Fuzzy Differential Sandwich Theorems Involving the Fractional Integral of Confluent Hypergeometric Function

Abstract

The operator defined as the fractional integral of confluent hypergeometric function was introduced and studied in previously written papers in view of the classical theory of differential subordination. In this paper, the same operator is studied using concepts from the theory of fuzzy differential subordination and superordination. The original theorems contain fuzzy differential subordinations and superordinations for which the fuzzy best dominant and fuzzy best subordinant are given, respectively. Interesting corollaries are obtained for particular choices of the functions acting as fuzzy best dominant and fuzzy best subordinant. A nice sandwich-type theorem is stated combining the results given in two theorems proven in this paper using the two dual theories of fuzzy differential subordination and fuzzy differential superordination.

1. Introduction

The concept of fuzzy differential subordination and its dual, and the concept of fuzzy differential superordination were introduced in the last decade as a result of a trend adapting the notion of fuzzy set to different topics of research. Even if the notion of fuzzy sets did not look promising when it was first introduced by Lotfi A. Zadeh in his paper published in 1965 [1], in the recent years it became part of many branches of science and scientific research. Mathematical sciences also aimed at introducing and using fuzzifications of the already established classical theories in different fields of research. The review paper published in 2017 [2] shows some parts of the history of the fuzzy set notion and how Zadeh’s new concept has revolutionized soft computing and artificial intelligence as well as other fields of science. Another review paper published as part of a Special Issue dedicated to celebrating the centenary of Zadeh’s birth [3] shows other aspects from the development process of fuzzy logic based on the notion of fuzzy sets.

Fuzzy sets theory was connected to geometric function theory in 2011 when the notion of fuzzy subordination was introduced [4] having been inspired by the theory of differential subordination initiated by Miller and Mocanu in 1978 [5] and 1981 [6]. The core of the theory of differential subordination was gradually adapted to fuzzy set notions in the following years [7,8,9] following the main lines of research as they can be found in the monograph published in 2000 [10]. The dual notion of fuzzy differential superordination was introduced in 2017 [11]. Obtaining fuzzy subordination and superordination results involving operators was a topic approached early in the study of fuzzy subordinations [12] and continued to be inspiring for researchers over the following years with the addition of the notion of fuzzy differential superordination as can be seen in [13,14,15,16,17,18,19,20,21] to give only some examples of published papers, although there are a lot more. The topic is still exciting for the imagination of researchers, with many papers being published in the last two years and these are just some examples [22,23,24,25,26].

The study presented in this paper is within a general context of geometric function 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 $f\left(z\right)=a+a_n{z}^n+a_{n+1}{z}^{n+1}+....$, $z\in U$.

A function with beautiful applications in defining operators is the fractional integral of order $\lambda$ given as:

Definition 1. 

([27]) The fractional integral of order λ ($\lambda >0$) is defined for a function f by

$$D_z^{-\lambda }f\left(z\right)=\frac{1}{\Gamma \left(\lambda \right)}{\int }_0^z\frac{f\left(t\right)}{{\left(z-t\right)}^{1-\lambda }}dt,$$

where f is an analytic function in a simply connected region of the z-plane containing the origin, and the multiplicity of ${\left(z-t\right)}^{\lambda -1}$ is removed by requiring $log\left(z-t\right)$ to be real, when $\left(z-t\right)>0.$

An impressive number of research papers have been published in recent years as a result of studies involving fractional integral and hypergeometric function. Only a few are listed at references [28,29,30,31,32,33,34,35].

The definitions of the notions used in the present investigation are recalled next.

Confluent (or Kummer) hypergeometric function is defined as:

Definition 2. 

([10], p. 5) Let m and n be complex numbers with $n\ne 0,-1,-2,\dots$ and consider

$$\varphi \left(m,n;z\right){=}_1{F}_1\left(m,n;z\right)=1+\frac{m}{n}\frac{z}{1!}+\frac{m\left(m+1\right)}{n\left(n+1\right)}\frac{z^2}{2!}+\dots ,z\in U.$$

(1)

This function is called confluent (Kummer) hypergeometric function, is analytic in $\mathbb{C}$ and satisfies Kummer’s differential equation

$$z{w}^{″}\left(z\right) + \left(n-z\right)w^{\prime }\left(z\right) - mw\left(z\right) =0.$$

The operator introduced in [35] using the fractional integral of confluent hypergeometric function is given in the following definition:

Definition 3. 

