Fuzzy Differential Subordinations Based upon the Mittag-Leffler Type Borel Distribution

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series Bλ,α,β(z) of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function Eα,β(z). Using the above-mentioned operator Bλ,α,β, we also introduce and study a class Mλ,α,βFη of holomorphic and univalent functions in the open unit disk Δ. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function pΨq(z), whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features.


Introduction and Motivation
Our main objective in this article is to investigate several potentially useful results that are based upon second-order fuzzy differential subordinations and their applications in Geometric Function Theory of Complex Analysis and that are intimately connected with the Mittag-Leffler-type Borel distribution series.
The motivation for this investigation was derived from a number of recent works that made use of Borel and other types of probability distributions in the study of such members of the family of holomorphic functions, such as univalent starlike functions and univalent convex functions, which are defined and normalized in the open unit disk in the complex z-plane.
We choose here to mention the works of El-Deeb et al. [1], Murugusundaramoorthy and El-Deeb [2], Srivastava et al. [3], and Wanas and Khuttar [4], in which use was made of the Borel distribution series, involving many different special functions and orthogonal polynomials, in their study of subclasses of normalized holomorphic functions in the open unit disk.
On the other hand, various applications of the concept of fuzzy sets and fuzzy systems in conjunction with the principle of differential subordination between analytic functions can be found in the works of El-Deeb et al. (see [5,6]), Lupaş et al. (see [7][8][9]), Oros and Oros (see [10][11][12]), and Wanas [13]. Some other recent publications that are worth mentioning here include those by Eş [14], Laengle et al. [15], Lupaş [16], Oros [17], and Venter [18]. In particular, the recently-published article by Laengle et al. [15] includes an interesting and useful bibliometric and bibliograhic account of notable developments on Fuzzy Sets and Fuzzy Systems over the past 40 years.
The organization of this paper is as follows. In Section 2, we present the definitions and preliminaries that provide the foundation of our paper. Section 3 includes several lemmas that are needed in proving our main results in Section 4. Some corollaries and consequences of our main results are also deduced in Section 4. In Section 5, we present a number of remarks and observations based upon our work. Finally, in the concluding section (Section 6), some potential directions for related further research are presented.

Definitions and Preliminaries
Let C and N denote the set of complex numbers and the set of positive integers, respectively. For Ω ⊂ C, we denote by H(Ω) the class of holomorphic functions in Ω.
For d ∈ N, we denote by A d the class of functions defined by where ∆ is the open unit disk given by ∆ := {z : z ∈ C and |z| < 1}.
In particular, we write A := A 1 . Finally, we let and we denote by S, S * , and C the classes of functions in A, which are, respectively, univalent, starlike, and convex in ∆, so that, by definition, we have and For our present investigation, we need the following definitions.
Definition 1 (see [19][20][21]). Given two functions f 1 and f 2 , which are analytic in ∆, we say that f 1 is subordinate to f 2 , denoted by provided that a Schwarz function w exisits, which is analytic in ∆ and saisfies the condition given by Moreover, in the case where f 2 is univalent in ∆, then the following equivalence holds true: Remark 1. The widely-applied principle of differential subordination between analytic functions happens to provide an interesting and useful generalization of various inequalities involving complex variables. In fact, the monograph on this subject by Miller and Mocanu [20] (see also [19] for recent developments on differential subordinations and differential superordinations) is a good source to learn about the theories and applications of differential subordinations and differential superordinations.
The following definitions and propositions present the notion of fuzzy differential subordination.
Definition 2 (see [22]). Assume that the set X is non-empty, that is, X = ∅. Then an application F X : X → [0, 1] is called a fuzzy subset of the non-empty set X . More precisely, a pair (B, is said to be a fuzzy subset of X . The set B is referred to as the support of the fuzzy set (B, F B ), written as supp(B, F B ), and the set function set function F B is called the membership function of the fuzzy set (B, F B ).

