**1. Introduction**

The concept of fuzzy differential subordination and its dual, the concept of fuzzy differential superordination were introduced in the last decade as a result of the trend for adapting the notion of fuzzy set to different topics of research. Even if the notion of fuzzy set didn't 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 for 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 and technique. Another review paper published as part of a special issue dedicated to celebrating Zadeh's birth centenary [3] shows other aspects from the development process of fuzzy logic based on the notion of fuzzy set.

Fuzzy sets theory was connected to geometric function theory in 2011 when the notion of fuzzy subordination was introduced [4] with as inspiration 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 sets notions in the subsequent years [7–9] following the main lines of research as found in the monograph published in 2000 [10]. Obtaining fuzzy subordination and superordination results involving operators was a topic approached early in the study of fuzzy subordinations. Fuzzy differential subordinations were first obtained using a convolution product of S ˘al ˘agean operator and Ruscheweyh derivative in 2013 [11]. The study following this line of research was continued and, in 2015, fuzzy differential subordinations were investigated using Ruscheweyh operator [12], and then using a multiplier transformation in 2016 [13]. Fuzzy differential subordinations for prestar-like functions of complex order and some applications were published in 2017 [14] and in the same year, the dual notion of fuzzy differential superordination was introduced [15]. After this event, the two notions were used together in many investigations giving applications such as in [16] and adding operators to the research such

**Citation:** Alb Lupa¸s, A. Applications of the Fractional Calculus in Fuzzy Differential Subordinations and Superordinations. *Mathematics* **2021**, *92*, 2601. https://doi.org/10.3390/ math9202601

Academic Editors: Sorin Nadaban and Ioan Dzitac

Received: 17 September 2021 Accepted: 14 October 2021 Published: 15 October 2021

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as the Srivastava–Attitya operator in [17]. In recent years, theories of fuzzy differential subordination and fuzzy differential superordination developed nicely. New fuzzy differential subordinations associated with integral operators were investigated in [18,19] and a linear operator was considered for the study in [20]. New applications of S ˘al˘agean and Ruscheweyh operators for obtaining fuzzy differential subordinations were investigated in [21] and fuzzy differential subordinations based upon the Mittag–Leffler-type Borel distribution also emerged [22].

The present paper considers for the investigation from a fuzzy point of view an operator obtained using fractional integrals of confluent hypergeometric function. The investigation of such an operator for obtaining new fuzzy differential subordinations and superordinations results is motivated by recent fuzzy related investigations that considered a fractional integral associated with generalized Mittag–Leffler function [23], a fuzzy Atangana–Baleanu fractional derivative operator in [24] and fuzzy differential subordinations obtained using a hypergeometric integral operator [25].

Fractional calculus has had tremendous development in recent years proving to have applications in many research domains such as physics, engineering, turbulence, electric networks, biological systems with memory, computer graphics, etc. As an example, the Korteweg–de Vries equation, developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves, is studied using a new integral transform where the fractional derivative is proposed in the Caputo sense [26]. Related to biological systems, examples can be given as the new study on the mathematical modelling of the human liver with the Caputo–Fabrizio fractional derivative proposed in [27] and fractional calculus analysis of the transmission dynamics of the dengue infection seen in [28].

Numerous interesting approaches were taken into account regarding fractional integral calculus and new results have emerged in an impressive number of papers published lately. A study of fractional integral operators involving a certain generalized multi-index Mittag–Leffler function was conducted in [29]. Integral inequalities were investigated using integral operators defined with fractional integral of Gauss hypergeometric function [30] and using *γ*-convex functions and a generalized fractional integral operator based on Raina's function [31]. Subclasses of analytic functions defined using a fractional integral operator were introduced and studied in [32,33]. Applications of differential subordination theory to analytic and *p*-valent functions defined by a generalized fractional differintegral operator were presented in [34] and a new fractional integral operator is used in the study on the Mittag–Leffler-confluent hypergeometric function [35].

Following the research line in which fuzzy differential subordinations and superordinations are connected to operators, the interesting fractional integral of confluent hypergeometric function introduced and investigated in [36] using the classical theories of differential subordinations and superordinations is further considered from this new perspective involving fuzzy set theory notions. The purpose of the investigation is to present new fuzzy differential subordinations and superordinations which lead to interesting corollaries when using functions with remarkable geometric properties known from geometric function theory as fuzzy best dominant and fuzzy best subordinant, respectively.
