Applications of Riemann–Liouville Fractional Integral of q -Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results

: Studies regarding the two dual notions are conducted in this paper using Riemann– Liouville fractional integral of q -hypergeometric function for obtaining certain fuzzy differential subordinations and superordinations. Fuzzy best dominants and fuzzy best subordinants are given in the theorems investigating fuzzy differential subordinations and superordinations, respectively. Moreover, corollaries are stated by considering particular functions with known geometric properties as fuzzy best dominant and fuzzy best subordinant in the proved results. The study is completed by stating fuzzy differential sandwich theorems followed by related corollaries combining the results previously established concerning fuzzy differential subordinations and superordinations.


Introduction
Fractional calculus and q-hypergeometric function are known to provide notable results when applied in studies regarding geometric function theory. Fuzzy differential subordination notion was introduced in geometric function theory as generalization of the classical concept of differential subordination in 2011. Later, in 2017, the dual notion of fuzzy differential superordination was also introduced generalizing the notion of differential superordination.
The important contribution fractional calculus and q-calculus has in the development of geometric function theory is indisputable and this is comprehensively shown in review article published by Srivastava [1]. In the same article, the importance of hypergeometric functions added to the studies is highlighted and also numerous fractional and q-calculus operators are listed emphasizing their role in the study of univalent functions.
The concept of fuzzy set introduced by Lotfi A. Zadeh in 1965 [2] has tremendous applications in various fields of science and technique. Extensions of many branches of mathematics have developed in the fuzzy context created by introducing the notion of fuzzy set into the studies. Applications of the notion related to integro-differential equations can be seen in [3,4]. Aspects of fuzzy linear fractional programming embedding the concept are presented in [5]. Proportional Integral Derivative (PID) and Fuzzy-PID control are used in [6] and an evolution of the concept of fuzzy normed linear spaces is presented in [7]. Some highlights of the applications of the fuzzy set concept can be read in [8,9].
The famous notion of subordination was interpreted in the fuzzy concept in 2011 [10] and fuzzy differential subordination was introduced in 2012 [11] as an extension of the classical notion of differential subordination due to Miller and Mocanu [12,13]. The Definition 1 ([53,54]). The Riemann-Liouville fractional integral of order λ (λ > 0) for an analytic function f is defined by The q-hypergeometric function is given by: Definition 2 ([55]). The q-hypergeometric function φ(m, n; q, z) is defined by φ(m, n; q, z) = and 0 < q < 1.
Using Definitions 1 and 2, Riemann-Liouville fractional integral of q-hypergeometric function in introduced as: . Let m, n be complex numbers with m = 0, −1, −2, ... and λ > 0, 0 < q < 1. We define the Riemann-Liouville fractional integral of q-hypergeometric function The Riemann-Liouville fractional integral of q-hypergeometric function has the following form, after a simple calculation: We note that D −λ z φ(m, n; q, z) ∈ H[0, λ]. Next, the fuzzy context where the research was conducted is shown.
The support of the fuzzy set (A, F A ) is the set A and the membership function of Definition 5 ([10]). Consider D ⊂ C, the functions f , g ∈ H(D) and z 0 ∈ D a fixed point. The function f is fuzzy subordinate to g, written f ≺ F g, if the following conditions are satisfied: Definition 6 ([11], Definition 2.2). Consider h a univalent function in U and ψ : C 3 × U → C, such that h(0) = ψ(a, 0; 0) = a. When p is analytic in U, such that p(0) = a and the fuzzy differential subordination is satisfied then p is a fuzzy solution of the fuzzy differential subordination. The univalent function q is a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, if F p(U) p(z) ≤ F q(U) q(z), z ∈ U, for all p satisfying (3). A fuzzy dominant q that satisfies F q(U)q (z) ≤ F q(U) q(z), z ∈ U, for all fuzzy dominants q of (3) is the fuzzy best dominant of (3).

Definition 7 ([17]).
Consider h an analytic function in U and ϕ : C 3 × U → C. When p and ϕ(p(z), zp (z), z 2 p (z); z) are univalent functions in U and the fuzzy differential superordination is satisfied then p is a fuzzy solution of the fuzzy differential superordination. An analytic function q is fuzzy subordinant of the fuzzy differential superordination if for all p satisfying (4). A univalent fuzzy subordinant q that satisfies F q(U) q ≤ F q(U) q for all fuzzy subordinant q of (4) is the fuzzy best subordinant of (4).

Definition 8 ([11]). Q represents the set of all analytic and injective functions f on U\E
The following lemmas are needed as tools in the proof of the theorems stated in the next section.
In the next section, fuzzy differential subordinations and superordinations will be investigated using the operator presented in (1) with his particular form as seen in (2). For the fuzzy differential subordinations and superordinations under investigation, fuzzy best dominant and fuzzy best subordinant are obtained, respectively. As applications of the results contained in the theorems, corollaries emerge when fuzzy best dominant and fuzzy best subordinant are considered particular functions known in geometric function theory to have certain geometric properties very useful in studies related to differential subordinations and superordinations. Two sandwich-type theorems and several corol-laries associated are also stated connecting the dual results concerning fuzzy differential subordinations and fuzzy differential superordinations obtained in this paper.
If the following fuzzy differential subordination is satisfied by g for β, ψ, δ, ε ∈ C, δ = 0, then and the fuzzy best dominant is g.
, z ∈ U and the fuzzy best dominant is g.
The fuzzy differential superordination (19) can be written as and applying Lemma 2, we obtain , z ∈ U, and the fuzzy best subordinant is g.

Conclusions
This paper presents new fuzzy differential subordinations and superordinations obtained by involving in the studies the operator presented in (1). The first theorem refers to a new fuzzy differential subordination for which the fuzzy best dominant is provided. Two corollaries follow as applications of the new result by considering two particular functions with nice geometric properties as fuzzy best subordinant of the first fuzzy differential subordination investigated in this study. Next, similar results are presented regarding a fuzzy differential superordination for which the fuzzy best dominant is obtained and two corollaries follow as applications. The sandwich-type theorem connecting Theorems 1 and 2 is stated as Theorem 3 and the corollaries formulated as a consequence of this theorem appear naturally by combining the results presented in the corollaries related to Theorems 1 and 2. Finally, a special fuzzy differential subordination is investigated in Theorem 4 and the dual result concerning a fuzzy differential superordination is contained in Theorem 5. Naturally, specific corollaries also accompany these theorems.
As future applications of the results presented in this paper, they could serve as inspiration for Riemann-Liouville fractional integral to be applied on other q-calculus functions and operators. Symmetry properties might be investigated regarding the operator shown in (1) considering the fuzzy subordinations and superordinations obtained here. In addition, classes of analytic functions could be introduced due to the geometric properties which can be interpreted from the corollaries. Those classes could also be investigated related to symmetry properties given by the use of a fractional operator in their definition.
Keeping in mind the notable applications of fuzzy sets theory and fractional calculus in real life contexts as it can be seen for example in [59][60][61], hopefully, this new fractional q-operator will find applications in future studies.