Remark 2.
Symmetry type properties and symmetry type features are known to be generally present in most works dealing with fuzzy sets and fuzzy systems. Our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetric or asymmetric features.
We now make use of moduli of complex-valued functions in order to introduce and apply the concept of membership functions on the set C of complex numbers given by z = x + iy (x, y ∈ R) and |z| = x 2 + y 2 0 (z ∈ C).
Definition 3 (see [9], p. 120). Let F : C → R + be a function such that Denote by F C (C) = {z : z ∈ C and 0 < |F(z)| 1} =: supp(C, F C ) the fuzzy subset of the set C of complex numbers. We call the following subset: the fuzzy unit disk. It is observed that (C, F C ) is the same as its fuzzy unit disk ∆ F (0, 1).
Proposition 1 (see [9,10]). Each of the following assertions holds true: For f , g ∈ H(Ω), we now use the following notations: and Definition 4 (see [10]). For a given fixed point z 0 ∈ Ω, let f , g ∈ H(Ω). Then we say that f is fuzzy subordinate to g, written as f ≺ F g or f (z) Proposition 2 (see [10]). Assume that z 0 ∈ Ω is a fixed point and the functions f , g ∈ H(Ω). If where f (Ω) and g(Ω) are defined by (2) and (3), respectively.
Definition 5 (see [11]). Assume that Φ : Let the function p be analytic in ∆, with p(0) = α and satisfy the following second-order fuzzy differential subordination: Then p is said to be a fuzzy solution of the fuzzy differential subordination. Moreover, if for all functions p that satisfy (4), then we say that the univalent function q is a fuzzy dominant of the fuzzy solutions for the fuzzy differential subordination.
A fuzzy dominant q satisfying the following condition: for all fuzzy dominants q of (4), is called the fuzzy best dominant of (4).

Remark 3.
In the literature on probability theory, a nice relationship between Poisson processes and Borel distributions can be found. In addition, Borel distributions are also closely related to the Galton-Watson branching processes. Details can be found in, for example, [23,24].
Various families of linear or convolution operators are known to play important roles in the Geometric Function Theory of Complex Analysis and its related fields. One can indeed express derivative and integral operators as convolutions of some families of analytic functions. This kind of formalism makes further mathematical investigation much easier and also aids in the better understanding of the geometric properties of the operators involved.
We now introduce the familiar Mittag-Leffler function E α (z) and its two-parameter version E α,β (z) are defined, respectively, by The Mittag-Leffler function E α (z) and its two-parameter version E α,β (z) are known to contain, as their special cases, a number of elementary functions, such as the exponential, trigonometric, and hyperbolic functions. In fact, these Mittag-Leffler functions happen to be the most commonly-used special cases of the Fox-Wright function p Ψ q (z) with p numerator parameters and q denominator parameters, which is defined by the following series see, for example, Ref. [28] p. 67, Equation (1.12.68) and Ref. [29] p. 21, Equation 1.2 (38); see also Ref. [30] : Indeed, by comparing the definitions in (5) and (6), it can be seen that Remark 4. In the vast and widely-scattered literature on mathematical, physical and engineering sciences, one can find infinitely many usages of the celebrated Gauss hypergeometric function 2 F 1 , the Kummer (or confluent) hypergeometric function 1 F 1 , the Clausen hypergeometric function 3 F 2 , and various other mathematical functions of the hypergeometric type, all of which are contained in the generalized hypergeometric function p F q , involving p numerator parameters a 1 , · · · , a p and q denominator parameters b 1 , · · · , b q , as special cases (see, for details, [29,31]). The Fox-Wright function p Ψ q (z) defined by (6) does, in fact, provide a further generalization of the generalized hypergeometric function p F q (z), involving p numerator parameters a 1 , · · · , a p and q denominator parameters b 1 , · · · , b q , given by p F q   α 1 , , · · · , α p ; (α 1 , 1), · · · , (α p , 1); (β 1 , 1), · · · , (β q , 1); The relatively more familiar Bessel-Wright function J µ ν (z) is also a very specialized case of the Fox-Wright function p Ψ q (z) defined by (6).
In view of Remark 4, it is clear that almost all of the special functions of hypergeometric class as well as most (if not all) of the Mittag-Leffler-type functions, including those that we have used in our present investigation, belong to the much wider family of the Fox-Wright function p Ψ q (z) , whose p numerator parameters and q denominator parameters possess some kind of symmetry behavior in the sense that the Fox-Wright function p Ψ q (z) remains invariant (or unchanged) when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.
It is easy to rewrite the second definition in (5) as follows: In the solutions of many real-world problems, which are modeled by means of fractional-order differential, integral and integro-differential equations, Mittag-Lefflertype functions are known to arise naturally. Some important examples include fractionalorder generalizations of the kinetic equation, random walks, Lévy flights, super-diffusive transport and the study of complex systems. Potentially useful properties of the Mittag-Leffler-type functions E α (z) and E α,β (z) can be found in, for example, [28,[32][33][34][35][36][37][38].
The Mittag-Leffler function E α,β (z) does not belong to the normalized analytic function class A. We, therefore, normalize the Mittag-Leffler function E α,β (z) as follows: α ∈ C (α) > 0 and β, z ∈ C. For convenience, hereafter we only consider the case when the parameters α and β are real-valued and for z ∈ ∆.
Named after the French mathematician, Félix Édouard Justin Émile Borel (1871-1956), the widely-and extensively-studied Borel distribution is a discrete probability distribution that arises in such contexts as (for example) branching processes and queueing theory. We recall that a discrete random variable x defines a Borel distribution if it takes on the values 1, 2, 3, . . . , together with the probabilities given by respectively, λ being the parameter of the Borel distribution.
Recently, Wanas and Khuttar [4] applied the Borel distribution (BD) in their study of certain convexity and other geometric properties of analytic functions. Its probability mass function, given by was studied by Wanas and Khuttar [4]. Following the work of Wanas and Khuttar [4], we introduce the following series M(λ; z) in which the coefficients involve probabilities of the Borel distribution (BD): We now recall the following Mittag-Leffler type Borel distribution that was studied by Murugusundaramoorthy and El-Deeb [2] (see also [1,3]): where the two-parameter Mittag-Leffler function E α,β (z) is defined in (5). Thus, by using (9) and (10), and by means of the Hadamard product (or convolution), we now define the Mittag-Leffler-type Borel distribution series as follows: Moreover, by making use of the Hadamard product (or convolution), for 0 denotes the set of non-positive integers and We have used, in this paper, such special functions as the Mittag-Leffler type functions in our study of a general Borel type probability distribution, which may be conditioned to be symmetric or asymmetric.