([35]) Let $m,$$n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ We define the fractional integral of confluent hypergeometric function

$$D_z^{-\lambda }\varphi \left(m,n;z\right)=\frac{1}{\Gamma \left(\lambda \right)}{\int }_0^z\frac{\varphi \left(m,n;t\right)}{{\left(z-t\right)}^{1-\lambda }}dt=$$

(2)

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

Remark 1. 

([35]) The fractional integral of confluent hypergeometric function can be written

$$D_z^{-\lambda }\varphi \left(m,n;z\right)=\frac{\Gamma \left(n\right)}{\Gamma \left(m\right)}\sum _{k=0}^{+\infty }\frac{\Gamma \left(m+k\right)}{\Gamma \left(n+k\right)\Gamma \left(\lambda +k+1\right)}{z}^{k+\lambda },$$

(3)

after a simple calculation. Evidently $D_z^{-\lambda }\varphi \left(m,n;z\right)\in \mathcal{H}\left[0,\lambda \right].$

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

Definition 4. 

([36]) 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 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 2. 

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

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 5. 

([4]) 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 fuzzy subordinate to g and write $f{\prec }_{\mathcal{F}}g$ or $f\left(z\right){\prec }_{\mathcal{F}}g\left(z\right)$, if they satisfy the conditions:

(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 6. 

([8], 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 (second-order) fuzzy differential subordination

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

(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 $F_{p\left(U\right)}p\left(z\right)\le F_{q\left(U\right)}q\left(z\right)$, $z\in U$, for all p satisfying (4). 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 (4) is said to be the fuzzy best dominant of (4).

Definition 7. 

([11]) Let $\phi :{\mathbb{C}}^3\times U\to \mathbb{C}$ and let h be analytic in U. If p and $\phi (p\left(z\right),z{p}^{\prime }\left(z\right),$$z^2{p}^{″}\left(z\right);z)$ are univalent in U and satisfy the (second-order) fuzzy differential superordination

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

(5)

i.e.,

$$h\left(z\right){\prec }_{\mathcal{F}}\phi (p\left(z\right),z{p}^{\prime }\left(z\right),z^2{p}^{″}\left(z\right);z),z\in U,$$

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

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

for all p satisfying (5). A univalent fuzzy subordination $\tilde{q}$ that satisfies $F_{q\left(U\right)}q\le F_{q\left(U\right)}\tilde{q}$ 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 up to a relation of U.

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

Definition 8. 

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

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 $q\left(U\right)$ 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 and

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.

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 $q\left(U\right)$. Suppose that

1. $Re\left(\frac{{\nu }^{\prime }\left(q\left(z\right)\right)}{\varphi \left(q\left(z\right)\right)}\right)>0$ for $z\in U$ and

2. $\psi \left(z\right) = z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right)$ is starlike univalent in U.

If $p\left(z\right) \in \mathcal{H}\left[q\left(0\right),1\right]\cap Q$, with $p\left(U\right) \subseteq D$ and $\nu \left(p\left(z\right)\right) + z{p}^{\prime }\left(z\right)\varphi \left(p\left(z\right)\right)$ is univalent in U and

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

then

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

and q is the fuzzy best subordinant.

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, which are some inequalities, the study of special functions, given their symmetry properties, could provide interesting results. Studies on the symmetry properties for different types of functions associated with the concept of quantum computing could also be investigated in a future paper.

2. Main Results

The first fuzzy subordination result obtained using the operator given by (2) is the following theorem:

Theorem 1.

Let the function q be analytic and univalent in U such that $q\left(z\right) \ne 0$ and $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left(U\right)$, for all $z\in U$, where m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Suppose that $\frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}$ is univalent starlike in U. Let

$$Re\left(z\frac{q^{″}\left(z\right)}{q\left(z\right)}-z\frac{q^{\prime }\left(z\right)}{q\left(z\right)} + \frac{2\mu }{\beta }{q}^2\left(z\right) + \frac{\xi }{\beta }q\left(z\right) + 1\right) > 0,$$

(6)

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0$, $z\in U$ and

$${\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right):=\beta \frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{″}}{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}+\mu {\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)}^2+$$

(7)

$$\left(\xi -\beta \right)\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}+\beta +\alpha .$$

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

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

(8)

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$ then

$$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}q\left(z\right),$$

(9)

and the fuzzy best dominant is the function q.

Proof. 

