1. Introduction
The dual ideas of third-order fuzzy differential subordination and superordination, a recently considered type of fuzzy differential subordination, are the focus of this work. Lotfi A. Zadeh first proposed the idea of the fuzzy set in 1965 [
1], which was included into the differential subordination theory leading to the emergence of the fuzzy differential subordination notion in 2012 [
2]. The fuzzy differential subordination theory adheres to the general differential subordination idea as investigated in [
3,
4]. The investigation of fuzzy differential subordination and superordination continues to deliver intriguing results in recent publications. Mittag–Leffler-type distributions are included in the research on the fuzzy differential subordination theory seen in [
5,
6]. Different types of operators are employed for obtaining the results seen in [
7,
8]. Quantum calculus operators provide the tools for the new fuzzy differential subordination results obtained in [
9,
10], and the dual theories of fuzzy differential subordination and superordination are used for developing sandwich-type results involving quantum calculus operators in [
11]. All those results involving different types operators, including the ones involving quantum calculus, have motivated the research presented in this paper.
The context for obtaining the innovative conclusions seen in this work was given by the emergence in [
12] of the idea of third-order fuzzy differential subordination. Expanding upon the notion put forth in [
13], the authors of the work introduced the idea of third-order fuzzy differential subordination. The key notions to be used for the studies pertaining to this line of research for the fuzzy differential subordination theory, including the admissible functions class and basic theorems, were provided in [
12], along with the main principles of third-order fuzzy subordination. As the concept of differential superordination was first proposed as a dual idea to that of differential subordination, in light of this concept, the idea of third-order fuzzy differential superordination was first presented in [
14], where the dual problem of the third-order fuzzy differential superordination was studied by describing the fundamental ideas connected with the notion of third-order fuzzy differential superordination.
Since no other papers are published so far on the idea of using the two new dual fuzzy theories in order to obtain third-order fuzzy sandwich-type results, the present work adds valuable knowledge for enhancing the new lines of research recently initiated. The framework for the investigation is well known in geometric function theory.
is the class of analytic functions in and , with and .
Certain subclasses of
are famous and indispensable for research, such as the following:
where
, and
where
, and
with
and
.
Designate the class of convex functions as
Given
, if there exists
, then
is
subordinate to
, denoted
, if
, and
for every
, and
. The following is what must be met such that
we have the function
:
For two functions
given by
the convolution of
and
is defined as
Numerous papers refer to geometric properties of different families of special functions, particularly the generalized hypergeometric functions (see [
15,
16,
17,
18]) and the Bessel functions (see [
19,
20,
21,
22]). The Lommel functions of the first and second kind appear as specific solutions of particular second-order differential equations in the theory of Bessel functions (see, for example, [
23,
24,
25]). We recall now the Lommel function, denoted by
and given by
which is a particular solution of the inhomogeneous Bessel differential equation
where
, and
stands for the Euler gamma function. It is clear that the function
is analytic for all
.
Next, the normalized Lommel function
is considered as follows:
Using the shifted factorial
defined as
the function
can be represented by the following series representation:
For simplicity, let
and
. Thus, the function
can be defined as follows:
The function is analytic for all and , and it is clear that .
Using the notions presented above, the new operator used for the investigation is defined by means of a Hadamard product as follows:
Remark 1. We note that by taking in (6), the operator is obtained, defined as follows:which is related to Bessel functions of the first kind (see [21]). The operator
, satisfies
and
2. Preliminaries
The new findings in the section that follows will be supported by the subsequent investigations.
Definition 1 ([
13] (p. 441, Definition 2))
. Denote by the set of all functions that are analytic and injective on whereand are such that for . is called the exception set, and the subclass of when can be denoted by and . From the theory of fuzzy differential subordination, we use the following:
Definition 2 ([
1])
. A fuzzy set comprises of the pair , where ς is a set, , and is a membership function. Definition 3 ([
26])
. A fuzzy subset of ς is a pair , where is known as the membership function of the fuzzy set and is called fuzzy subset. The notations that are listed next are used for the investigation proposed here. Let . Indicate by
- (i)
. Then, , where and .
- (ii)
If . Then, where and .
