**2. Preliminaries**

The research presented in this paper is done in the general environment known in the theory of differential subordination given in the monograph [25] combined with fuzzy set notions introduced in [4,7].

The unit disc of the complex plane is denoted by *U*. H(*U*) stands for the class of holomorphic functions in *U*. Consider the subclass, A*n* = { *f* ∈ H(*U*) : *f*(*z*) = *z* + *an*+1*zn*+<sup>1</sup> + ··· , *z* ∈ *<sup>U</sup>*}, with A1 = A.

For *a* ∈ C, *n* ∈ N∗ the following subclass of holomorphic functions is obtained: H[*<sup>a</sup>*, *n*] = { *f* ∈ H(*U*) : *f*(*z*) = *a* + *anz<sup>n</sup>* + *an*+1*zn*+<sup>1</sup> + ··· , *z* ∈ *<sup>U</sup>*}, with H0 = H[0, 1]. 

For *α* < 1, let S∗(*α*) = { *f* ∈ A : Re*z f* (*z*) *f*(*z*) > *α*} denote the class of starlike functions of order *α*. For *α* = 0, the class of starlike functions is denoted by S∗.

For *α* < 1, let K(*α*) = { *f* ∈ A :Re *z f* (*z*) *f* (*z*) + 1> *α*} denote the class of convex functions of order *α*. For *α* = 0, the class of convex functions is denoted by K.

The subclass of close-to-convex functions is defined as: C = { *f* ∈ H(*U*) : ∃ *ϕ* ∈ K, Re- *f* (*z*) *ϕ*(*z*) > 0, *z* ∈ *<sup>U</sup>*}.

It is also said that function *f* is close-to-convex with respect to function *ϕ*.

**Definition 1** ([4])**.** *Let D* ⊂ C *and z*0 ∈ *D be a fixed point. We take 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 there exists a function F* : C → [0, 1]*, such that*


**Remark 1.** *(a) Such a function F* : C → [0, 1] *can be considered <sup>F</sup>*(*z*) = |*z*| <sup>1</sup>+|*z*| *, <sup>F</sup>*(*z*) = 1 <sup>1</sup>+|*z*| *, <sup>F</sup>*(*z*)√|*z*|

 = <sup>1</sup>+√|*z*| . *(b) Relation (ii) is equivalent to f*(*D*) ⊂ *g*(*D*).

**Definition 2** ([7], Definition 2.2)**.** *Let ψ* : C<sup>3</sup> × *D* → C*, a* ∈ C, *and let h be univalent in U, with h*(*<sup>z</sup>*0) = *a, g be univalent in D, with g*(*<sup>z</sup>*0) = *a, and p be analytic in D, with p*(*<sup>z</sup>*0) = *a*. *Likewise, ψ*(*p*(*z*), *zp*(*z*), *z*2 *p*(*z*); *z*) *is analytic in D and F* : C → [0, 1]*, <sup>F</sup>*(*z*) = |*z*| <sup>1</sup>+|*z*| *. If p is analytic in D and satisfies the (second-order) fuzzy differential subordination*

$$F\left(\psi(p(z), zp'(z), z^2p''(z); z)\right) \le F(h(z)), \quad z \in \mathcal{U},\tag{1}$$

*i.e., ψ*(*p*(*z*), *zp*(*z*), *z*2 *p*(*z*); *z*) ≺F *h*(*z*)*, or*

$$\frac{|\psi(p(z), zp'(z), z^2p''(z); z)|}{1 + |\psi(p(z), zp'(z), z^2p''(z); z)|} \le \frac{|h(z)|}{1 + |h(z)|}, \ z \in D,\tag{2}$$

*then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of fuzzy solutions of the differential subordination, or more simply, a fuzzy dominant, if* |*p*(*z*)| <sup>1</sup>+|*p*(*z*)| ≤ |*q*(*z*)| <sup>1</sup>+|*q*(*z*)| *, or p*(*z*) ≺F *q*(*z*), *z* ∈ *D, for all p satisfying (1) or (2). A fuzzy* *dominant q that satisfies* |*q*(*z*)| <sup>1</sup>+|*q*(*z*)| ≤ |*q*(*z*)| <sup>1</sup>+|*q*(*z*)| *, or q*(*z*) ≺F *q*(*z*), *z* ∈ *D, for all fuzzy dominants q of (1) or (2) is said to be the fuzzy best dominant of (1) or (2). Note that the fuzzy best dominant is unique up to a rotation in D*.

**Lemma 1** ([25], Theorem 2.2)**.** *Let δ*, *ω* ∈ C, *ω* = 0*, and h be a convex function in D, and F* : C → [0, 1]*, <sup>F</sup>*(*z*) = |*z*| <sup>1</sup>+|*z*| *, z* ∈ *D. We suppose that the Briot–Bouquet differential equation*

$$q(z) + \frac{zq'(z)}{\delta + \omega q(z)} = h(z), \; z \in D, \; q(z\_0) = h(z\_0) = a$$

*has a solution q* ∈ H(*D*)*, which verifies q*(*z*) ≺F *h*(*z*)*, z* ∈ *D, or* |*q*(*z*)| <sup>1</sup>+|*q*(*z*)| ≤ |*h*(*z*)| <sup>1</sup>+|*h*(*z*)| *, z* ∈ *D*.

