcal{U})}q(z)$$

*and q is the fuzzy best dominant.*

**Lemma 2** ([11])**.** *Let the function q be convex univalent in the open unit disc U and ν and φ be analytic in a domain D containing q*(*U*)*. Suppose that*

1.  $\operatorname{Re}\left(\frac{\operatorname{v}'(q(z))}{\Phi(q(z))}\right) > 0$  for  $z \in \mathcal{U}$  and  $\\2$ .  $\psi(z) = zq'(z)\phi(q(z))$  is starlike univalent in  $\mathcal{U}$ .  $\operatorname{If}\ p(z) \in \mathcal{H}[q(0), 1] \cap Q$ , with  $p(\mathcal{U}) \subseteq \mathcal{D}$  and  $\nu(p(z)) + zp'(z)\phi(p(z))$  is univalent in  $\mathcal{U}$  and

$$F\_{q(L)}\nu(q(z)) + zq'(z)\phi(q(z)) \le F\_{p(L)}\nu(p(z)) + zp'(z)\phi(p(z)),$$

*then*

$$F\_{q(\mathcal{U})}q(z) \le F\_{p(\mathcal{U})}p(z)$$

*and q is the fuzzy best subordinant.*
