Fuzzy Differential Subordination Associated with a General Linear Transformation

: In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator BS m , t α , λ :A n → A n in the open unit disc U . The second step is to develop fuzzy differential subordination for the operator BS m , t α , λ . By considering linear transformations of the operator BS m , t α , λ , we deﬁne a new fuzzy class of analytic functions in U which we denote by T λ , t (cid:122) ( m , α , δ ) . Several innovative results are found using the concept of fuzzy differential subordination and the operator BS m , t α , λ for the function f in the class T λ , t (cid:122) ( m , α , δ ) . In addition, we explore a number of examples and corollaries to illustrate the implications of our key ﬁndings. Finally, we highlight several established results to demonstrate the connections between our work and existing studies.


Introduction and Definitions
The history of fuzzy sets theory began in 1965 with the publication of "Fuzzy Sets" [1] by Zadeh, which was first received with distrust but is now mentioned in more than 95,000 publications.Many links between fuzzy sets theory and other areas of mathematics have been developed due to the widespread interest in this topic among mathematicians.The excellent review article [2] from 2017 is a dedication to Zadeh's work and explains how the fuzzy sets concept has developed over time and how it is connected to many various areas of mathematics, science, and technology.This issue celebrates the centennial of Zadeh birth with a number of excellent review articles, including one [3] that provides background on the evolution of fuzzy sets theory and shines a light on the work of Dzitac, a former student and colleague of Zadeh.In 2008, he collaborated on a book [4] with Zadeh, forever linking both of their names.
One of the most recent research techniques in the theory of single complex variable functions is the differential subordination method.It was investigated in [5] and introduced by Miller and Mocanu in [6,7].This technique allows novel findings to be rapidly acquired while simultaneously presenting certain well-established outcomes in the field.One of the more common results of the differential subordination approach is differential inequalities.Numerous papers and monographs on the theory of single functions of complex variables have been published as a direct consequence of the advancement of this method.
According to [8], "Knowing the properties of differential expression for a function, we can determine the properties of that function on a given interval."This is the rationale behind the development of the differential subordination theory.In publishing their works [8,9], the authors intended to establish a new line of inquiry in mathematics by merging concepts from the domain of complex functions with those from fuzzy sets theory.As previously stated, the authors support their claim that a function's characteristics can be ascertained on a certain fuzzy set by understanding the characteristics of a differential expression on that set.The case of actual functions has been left as an "open problem" by the authors, who only examined the case of a single complex function.
Fuzzy subordination was first mentioned in [8].The concept of fuzzy differential subordination has been defined in [9].The fuzzy differential subordination produced by the differential operator was studied in [10][11][12].
This kind of research is crucial for improving our comprehension of the relationships between various mathematical ideas and for creating new tools and approaches to solve mathematical difficulties.
Motivated by the studies of [8,9], our aim in this paper is to establish properties of differential subordination and fuzzy differential subordination associated with linear combinations of the Al-Oboudi differential operator and the Babalola convolution operator as defined in the open unit disc.
We refer to the set of all analytic functions (AFs) f in U= {τ ∈ C : |τ| < 1} as H(U) and to the class of all normalized analytic functions as A, (A 1 =A).The Taylor series for each f ∈ A n is of the following form: The family of all convex functions of order α for 0 ≤ α < 1 is represented by C(α), and is defined as When α = 0, then the class C of convex functions is obtained.We subsequently discuss the background works that generate the notion of fuzzy differential subordinations and their corresponding definitions.

Definition 1 ([1]
).Let Y be a non-empty set, let F L :Y→ [0, 1], and let Then, a pair (L, F L ) is a fuzzy subset of Y.
Remark 1.The function that determines membership in the fuzzy set (L, F L ) is termed F L , and the set L is known as the support of the fuzzy set (L, F L ).In addition, it is possible to indicate that L = Supp(L, F L ). ( Remark 2. Suppose that L ⊂Y; then, Definition 2 ([13]).Let U ⊂ C. For a fixed point, let τ 0 ∈ U and let the functions f , g ∈ H(U).
Then, we can say that f is fuzzy subordinate to g and write if the following conditions are satisfied: and Definition 3 ([6]).Let us say that ψ : C 3 ×U→ C and that Let h be univalent in U with h(0) = b.If ϕ is analytic in U with ϕ(0) = b and satisfies the second-order fuzzy differential subordination then ϕ is referred to as a fuzzy solution of the fuzzy differential subordination.
Remark 3. Any univalent function q satisfying (3) is called fuzzy dominant with respect to the fuzzy solutions of the fuzzy differential subordination Then, the fuzzy dominant q that satisfies F q(U) q(τ) ≤ F q(U) q(τ), τ ∈ U is referred as the fuzzy best dominant for all fuzzy dominants of (3).
Real and complex order integrals and derivatives have shown promise in mathematical modeling and analysis of practical issues in the sciences, and this work has made an impact on the study of geometric functions.A novel model of the human liver [14] and an examination of the dynamics of dengue transmission [15] are only two examples of the kind of research that can be considered part of the aforementioned field; and see [16][17][18][19].The family of integral operators connected to the first-kind Lommel functions was introduced in [20], and has important applications in both pure and applied mathematics.As a consequence of the existence of differential and integral operators, functional analysis and operator theory can be used in the study of differential equations.Here, we employ the characteristics of differential operators to solve differential equations using the operator technique; such operators may be involved in the solution of partial differential equations, although this needs more study.The Babalola convolution operator is well recognized for its attractive results in geometric function theory.Its nature and several of its distinguishing characteristics are described below.