Define $p\left(z\right):=\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$, $z\in U$, $z\ne 0$. Differentiating it we obtain $p^{\prime }\left(z\right)=\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}+z\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{″}}{D_z^{-\lambda }\varphi \left(m,n;z\right)}-z{\left(\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)}^2$ and

$$\frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}=z\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{″}}{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}-z\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}+1.$$

(10)

Considering $\theta \left(w\right):=\mu w^2+\xi w+\alpha$ and $\varphi \left(w\right):=\frac{\beta }{w}$, it is evident that $\theta$ is analytic in $\mathbb{C}$, $\varphi$ is analytic in $\mathbb{C}\backslash \left\{0\right\}$ and that $\varphi \left(w\right) \ne 0,$ $w\in \mathbb{C}\backslash \left\{0\right\}.$

By setting $Q\left(z\right) = z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right) = \beta \frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}$ and $h\left(z\right) =$$Q\left(z\right)+\theta \left(q\left(z\right)\right) = \beta \frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}\mu {\left(q\left(z\right)\right)}^2$$+\xi q\left(z\right) + \alpha$, we deduce that $Q\left(z\right)$ is univalent starlike in U.

Differentiating we obtain $h^{\prime }\left(z\right) =\beta z\frac{q^{″}\left(z\right)}{q\left(z\right)}-\beta z{\left(\frac{q^{\prime }\left(z\right)}{q\left(z\right)}\right)}^2+\beta \frac{q^{\prime }\left(z\right)}{q\left(z\right)} + \xi q^{\prime }\left(z\right) + 2\mu q\left(z\right)q^{\prime }\left(z\right)$ and $\frac{z{h}^{\prime }\left(z\right)}{Q\left(z\right)}=z\frac{q^{″}\left(z\right)}{q\left(z\right)}-z\frac{q^{\prime }\left(z\right)}{q\left(z\right)}+\frac{2\mu }{\beta }{q}^2\left(z\right)+\frac{\xi }{\beta }q\left(z\right) + 1.$

We deduce that $Re\left(\frac{z{h}^{\prime }\left(z\right)}{Q\left(z\right)}\right) = Re\left(z\frac{q^{″}\left(z\right)}{q\left(z\right)}-z\frac{q^{\prime }\left(z\right)}{q\left(z\right)}+\frac{2\mu }{\beta }{q}^2\left(z\right)+\frac{\xi }{\beta }q\left(z\right) + 1\right)$$>0$.

Using (10), we obtain $\beta \frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}+\mu {\left(p\left(z\right)\right)}^2+\xi p\left(z\right) + \alpha =$$\beta \frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{″}}{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}\mu {\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)}^2+ \left(\xi -\beta \right)\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}+\alpha +\beta .$

Using (8), we have $F_{p\left(U\right)}\left(\beta \frac{z{p}^{\prime }\left(z\right)}{p\left(z\right)}+\mu {\left(p\left(z\right)\right)}^2+\xi p\left(z\right) + \alpha \right)\le$

$F_{q\left(U\right)}\left(\beta \frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}+\mu {\left(q\left(z\right)\right)}^2+\xi q\left(z\right) + \alpha \right).$

By an application of Lemma 1, we obtain $F_{p\left(U\right)}p\left(z\right) \le F_{q\left(U\right)}q\left(z\right)$, $z\in U,$ i.e.,

$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}q\left(z\right)$, $z\in U$ and q is the fuzzy best dominant. □

Corollary 1.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Assume that (6) holds. If

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

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

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$ $-1\le N<M\le 1$, where ${\psi }_{\lambda }^{m,n}$ is defined in (7), then

$$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}\frac{1+Mz}{1+Nz},$$

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

Proof. 

For $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $-1\le N<M\le 1,$ Theorem 1 give the corollary. □

Corollary 2.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Assume that (6) holds. If

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

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

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $0<\gamma \le 1$, $\beta \ne 0,$ where ${\psi }_{\lambda }^{m,n}$ is defined in (7), then

$$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right) \le F_{q\left(U\right)}{\left(\frac{1+z}{1-z}\right)}^{\gamma },$$

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

Proof. 

Theorem 1 gives Corollary for $q\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\gamma }$, $0<\gamma \le 1.$ □

Theorem 2.

