1. Introduction
Fuzzy set theory is a new branch of mathematics developed by Zadeh in 1965 [
1], using the idea of fuzzy sets. It has advanced rapidly, finding use in many areas of science and technology today. Many different branches of mathematics have taken up new directions with the use of the fuzzy set notion. As a generalization of the traditional idea of differential subordination given by Miller and Mocanu [
2,
3], it was utilized to introduce the novel notions of fuzzy subordination [
4] and fuzzy differential subordination [
5]. The fundamentals of differential subordination are outlined in [
6,
7] and the development of the concept of fuzzy differential subordination is detailed in [
8]. These advances have shown the strength and adaptability of fuzzy set theory as a mathematical tool, and they have paved the way for new studies in geometric function theory and other fields of mathematics. Due to its similarities to fuzzy set theory, the same authors first put forward the classical theory of differential subordination in 1978 [
2] and 1981 [
3]. As was indicated before, this most likely entailed expanding the classical theory to include the idea of fuzzy differential subordination.
As well as showing potential in the mathematical modeling and analysis of actual scientific problems, the theory of real and complex-order integrals and derivatives has aided the study of geometric functions theory (GFT). There have been some remarkable discoveries made in the study of differential and integral operators recently, having applications to many areas of physics and mathematics. Two such instances include research analyzing the dynamics of dengue virus transmission [
9] and a novel model of the human liver that employs Caputo–Fabrizio fractional derivatives with the exponential kernel suggested in [
10]; for more information, see [
11,
12].
At the beginning of 2012, a new research direction was launched by combining the concept of differential subordination with the complex function domain in fuzzy set theory, which is named fuzzy differential subordination. For more details, see [
5,
6,
13,
14,
15,
16,
17,
18,
19]. These articles go further into the subject of fuzzy differential subordination and how it applies to different operators. The use of the concept of fuzzy differential subordination for certain classes of operators, such as those found in geometric function theory (GFT), was likely further investigated. These developments could have led to new insights into and approaches for examining the properties of complex functions. Fuzzy differential subordination is the focus of every paper, exploring its application to various mathematical problems. Due to this, we will examine fuzzy differential subordination in this article, along with a newly developed operator and a set of analytical functions.
In the study presented in this article, fuzzy set theory and GFT interact within a single framework. This sort of study is essential for explaining the connections between distinct mathematical concepts and developing new methods for solving mathematical problems.
Here, we shall call
the set of all analytic functions on the open unit disc
. Let
be the set of normalized analytic functions. For
and
then
The class of convex functions of order
(
denoted by
and defined as:
when
, then the class
C of convex functions is obtained.
This research is in line with recent efforts in the field of fuzzy differential subordination that develop and investigate a novel fuzzy class by introducing new operators. The current research is divided into five parts. In
Section 1, we covered the fundamentals of fuzzy sets, differentiable operators, and analytic functions. To set up the conditions for the rest of the study, we shall define fuzzy differential subordination and explore several well-established ones in
Section 2. In the end, we talk about a brand-new operator
for analytic functions. It is based on the generalized Sălăgean differential operator, the Noor integral operator, and the Riemann–Liouville fractional integral. The class of fuzzy analytic functions is defined in
Section 2 by using this operator and fuzzy differential subordination. In
Section 3, we give some lemmas for our planned research on a new class of analytic functions connected with fuzzy differential subordination. Using the aforementioned lemmas, we demonstrate several helpful findings and some examples in
Section 4. It is interesting to discuss the applications of the fuzzy set and system in the management sciences, modelling, optimization statistics, and probability theory. For further information, see [
20,
21,
22] and references therein.
2. Definitions
Subsequently, we examine the definitions and supporting studies that facilitated the understanding of fuzzy differential subordination.
Definition 1 ([
8]).
Let X be a non-empty set. A pair is called a fuzzy subset of X, whereandare the support of the fuzzy set. The function is called the membership function of the fuzzy subset and is denoted by Remark 1. The 0 shows the smallest membership of to , while 1 indicates the highest membership of to for a fuzzy subset.
Definition 2 ([
4]).
Let fixed point and let We take the functions , and f is called a fuzzy subordinate to g, written asif there exists a function such thatand Remark 2. If g is univalent, then and
Remark 3. Such a function can be considered Remark 4. If , the conditions become and , which is equivalent to the classical definition of subordination.
Definition 3 ([
5]).