Let
. We say
and
Definition 4 ([
26])
. Let two fuzzy subsets of η, and . We state that the fuzzy subsets and are equal if , , and we denote by . The fuzzy subset is contained in the fuzzy subset if , and denote by . Definition 5 ([
26])
. Let and . is fuzzy subordinate to and written as or if each of the following is satisfied: Proposition 1 ([
26])
. Let and . If , thenwhere and are given by (9) and (10), respectively. Definition 6 ([
13])
. Let consider a function where and . Functions , are called admissible functions and belong to the so-called class of admissible functions denoted by ifis satisfied, withwhere , and . In particular, if we set
then
In this case, we set
and, in the special case when the set
, the resulting class is simply denoted by
.
Definition 7 ([
14] (Definition 5))
. Let Ω
be a set in , , . Denote by the set of functions , satisfyingwherewith , and . Condition (11) is called the admissibility condition. The definition of the concept of fuzzy dominance for the solutions of a third-order fuzzy differential subordination is given in [
12] in terms of a function
that satisfies
whenever
satisfies
i.e.,
with
,
and
,
being called a solution of the fuzzy differential subordination (
12). For all dominants
of (
12), the fuzzy best dominant is a fuzzy dominant
satisfying
. It is known that the fuzzy best dominant is unique up to a rotation of
.
The dual concept of a fuzzy subordinant of a third-order fuzzy differential superordination is introduced in [
14] as being a function
satisfying
or, equivalently written, as
, whenever
satisfies
i.e.,
with
,
and
,
being called a solution for the third-order fuzzy differential superordination (
13). For all fuzzy subordinants
of (
13), the best fuzzy subordinant is a fuzzy subordinant
satisfying
or, equivalently written, as
.
The next two results listed as lemmas, proved in [
12] and [
14], respectively, serve as tools for obtaining the new outcome of the next sections.
Lemma 1 ([
12] (Theorem 3.4))
. Let with , and consider the function and a function for whichwhere , and . If Ω
is a set in , with satisfyingthen Lemma 2 ([
14] (Theorem 1))
. Let , function given by and , satisfyingwhere , .If and are univalent in thenor, equivalently, aswhich implies that In the following sections, by utilizing the third-order differential subordination results in accordance to Antonino and Miller [
13] in the unit disk
and the third-order fuzzy differential subordination and superordination results introduced by Oros et al. [
12,
14], we define certain suitable classes of admissible functions and study some third-order fuzzy differential subordination and superordination properties of univalent functions connected with the Lommel function
defined by (
6). The results obtained using the two dual theories are linked at the end of the study by sandwich-type results.
3. Third-Order Fuzzy Differential Subordination Results
Throughout the study, unless otherwise indicated, we will suppose that .
Certain new third-order fuzzy differential subordinations are obtained in this section. The following definition applies to the class of admissible functions for this purpose:
Definition 8. Let and . Functions that satisfywheneverandwhere , and , form the class of admissible functions denoted as . Our first result is now stated and proved as Theorem 1 below.
Theorem 1. Consider . If and satisfyandthenor Proof. Utilizing (
7) and (
20), we have
Further computations show that
and
Now, we specify how
transforms to
by
and
By applying Lemma 1, conditions (
20)–(
27), we write
Additionally, applying (
24)–(
26), it is simple to obtain
and
Accordingly, the
admissibility condition in Definition 8 corresponds to the
admissibility condition in Definition 6. Thus, by applying (
18) and Lemma 1, we write
or, equivalently,
,
The proof is completed. □
Example 1. By taking , in Theorem 1 we obtainwhich is analytic in andthenor When behaves in an unknown manner on , the following result is generated by Theorem 1.
Corollary 1. Consider and take with . Consider , for certain when . If and satisfy Proof. Following Theorem 1, we obtain
The assertion of Corollary 1 is obtained from
□
If is a simply connected domain, then for a well-chosen conformal mapping of onto . In this situation, the class is written as . The next listed results are direct outcomes of Theorem 1 and Corollary 1.
Theorem 2. Suppose that . If and satisfy Condition (18), thenimplies thati.e., Corollary 2. Consider and take with . Consider for certain when . If and satisfyandthen The fuzzy best dominant of the fuzzy differential subordination (
19) or (
34) is obtained by our next theorem.
Theorem 3. Suppose that be univalent in . Also, let and ψ be given by (27). Lethave a solution , which satisfies the conditions in (18). If satisfies Condition (33) andis analytic in , thenand is the fuzzy best dominant. Proof. We determine that
is a dominant of (
34) by using Theorem 1.
is also a solution of (
34) since it satisfies (
36). As a result, all dominants will dominate
. As a result, the fuzzy best dominant is
. □
In the particular case when and, in view of Definition 8, the class of admissible functions, which we denote by , is described below.