Let q be analytic and univalent in U such that $q\left(z\right)\ne 0$ and $\frac{z{q}^{\prime }\left(z\right)}{q\left(z\right)}$ be univalent starlike in U. Assume that

$$Re\left(\frac{2\mu }{\beta }{q}^2\left(z\right)q^{\prime }\left(z\right)+\frac{\xi }{\beta }q\left(z\right)q^{\prime }\left(z\right)\right) > 0,\mathit{for}\mu ,\beta ,\xi \in \mathbb{C},\beta \ne 0.$$

(11)

If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right]\cap Q$ and ${\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right)$ is univalent in U, where ${\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right)$ is as defined in (7) and $\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$, then

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

(12)

implies

$$F_{q\left(U\right)}q\left(z\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)},z\in U,$$

(13)

and the fuzzy best subordinant is the function q.

Proof. 

Define $p\left(z\right):=\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$, $z\in U$, $z\ne 0$.

Considering $\nu \left(w\right):=\mu w^2+\xi w+\alpha$ and $\varphi \left(w\right):=\frac{\beta }{w}$ it is evident that $\nu$ is analytic in $\mathbb{C}$, $\varphi$ is analytic in $\mathbb{C}\backslash \left\{0\right\}$ and that $\varphi \left(w\right) \ne 0$, $w\in \mathbb{C}\backslash \left\{0\right\}.$

In this conditions $\frac{{\nu }^{\prime }\left(q\left(z\right)\right)}{\varphi \left(q\left(z\right)\right)}=\frac{q^{\prime }\left(z\right)q\left(z\right)\left[\xi + 2\mu q\left(z\right)\right]}{\beta }$, which imply $Re\left(\frac{{\nu }^{\prime }\left(q\left(z\right)\right)}{\varphi \left(q\left(z\right)\right)}\right) =$ $Re\left(\frac{2\mu }{\beta }{q}^2\left(z\right)q^{\prime }\left(z\right) + \frac{\xi }{\beta }q\left(z\right)q^{\prime }\left(z\right)\right) > 0$, for $\xi ,\beta ,\mu \in \mathbb{C},$$\beta \ne 0.$

We obtain

$$F_{q\left(U\right)}\left(\frac{\beta z{q}^{\prime }\left(z\right)}{q\left(z\right)}+\mu {\left(q\left(z\right)\right)}^2+\xi q\left(z\right) + \alpha \right)\le$$

$$F_{p\left(U\right)}\left(\frac{\beta z{p}^{\prime }\left(z\right)}{p\left(z\right)}+\mu {\left(p\left(z\right)\right)}^2+\xi p\left(z\right) + \alpha \right).$$

Applying Lemma 2, we obtain

$$F_{q\left(U\right)}q\left(z\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),z\in U,$$

and q is the fuzzy best subordinant. □

Corollary 3.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Assume that (11) holds. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right] \cap Q$ and

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

$$F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right),$$

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$-1\le B<A\le 1$, where ${\psi }_{\lambda }^{m,n}$ is defined in (7), then

$$F_{q\left(U\right)}\left(\frac{1+Mz}{1+Nz}\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),$$

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

Proof. 

For $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $-1\le N<M\le 1$ in Theorem 2 we obtain the corollary. □

Corollary 4.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Assume that (11) holds. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right] \cap Q$ and

$$F_{q\left(U\right)}\left(\frac{2\beta \gamma z}{1-z^2}+\mu {\left(\frac{1+z}{1-z}\right)}^{2\gamma }+\xi {\left(\frac{1+z}{1-z}\right)}^{\gamma }+\alpha \right)\le$$

$$F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right),$$

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$0<\gamma \le 1$, where ${\psi }_{\lambda }^{m,n}$ is defined in (7), then

$$F_{q\left(U\right)}{\left(\frac{1+z}{1-z}\right)}^{\gamma }\le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),$$

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

Proof. 

For $q\left(z\right) = {\left(\frac{1+z}{1-z}\right)}^{\gamma }$, $0<\gamma \le 1$ in Theorem 2 we obtain the corollary. □

Theorems 1 and 2 combined give the following sandwich theorem.

Theorem 3.