A Set of Lemmas
Each of the following lemmas will be needed in proving our main results.
Lemma 3 (see [12] Theorem 2.7). Let the function g be convex in ∆ and suppose that If the function p given by belongs to the class H(∆) and This result is sharp, that is, the equality holds true for a suitably specified function.
In Section 4 below, we obtain several fuzzy differential subordinations that are associated with the operator B(λ, α, β) by using the method of fuzzy differential subordination.

Theorem 1.
Let the function k be in the normalized convex function class C on ∆ and suppose that then the following fuzzy differential subordination: This result is sharp, that is, the equality holds true for a suitably specified function.

Proof.
Since by differentiating both sides with respect to z, we obtain which, by differentiating with respect to z, yields By using (16), the fuzzy differential subordination (14) can be written as follows: We now set q(z) = B(λ, α, β)G(z) (18) such that q ∈ H [1, n]. Thus, by substituting from (18) into (17), we have By applying Lemma (3), we find that and k is the fuzzy best dominant. This completes our proof of Theorem 1.
Let the operator I λ be given by (13). Then, where Proof. Since the function h belongs to the normalized convex function class C in ∆, by using the same technique as in the proof of Theorem 1, we find from the hypothesis of Theorem 2 that where q(z) is defined in (18). Thus, by using Lemma 2, we obtain where the function k(z), given by belongs to C on ∆ and k(∆) is symmetric with respect to the real axis. Consequently, we have and This evidentally completes the proof of Theorem 2.
If f ∈ A and satisfies the following fuzzy differential subordination: The result is sharp, that is, the assertion holds true for a suitably specified function.

Proof.
For We, thus, see that the following inequality: Now, by applying Lemma 3, we have The result is easily seen to be sharp, that is, the result holds true for a suitably specified function. The proof of Theorem 3 is, thus, completed.

Theorem 4.
Let h ∈ H(∆), with h(0) = 1, such that If f ∈ A and the following fuzzy differential subordination holds true: where the function k(z), given by is convex and it is the fuzzy best dominant.

Corollary 1.
Let be in the normalized convex function class C in ∆, with h(0) = 1 and 0 β < 1. If the function f ∈ A satisfies the following fuzzy differential subordination: then the function k(z), given by