Definition 4 ([21]
).Let f be an analytic function in A. The Babalola convolution operator, denoted as B m t , is defined by where and where Equivalently, From ( 4), we have where The Al-Oboudi differential operator, studied in [23], is a generalization of the Salagean differential operator.
After a few simple calculations, we have The operator that is utilized to obtain the original results of this study is defined in the following.Definition 6.Let α ≥ 0, m ∈ N 0 = N ∪ {0}, and n ∈ N, and denote by BS m,t α,λ the operator provided by BS m,t α,λ :A n →A n : Remark 10.
Remark 12.For t = m = 1 and λ = 1 in (7), we have where f (U) is the membership function for the fuzzy set F f (U) , and is connected to the function f .The membership function of the fuzzy set ( f + g)(U) connected to the function f + g coincides with the half of the sum of the membership functions of the fuzzy set f (U), that is, First, using the operator provided by the definition above, a novel class of fuzzy analytic functions is defined.
where δ ∈ (0, 1], α ≥ 0, m ∈ N 0 , and n ∈ N. This study follows a notable current trend in the study of fuzzy differential subordination, namely, the creation and study of new fuzzy classes of functions using new operators.Based on the recently discovered linear differential operator BS m,t α,λ , a novel class T λ,t (m, α, δ) of fuzzy differential subordinations is generated in Section 1.In Section 2, we provide the known lemmas that establish our main results.The main results of the paper are presented in Section 3. In this section, we prove the convexity of the newly formed class and obtain fuzzy differential subordination via the operator BS m,t α,λ .These primary findings provide interesting corollaries, including the fuzzy best dominants for the investigated fuzzy differential subordination.We provide several examples to illustrate the value of these new results.In the last portion, we provide our final remarks.

Preliminaries
To prove our main results, we apply the following lemmas.

Lemma 1 ([6]). Suppose that h ∈
Lemma 2 ([26]).Suppose that γ ∈ C * is a complex number, Reγ ≥ 0, and h is a convex function with h(0 an analytic function in U, and then is the fuzzy best dominent and is convex.

Lemma 3 ([26]
). Suppose that g represents a convex function in U; moreover, suppose that be analytic in U, and and this result is sharp.
Theorem 2. Suppose that g is a convex function in U and is defined as Then, the fuzzy differential subordination i.e., BS m,t α,λ f (τ) i.e., BS m,t α,λ G(τ) ≺ F g(τ), τ ∈ U, and this result is sharp.
Proof.As a consequence of our definition of the function G(τ), we have Differentiating Equation (13) with respect to τ, we obtain and have Differentiating (14), we have From Equation (15), the fuzzy differential subordination is Let and let ϕ ∈ H [1, n].By substituting (17) into (16), we obtain Lemma 3 allows us to have The most effective best dominant is g, meaning that we have and let m − t > −1, c > 0 and Then, where Proof.We can use the same justifications as in the proof of Theorem 2, as the function h presented in the theorem is convex.When we interpret the premise of Theorem 3, we can see that where ϕ(τ) is provided by (17).By applying Lemma 2, the following fuzzy inequality is obtained: where It is understood that g(U) is symmetric with regard to the real axis using the notion of convexity for function g, and we can write and From (19), it is possible to deduce inclusion (18).
Theorem 4. Let the function g be a convex function with g(0) = 1 and and let m − t > −1.Then, we obtain the following fuzzy differential subordination: and the result is sharp.

Proof.
Let as From Lemma 1, we know that is a convex function and verifies the differential equation related to the following fuzzy differential subordination (22): Therefore, it is the fuzzy best dominant.Taking the derivative, we obtain From Lemma 3, we have Thus, we obtain BS m,t α,λ f (τ) τ ≺ F q(τ), τ ∈ U.
Corollary 1. Suppose that that is, is convex and fuzzy best dominant.
Following the application of Lemma 3, we have the required result.

Conclusions
In this article, fuzzy differential subordination is studied in relation to geometric function theory.First, we develop a new operator BS m,t α,λ :A n → A n in the open unit disc U.Then, taking this operator into consideration, we create fuzzy differential subordination.Next, we define a particular fuzzy class of analytic functions in U, which we call T λ,t (m, α, δ).Using the idea of fuzzy differential subordination and the operator BS m,t α,λ for the function f in the class T λ,t (m, α, δ), many novel results can be proved.When λ = 1 and t = m, all the results provided in this article reduce to known results proved previously in [11].
For conclusions that offer coefficient estimates, distortion theorems, or closure theorems, as is typical in geometric function theory, further research on the newly introduced class may be needed.Additionally, the introduction of this class can serve as an inspiration for future research that introduces and characterizes additional intriguing fuzzy classes.In order to identify additional feasible values of δ for accurate definitions of fuzzy classes, the constraint placed on δ ∈ (0, 1] should be further examined.