Let $q_1$ and $q_2$ be analytic and univalent in U such that $q_1\left(z\right) \ne 0$ and $q_2\left(z\right) \ne 0$, for all $z\in U$, with $\frac{z{q}_1^{\prime }\left(z\right)}{q_1\left(z\right)}$ and $\frac{z{q}_2^{\prime }\left(z\right)}{q_2\left(z\right)}$ being univalent starlike. Assume that $q_1$ satisfies (6) and $q_2$ satisfies (11). If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right] \cap Q$ and ${\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right)$ is as defined in (7) univalent in U, $\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots ,$ then

$$F_{q_1\left(U\right)}\left(\frac{\beta z{q}_1^{\prime }\left(z\right)}{q_1\left(z\right)}+\mu {\left(q_1\left(z\right)\right)}^2+\xi q_1\left(z\right)+\alpha \right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right)$$

$$\le F_{q_2\left(U\right)}\left(\frac{\beta z{q}_2^{\prime }\left(z\right)}{q_2\left(z\right)}+\mu {\left(q_2\left(z\right)\right)}^2+\xi q_2\left(z\right) + \alpha \right),$$

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$ implies

$$F_{q_1\left(U\right)}{q}_1\left(z\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\le F_{q_2\left(U\right)}{q}_2\left(z\right),$$

and $q_1$ and $q_2$ are respectively the fuzzy best subordinant and the fuzzy best dominant.

For $q_1\left(z\right)=\frac{1+M_{1}z}{1+N_{1}z}$, $q_2\left(z\right)=\frac{1+M_{2}z}{1+N_{2}z}$, where $-1\le N_2<N_1<M_1<M_2\le 1$, we obtain the following corollary.

Corollary 5.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Assume that (6) and (11) hold. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right] \cap Q$ and

$$F_{q_1\left(U\right)}\left(\frac{\beta \left(M_1-N_1\right)z}{\left(1+M_{1}z\right)\left(1+N_{1}z\right)}+\mu {\left(\frac{1+M_{1}z}{1+N_{1}z}\right)}^2+\xi \frac{1+M_{1}z}{1+N_{1}z}+\alpha \right)$$

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

$$F_{q_2\left(U\right)}\left(\frac{\beta \left(M_2-N_2\right)z}{\left(1+M_{2}z\right)\left(1+N_{2}z\right)}+\mu {\left(\frac{1+M_{2}z}{1+N_{2}z}\right)}^2+\xi \frac{1+M_{2}z}{1+N_{2}z}+\alpha \right),$$

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$-1\le N_2\le N_1<M_1\le M\le 1$, where ${\psi }_{\lambda }^{m,n}$ is defined in (7), then

$$F_{q_1\left(U\right)}\left(\frac{1+M_{1}z}{1+N_{1}z}\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q_2\left(U\right)}\left(\frac{1+M_{2}z}{1+N_{2}z}\right),$$

hence $\frac{1+M_{1}z}{1+N_{1}z}$ and $\frac{1+M_{2}z}{1+N_{2}z}$ are the fuzzy best subordinant and the fuzzy best dominant, respectively.

For $q_1\left(z\right) ={\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}$, $q_2\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}$, where $0<{\gamma }_1<{\gamma }_2\le 1$, we obtain the following corollary.

Corollary 6.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Suppose that (6) and (11) hold. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$$\in \mathcal{H}\left[q\left(0\right),1\right] \cap Q$ and

$$F_{q_1\left(U\right)}\left(\frac{2\beta {\gamma }_{1}z}{1-z^2}+\mu {\left(\frac{1+z}{1-z}\right)}^{2{\gamma }_1}+\xi {\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}+\alpha \right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\mu ,\xi ,\alpha ;z\right)$$

$$\le F_{q_2\left(U\right)}\left(\frac{2\beta {\gamma }_{2}z}{1-z^2}+\mu {\left(\frac{1+z}{1-z}\right)}^{2{\gamma }_2}+\xi {\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}+\alpha \right),$$

for $\mu ,\beta ,\xi ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$0<{\gamma }_1<{\gamma }_2\le 1$, where ${\psi }_{\lambda }^{m,n}$ is defined in (7), then

$$F_{q_1\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right) \le F_{q_2\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}\right),$$

hence ${\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}$ and ${\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}$ are the fuzzy best subordinant and the fuzzy best dominant, respectively.

We also have

Theorem 4.

Let $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left(U\right),$$z\in U$, where $\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots ,$ and let the function $q\left(z\right)$ be convex and univalent in U such that $q\left(0\right)=\lambda$, $z\in U$. Assume that

$$Re\left(z\frac{q^{″}\left(z\right)}{q^{\prime }\left(z\right)}+\frac{\alpha +\beta }{\beta }\right)>0,$$

(14)

for $\alpha ,\beta \in \mathbb{C}$, β$\ne 0$, $z\in U,$ and

$${\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right):=\beta \frac{z^2{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{″}}{D_z^{-\lambda }\varphi \left(m,c;z\right)}-\beta {\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)}^2$$

(15)

$$+\left(\alpha +\beta \right)\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}.$$

If q satisfies the following fuzzy differential subordination

$$F_{{\psi }_{\lambda }^{m,n}\left(U\right)}\left({\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)\right) \le F_{q\left(z\right)}\left(\beta z{q}^{\prime }\left(z\right) + \alpha q\left(z\right)\right),$$

(16)

for$\beta ,\alpha \in \mathbb{C}$, β$\ne 0,$$z\in U,$ then

$$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}\left(q\left(z\right)\right),z\in U,$$

(17)

and the fuzzy best dominant is the function q.