Let h be univalent in Δ and , such that . When the fuzzy differential subordinationis satisfied for an analytic function φ in Δ, such that then φ is called a fuzzy solution of the fuzzy differential subordination. A fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination is a univalent function q, for which for all φ satisfying (2). The fuzzy best dominant of (2) is a fuzzy dominant such that for all fuzzy dominants q of (2). Definition 4 ([
13]).
Letwhere is the fuzzy set membership function that is connected to the function f. The membership functions of the fuzzy sets connected to the functions and have a one-to-one correspondence, i.e., Remark 5. Let also ; it is obvious that Definition 5. Let be given by (1) and , then the convolution of f and g is defined as: Now, by using the idea of fuzzy differential subordination and the linear combination of the generalized Sălăgean differential operator and the Noor integral operator together with the Riemann–Liouville fractional integral operator, we define a new class of analytic functions. So first, we give the definitions of the already-defined operators and references therein.
Definition 6 ([
23]).
Noor studied the Noor integral operator . Letand let be defined such that Remark 6. The following identity is held for :and The Al–Oboudi differential operator was studied in [
24] and this operator is the generalization of the Salagean differential operator.
Definition 7 ([
24]).
For and the operator is defined byAfter some simple calculation, we have Remark 7. The following is held for : In the next definition, we define the operator that will be used to obtain the results of this paper.
Definition 8 ([
25]).
Let and The operator is defined by Remark 8. is a linear operator, and if then, after some simple calculations, we have Remark 9. If then and for then
Remark 10. For then
Remark 11. For and in (5), then We also review the definition of the Riemann–Liouville fractional integral:
Definition 9 ([
26]).
If we apply the Riemann–Liouville fractional integral of order λ to an analytic function f, we define the following expression:where The fractional integral has been extensively exploited to construct novel operators, and these operators have given rise to intriguing subclasses of functions that have yielded practical and motivating results (see, [
27,
28,
29,
30,
31,
32,
33]). Using Ruscheweyh and Salagean differential operators, several studies (see, for example, [
34]) have generated fuzzy differential subordination. The Riemann–Liouville fractional integral in [
35] is used to find the derivative of a Gaussian hypergeometric function and for studying the applications of the fractional integral operator we refer [
36,
37,
38]. The fractional integral in [
39] is used to find the derivative of a confluent hypergeometric function. In this study, we first establish linear combinations of the Al–Abodi differential operator and the Noor integral operator. Using Riemann–Liouville fractional integral and linear combinations of these two operators, we define a new operator and consider it to define a class of analytic functions of fuzzy differential subordination.
Taking the motivation from the article [
25], and using the Definition 8 of Riemann–Liouville fractional integral
and Definition 9 of linear combination
we then define a differential integral operator
as follows:
Definition 10. The Riemann–Liouville fractional integral applied to the linear combination of differential operator is to introduce differential integral operator as follows:where The series of the Riemann–Liouville fractional integral is:and Remark 12. The operator discussed in this article is a more comprehensive version compared to the operator introduced in [25]. Definition 11. Let is a fuzzy class and contains all functions that fulfil the fuzzy inequalitywhere and . Remark 13. The analytic function class discussed in this study is more general in scope compared to the class introduced in [25]. 3. Set of Lemmas
In order to demonstrate our primary findings, we shall employ the following Lemmas.
Lemma 1 ([
2]).
Let . Ifthenis a convex function. Lemma 2 ([
17]).
Let , and h be a convex function with . If with an analytic in Δ andthen,with the convex functionis the fuzzy best dominant. Lemma 3 ([
17]).
Let g represent a convex function in Δ and letwhere and Letbe analytic in Δ andthen the sharp result will be 4. Main Results
Now, we establish a relation to show that the set of functions is convex.
Theorem 1. The set is convex.