*If the function p* ∈ H[*<sup>a</sup>*, *<sup>n</sup>*]*, ψ* : C<sup>2</sup> × *D* → C*, ψ*(*p*(*z*), *zp*(*z*)) = *p*(*z*) + *zp*(*z*) *<sup>δ</sup>*+*ωp*(*z*) *is analytic in D, with ψ*(*p*(*<sup>z</sup>*0), *z*0 *p*(*<sup>z</sup>*0)) = *h*(*<sup>z</sup>*0)*, z*0 ∈ *D, then*

$$
\psi\left(p(z), zp'(z)\right) \preccurlyeq \pi \, h(z), \, z \in D,\tag{3}
$$

*or*

$$\frac{|\psi(p(z), zp'(z))|}{1 + |\psi(p(z), zp'(z))|} \le \frac{|h(z)|}{1 + |h(z)|}\tag{4}$$

*implies*

$$p(z) \prec\_{\mathcal{F}} q(z), \text{ or } \frac{|p(z)|}{1+|p(z)|} \le \frac{|q(z)|}{1+|q(z)|}, \ z \in D\_{\star}$$

*and q is the fuzzy best dominant of the fuzzy differential subordination (3) or (4).*

The confluent (or Kummer) hypergeometric function has been investigated connected to univalent functions more intensely starting from 1985 when it was used by L. de Branges in the proof of Bieberbach's conjecture [26]. The applications of hypergeometric functions in univalent function theory is very well pointed out in the review paper, recently published by H.M. Srivastava [27].

**Definition 3** ([25])**.** *Let u and v be complex numbers with v* = 0, −1, −2, ... , *and consider the function defined by*

$$\phi(u,v;z) = \frac{\Gamma(v)}{\Gamma(u)} \sum\_{k=0}^{\infty} \frac{\Gamma(u+k)}{\Gamma(v+k)} \frac{z^k}{k!} = \tag{5}$$
 
$$\left[ \begin{array}{c} u(u+1) \ z^2 \end{array} \right]\_+ \quad \left[ \begin{array}{c} u(u+1) \dots (u+n-1) \ z^n \end{array} \right]\_+$$

$$1 + \frac{u}{v} \frac{z}{1!} + \frac{u(u+1)}{v(v+1)} \frac{z^2}{2!} + \dots + \frac{u(u+1)\dots(u+n-1)}{v(v+1)\dots(v+n-1)} \frac{z^n}{n!} + \dots, z^n$$

*where* (*e*)*k* = <sup>Γ</sup>(*e*+*k*) <sup>Γ</sup>(*e*) = *e*(*e* + <sup>1</sup>)(*e* + <sup>2</sup>)...(*<sup>e</sup>* + *k* − <sup>1</sup>)*, and* (*e*)0 = 1*, called the confluent (or Kummer) hypergeometric function is analytic in* C.

**Remark 2.** *(a) For z* = 0*, φ*(*<sup>u</sup>*, *v*; 0) = 1 *and φ*(*<sup>u</sup>*, *v*; *z*) = 0*, z* ∈ *U*, *(b) For u* = 0*, φ*(*<sup>u</sup>*, *v*; 0) = *uv* = 0.

The operator used for obtaining the original results presented in this paper was obtained using a confluent (or Kummer) hypergeometric function and a general operator studied in 1978 by S.S. Miller, P.T. Mocanu and M.O. Reade [28] by taking specific values for parameters *β*, *γ*, *α*, *δ*:

$$J(f)(z) = \left[\frac{\beta + \gamma}{z^{\gamma}\phi(z)} \int\_0^z f^{\alpha}(t)\,\eta(t)t^{\delta - 1}dt\right]^{\frac{1}{\beta}}.\tag{6}$$

A confluent (or Kummer) hypergeometric function was recently used in many papers for defining new interesting operators as it can be seen in [29–32].

Two more lemmas from differential subordination theory that are necessary in the proofs of the original results are listed next:

**Lemma 2** ([33], Theorem 4.6.3, p. 84)**.** *A necessary and sufficient condition for a function f* ∈ H(*U*) *to be close-to-convex is given by:*

$$\int\_{\theta\_1}^{\theta\_2} \text{Re}\left[1 + \frac{zf''(z)}{f'(z)}\right] d\theta > -\pi, \ z\_0 = r\_0 e^{i\theta\_0}.$$

*for all θ*1, *θ*2 *with* 0 ≤ *θ*1 < *θ*2 < 2*π, r* ∈ (0, <sup>1</sup>).

**Lemma 3** ([25], Theorem Marx–Strohhäcker, p. 9)**.** *If f* ∈ K *then Re z f* (*z*) *f*(*z*) > 12 *, i.e., f* ∈ S∗-12*, for z* ∈ *U*.