Proof. 

Define $p\left(z\right):=\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$, $z\in U$, $z\ne 0$. The function p is analytic in U and $p\left(0\right) = \lambda .$

Differentiating it we obtain $p^{\prime }\left(z\right) = z\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{″}}{D_z^{-\lambda }\varphi \left(m,n;z\right)}-z{\left(\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)}^2+\frac{{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$ and

$$z{p}^{\prime }\left(z\right)=\frac{z^2{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime \prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}-{\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)}^2+\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}.$$

(18)

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

Considering $Q\left(z\right) = z{q}^{\prime }\left(z\right)\varphi \left(q\left(z\right)\right)=\beta z{q}^{\prime }\left(z\right),$ and $h\left(z\right)=Q\left(z\right) + \theta \left(q\left(z\right)\right) = \beta z{q}^{\prime }\left(z\right) + \alpha q\left(z\right),$ we deduce that $Q\left(z\right)$ is univalent starlike in $U.$

We obtain $Re\left(\frac{z{h}^{\prime }\left(z\right)}{Q\left(z\right)}\right) =Re\left(z\frac{q^{″}\left(z\right)}{q^{\prime }\left(z\right)}+\frac{\alpha +\beta }{\beta }\right) >0$ and (18) give the following relation

$$\beta z{p}^{\prime }\left(z\right) + \alpha p\left(z\right) = \beta \frac{z^2{\left(D_z^{-\lambda }\varphi \left(a,c;z\right)\right)}^{″}}{D_z^{-\lambda }\varphi \left(a,c;z\right)}-\beta {\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(a,c;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(a,c;z\right)}\right)}^2+\left(\alpha +\beta \right)\frac{z{\left(D_z^{-\lambda }\varphi \left(a,c;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(a,c;z\right)},$$

which imply $F_{p\left(U\right)}\left(\beta z{p}^{\prime }\left(z\right)+\alpha p\left(z\right)\right)\le F_{q\left(U\right)}\left(\beta z{q}^{\prime }\left(z\right)+\alpha q\left(z\right)\right).$

By an application of Lemma 1 we obtain $F_{p\left(U\right)}p\left(z\right)\le F_{q\left(U\right)}q\left(z\right)$, $z\in U,$ i.e.,

$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}q\left(z\right)$, $z\in U,$ and q is the fuzzy best dominant. □

Corollary 7.

Let $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $z\in U,$$-1\le N<M\le 1,$$\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots .$ Assume that (14) holds. If

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

for $\beta ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$-1\le N<M\le 1$, where ${\psi }_{\lambda }^{m,n}$ is defined in (15), then

$$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}\left(\frac{1+Mz}{1+Nz}\right),$$

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

Proof. 

Consider in Theorem 4 $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $-1\le N<M\le 1.$ □

Corollary 8.

Let $q\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\gamma },$$\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots .$ Suppose that (14) holds. If

$$F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)\le F_{q\left(U\right)}\left(\frac{2\beta \gamma z}{1-z^2}{\left(\frac{1+z}{1-z}\right)}^{\gamma }+\alpha {\left(\frac{1+z}{1-z}\right)}^{\gamma }\right),$$

for $\beta ,\alpha \in \mathbb{C}$, $0<\gamma \le 1$, $\beta \ne 0,$ where ${\psi }_{\lambda }^{m,n}$ is introduced in (15), then

$$F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{\gamma }\right),$$

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

Proof. 

Using Theorem 4 for $q\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\gamma }$, $0<\gamma \le 1$, we obtain the corollary. □

Theorem 5.