Proof. For the required result, the function
must belong to the class
with
and
being non-negative such that
. We next show that
and
From Definition 4, we obtain that
As, if
then
as well as
This implies that and is convex. □
Theorem 2. Let g be a convex function in Δ and letalso , let , as well asthen, the fuzzy differential subordination This implies the sharp result Proof. By using
, we get
Differentiating Equation (10) with respect to
, we have
also,
Differentiating (11), we have
and the inequality (
9) may be formulated to express the fuzzy differential subordination:
Let
and
. Putting (13) in (12), we get
By Lemma 3, we have
where
g is the best dominant. □
In the following theorem, we proved the inclusion result for the class
Theorem 3. Suppose thatas . Let and thenwhere Proof. By repeating the procedures used to prove Theorem 2, while keeping in mind the presumption of Theorem 3 and that
is a convex function, we obtain
where
is given by (
13). Using the Lemma 2, we get
and
Since
is symmetric with respect to the real axis and
g is convex, therefore we may write
and
Inclusion (
14) follows obviously from (
15). □
Theorem 4. Let g be a convex function and define Let then we get the sharp fuzzy differential subordination Proof. Consider
we can write
for
. One differentiates the obtained expression
The inequality (
16) can be written as follows:
Using the Lemma 3, we have
□
Example 1. Considera convex function in Δ, and we obtain that and Define Take and, after a short computation, we obtain Now,and, differentiating it, Applying Theorem 4, we get the following fuzzy differential subordination: Induce the following fuzzy differential subordination: Theorem 5. Suppose that h is a holomorphic function such that , and . Let , and the fuzzy differential subordination holdsthen,whereis the best fuzzy dominant and a convex set. Proof. By Lemma 1, we get that
is a convex function and verifies the differential equation associated with the fuzzy differential subordination (17)
Therefore, it is the fuzzy best dominant and, by differentiating, we obtain
□
Corollary 1. Suppose thatis a convex function in for . Let and verify the fuzzy differential subordinationthen,and The function is convex and fuzzy best dominant.
Proof. Using the same steps of the proof of Theorem 5, and considering
the fuzzy differential subordination (18) becomes
According to the Lemma 2 for
,
where
and
□
Example 2. As well asthe function h is convex in Δ. For , and we get By Theorem 5
, we get the following fuzzy differential subordination:and induce the following fuzzy differential subordination: Theorem 6. Let , and function g is a convex in Δ with . Let satisfy As and , then we get sharp fuzzy differential subordination, Proof. Consider
differentiating it, we get
According to Lemma 3, we have
□
Theorem 7. Define a function , and function g is a convex in Δ with , , when and the fuzzy differential subordinationhold, then we get sharp fuzzy differential subordination Proof. Consider
differentiating it, we get
According to the Lemma 3, we have
□
Example 3. Consider thatis a convex function in Δ and , and Define Take and, after a short computation, we obtainThen,and Now,differentiating it, we have Applying Theorem 7, we get the following fuzzy differential subordination:induce the following fuzzy differential subordination Theorem 8. Suppose that h is a holomorphic function such that , and , when and the fuzzy differential subordinationhold, we get sharp fuzzy differential subordination,where the fuzzy best dominant is convex. Proof. Consider
differentiating it, we get
According to Lemma 3, we get
Applying the Lemma 1, we obtain that
where
is a convex function and it is a solution of the differential equation of the fuzzy differential subordination (
21) and
so
is the fuzzy best dominant. □
Example 4. Hence, the function h is convex in Δ.
For , and we get By Theorem 8, we get the following fuzzy differential subordination:induce the following fuzzy differential subordination: Theorem 9. Considering a convex function g with the property and definingwhen and the fuzzy differential subordinationhold, then we obtain the sharp result, Proof. Differentiating
we obtain
Using this notation, the fuzzy differential subordination can be written as:
According to the Lemma 3, we obtain the sharp result
□
5. Conclusions
In this research, we combined ideas from geometric function theory and fuzzy set theory. As the first step of our research, we developed an interesting operator of analytic functions in an open unit disc by combining the help of the Riemann–Liouville fractional integral operator. Then, we considered the operator , and defined a special fuzzy class of analytic functions represented by . Using the idea of fuzzy differential subordination, we found many new findings that are relevant to this class. Following our discussion of the main theorems and corollaries, we provided many examples related to these results.
The subclass provided in Definition 11 and the operator specified in Definition 10 may be used in further studies, and other studies involving neighborhood analysis, convexity, close-to-convexity, coefficient estimates, closure theorems, and distortion theorems may all be investigated. The dual theory of fuzzy differential superordination may provide similar results using the operator and the class , and these may be related to the results given here for sandwich-type theorems.
Author Contributions
Conceptualization, S.K., D.B. and F.M.O.T.; methodology, S.K. and D.B.; software, F.T.; validation, F.T., S.K. and D.B.; formal analysis, F.T. and F.M.O.T.; investigation, S.K. and F.M.O.T.; resources, D.B., F.T. and F.M.O.T.; data correction, S.K. and D.B.; writing original draft preparation, writing—review and editing, S.K. and F.M.O.T.; supervision, D.B. and S.K.; funding acquisition, D.B. All authors have read and agreed to the published version of the manuscript.
Funding
The research work of the fourth author is supported by Researchers Supporting Project number (RSP2023R401), King Saud University, Riyadh, Saudi Arabia.
Data Availability Statement
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interest.
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