Let q be convex and univalent in U such that $q\left(0\right)=\lambda$, where $\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$. Suppose that

$$Re\left(\frac{\alpha }{\beta }{q}^{\prime }\left(z\right)\right)>0,for\alpha ,\beta \in \mathbb{C},\beta \ne 0.$$

(19)

If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$$\in \mathcal{H}\left[q\left(0\right),1\right] \cap Q$ and ${\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)$ is univalent in U, where ${\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)$ is as defined in (15), then

$$F_{q\left(U\right)}\left(\beta z{q}^{\prime }\left(z\right)+\alpha q\left(z\right)\right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)$$

(20)

implies

$$F_{q\left(U\right)}q\left(z\right) \le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),z\in U,$$

(21)

and the best fuzzy subordinant is the function q.

Proof. 

Define $p\left(z\right):=\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}$, $z\in U$, $z\ne 0$. The function p is analytic in U and $p\left(0\right)=\lambda .$

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

In these conditions $\frac{{\nu }^{\prime }\left(q\left(z\right)\right)}{\varphi \left(q\left(z\right)\right)}=\frac{\alpha }{\beta }{q}^{\prime }\left(z\right)$, which imply $Re\left(\frac{{\nu }^{\prime }\left(q\left(z\right)\right)}{\varphi \left(q\left(z\right)\right)}\right)=Re\left(\frac{\alpha }{\beta }{q}^{\prime }\left(z\right)\right)>0$, for $\alpha ,\beta \in \mathbb{C},$$\beta \ne 0.$

Relation (20) can be written

$$F_{q\left(U\right)}\left(\beta z{q}^{\prime }\left(z\right)+\alpha q\left(z\right)\right)\le F_{p\left(U\right)}\left(\beta z{p}^{\prime }\left(z\right)+\alpha p\left(z\right)\right),z\in U.$$

Applying Lemma 2, we obtain

$$F_{q\left(U\right)}q\left(z\right)\le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),z\in U,$$

and q is the fuzzy best subordinant. □

Corollary 9.

Let $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $-1\le N<M\le 1,$$z\in U,$$\lambda >0,$ m, $n\in \mathbb{C}$$n\ne 0,-1,-2,\dots .$ Suppose that (19) holds. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right]\cap Q,$ and

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

for $\beta ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$-1\le N<M\le 1$, where ${\psi }_{\lambda }^{m,n}$ is introduced in (15), then

$$F_{q\left(U\right)}\left(\frac{1+Mz}{1+Nz}\right)\le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),$$

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

Proof. 

For $q\left(z\right)=\frac{1+Mz}{1+Nz}$, $-1\le N<M\le 1,$ in Theorem 5 we obtain the corollary. □

Corollary 10.

Let $q\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\gamma },$ $\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots .$ Suppose that (19) holds. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right]\cap Q$ and

$$F_{q\left(U\right)}\left(\frac{2\beta \gamma z}{1-z^2}{\left(\frac{1+z}{1-z}\right)}^{\gamma }+\alpha {\left(\frac{1+z}{1-z}\right)}^{\gamma }\right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right),$$

for $\beta ,\alpha \in \mathbb{C}$, $0<\gamma \le 1$, $\beta \ne 0,$ where ${\psi }_{\lambda }^{m,n}$ is introduced in (15), then

$$F_{q\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{\gamma }\right)\le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right),$$

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

Proof. 

Theorem 5 for $q\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\gamma }$, $0<\gamma \le 1,$ gives the corollary. □

Theorems 4 and 5 combined give the following sandwich theorem.

Theorem 6.

Let $q_1$ and $q_2$ be convex and univalent in U such that $q_1\left(z\right)\ne 0$ and $q_2\left(z\right)\ne 0$, for all $z\in U$. Assume that $q_1$ satisfies (14) and $q_2$ satisfies (19). If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right]\cap Q$, and ${\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)$ is as defined in (15) univalent in U, $\lambda >0,$ m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots ,$ then

$$F_{q_1\left(U\right)}\left(\beta z{q}_1^{\prime }\left(z\right)+\alpha q_1\left(z\right)\right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)\le F_{q_2\left(U\right)}\left(\beta z{q}_2^{\prime }\left(z\right)+\alpha q_2\left(z\right)\right),$$

for $\beta ,\alpha \in \mathbb{C},$$\beta \ne 0,$ implies

$$F_{q_1\left(U\right)}{q}_1\left(z\right)\le F_{D_z^{-\lambda }\varphi \left(U\right)}\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\le F_{q_2\left(U\right)}{q}_2\left(z\right),z\in U,$$

and $q_1$ and $q_2$ are respectively the fuzzy best subordinant and the fuzzy best dominant.

For $q_1\left(z\right)=\frac{1+M_{1}z}{1+N_{1}z}$, $q_2\left(z\right)=\frac{1+M_{2}z}{1+N_{2}z}$, where $-1\le N_2<N_1<M_1<M_2\le 1$, we obtain the following corollary.

Corollary 11.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Suppose that (14) and (19) hold for $q_1\left(z\right)=\frac{1+M_{1}z}{1+N_{1}z}$ and $q_2\left(z\right)=\frac{1+M_{2}z}{1+N_{2}z}$, respectively. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right]\cap Q$ and

$$F_{q_1\left(U\right)}\left(\frac{\beta \left(M_1-N_1\right)z}{{\left(1+N_{1}z\right)}^2}+\alpha \frac{1+M_{1}z}{1+N_{1}z}+\right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)$$

$$\le F_{q_2\left(U\right)}\left(\frac{\beta \left(M_2-N_2\right)z}{{\left(1+N_{2}z\right)}^2}+\alpha \frac{1+M_{2}z}{1+N_{2}z}\right),z\in U,$$

for $\beta ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$-1\le N_2\le N_1<M_1\le M_2\le 1$, where ${\psi }_{\lambda }^{m,n}$ is introduced in (15), then

$$F_{q_1\left(U\right)}\left(\frac{1+M_{1}z}{1+N_{1}z}\right)\le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q_2\left(U\right)}\left(\frac{1+M_{2}z}{1+N_{2}z}\right),$$

$z\in U,$ hence $\frac{1+M_{1}z}{1+N_{1}z}$ and $\frac{1+M_{2}z}{1+N_{2}z}$ are the fuzzy best subordinant and the fuzzy best dominant, respectively.

For $q_1\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}$, $q_2\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}$, where $0<{\gamma }_1<{\gamma }_2\le 1$, we obtain the following corollary.

Corollary 12.

Let m, $n\in \mathbb{C}$ with $n\ne 0,-1,-2,\dots$ and $\lambda >0.$ Suppose that (14) and (19) hold for $q_1\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}$ and $q_2\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}$, respectively. If $\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\in \mathcal{H}\left[q\left(0\right),1\right]\cap Q$ and

$$F_{q_1\left(U\right)}\left(\frac{2\beta {\gamma }_{1}z}{1-z^2}{\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}+\alpha {\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}\right)\le F_{{\psi }_{\lambda }^{m,n}\left(U\right)}{\psi }_{\lambda }^{m,n}\left(\beta ,\alpha ;z\right)$$

$$\le F_{q_2\left(U\right)}\left(\frac{2\beta {\gamma }_{2}z}{1-z^2}{\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}+\alpha {\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}\right),z\in U,$$

for $\beta ,\alpha \in \mathbb{C}$, $\beta \ne 0,$$0<{\gamma }_1<{\gamma }_2\le 1$, where ${\psi }_{\lambda }^{m,n}$ is introduced in (15), then

$$F_{q_1\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}\right)\le F_{D_z^{-\lambda }\varphi \left(U\right)}\left(\frac{z{\left(D_z^{-\lambda }\varphi \left(m,n;z\right)\right)}^{\prime }}{D_z^{-\lambda }\varphi \left(m,n;z\right)}\right)\le F_{q_2\left(U\right)}\left({\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}\right),$$

$z\in U,$ hence ${\left(\frac{1+z}{1-z}\right)}^{{\gamma }_1}$ and ${\left(\frac{1+z}{1-z}\right)}^{{\gamma }_2}$ are the fuzzy best subordinant and the fuzzy best dominant, respectively.

3. Discussion

The interesting operator presented in Definition 3 was previously defined and studied related to several aspects of differential subordination theory in [35] as the fractional integral of confluent hypergeometric function. In this paper, the study of the operator is continued using the recently introduced notions of fuzzy differential subordination and fuzzy differential superordination as a result of the preoccupation with adapting the classical notions of differential subordination and superordination to fuzzy sets theory. Fuzzy differential subordinations and fuzzy differential superordinations are presented in the original theorems giving their best fuzzy dominant and best fuzzy subordinant, respectively. Using particular functions, interesting corollaries are presented that could inspire future studies related to the univalence of the operator. A sandwich-type result is obtained in the last theorem combining the results proved using the two dual theories of fuzzy differential subordination and fuzzy differential superordination. Since the operator gives good results in studies done with both theories, it could be used for introducing new fuzzy classes of analytic functions and performing studies on those classes using both theories. Symmetry properties, coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity or close-to-convexity can be studied.