Definition 9. Let Ω
be a set in and . The class of admissible functions consists of the functions such thatwhenever , and Corollary 3. Let . If , thenIfthenorwhere and . Then, as follows, we define a new admissible class :
Definition 10. Suppose that Ω
is a set in and . The class of admissible functions consists of function which satisfieswheneverwhere , and . Theorem 4. Consider . If and satisfyandthen Proof. Utilizing (
7) and (
39), we have
Further computations show that
and
We now specify how
transforms to
by
By applying Lemma 1, conditions (
39)–(
42), we get
Using (
41) and (
42), we have
and
Accordingly, the
admissibility condition in Definition 10 corresponds to the
admissibility condition in Definition 6. Thus, by applying Lemma 1 and (
37), we write
or, equivalently,
,
The proof is completed. □
If is a simply connected domain, then for a well-chosen conformal mapping of onto . In this case, the class is written as . The following result is an immediate consequence of Theorem 4.
Corollary 4. Assume that . If and satisfies Condition (37), thenimplies that In the particular case when and, in view of Definition 8, the class of admissible functions, which we denote by , is described below.
Definition 11. Let Ω
be a set in and . The class of admissible functions consists of the functions such thatwhenever , and Corollary 5. Let . If satisfies Ifthenwhere and . In the special case when
the class
is simply denoted by
.
4. Third-Order Fuzzy Differential Superordination Results
We derive certain third-order fuzzy differential superordinations in this section. The following definition applies to the class of admissible functions for this purpose:
Definition 12. Consider , with and . Functions that satisfywheneverwhere , and , form the class of admissible functions denoted by . Theorem 5. Let . If and satisfyandis univalent in , thenimplies Proof. Let
be given by (
20) and
be given by (
27). Since
, (
28) and (
49) yield
From (
27), we deduce that the
admissibility condition in Definition 12 corresponds to the admissibility condition for
in Definition 8. Therefore, by applying the terms in (
48) and Lemma 2, we write
or, equivalently,
The proof is now complete. □
Example 2. . In Theorem 4, we obtainwhich is analytic in andthus,or If is a simply connected domain and for a well-chosen mapping of onto , then . The following theorem is derived from Theorem 5 using procedures similar to those in the previous section.
Theorem 6. Consider , and λ to be analytic in . If and satisfy the conditions in (48) andis univalent in , thenimplies For a well-chosen
, the next result establishes the existence of the best fuzzy subordinant of (
55).
Theorem 7. Consider λ to be analytic in , and ψ to be given by (27). Let the differential equationhave a solution . If and satisfy Condition (48) andis univalent in , thenimpliesand is the best fuzzy subordinant. Proof. The proof of Theorem 7 is similar to that of Theorem 3 so we omitted it. □
Next, we consider a new definition for the class of admissible functions, .
Definition 13. Consider , with and . Functions that satisfywheneverwhere , and , form the class of admissible functions denoted by . If is a simply connected domain and for a well-chosen conformal mapping of onto , then the class . The following theorem is derived from Theorem 7 using procedures similar to those in the previous section.
Theorem 8. Consider , and ψ be given by (27). Let the differential equationhave a solution . If and satisfy the conditions in (37) andis univalent in , thenimpliesand is the best fuzzy subordinant. Proof. We left out Theorem 8 since its proof is comparable to that of Theorem 4. □
5. Sandwich-Type Results
Two sandwich-type outcomes are shown in this section. Combining Theorems 1 and 5 yields the sandwich-type result that follows:
Theorem 9. Let the functions and be analytic functions in . Also, let be univalent in , with and . If and andwith Conditions (18) and (48) being satisfied, thenimplies that Similarly, combining Theorems 4 and 8, we obtain
Theorem 10. Let the functions . Also, let , with and . If and andwith conditions (18) and (48) being satisfied, thenimplies that Remark 2. By taking in (6), we obtain the operator defined as follows:which is related to Bessel functions. If this function is used, new results in third-order fuzzy differential subordination can be developed. 6. Conclusions
The theory of differential subordination and superordination advances as a result of the new findings from the inquiry presented in this paper. The basic concepts needed for this study, the Lommel function
, and the motivation for the topic’s inquiry are all included in the introduction in
Section 1.
Section 2 reports the research’s preliminary known results, whereas
Section 3 presents the primary findings. The fundamental theorems of the new third-order fuzzy differential subordination theory were established and validated. The findings in
Section 4 of this paper contribute to the field of third-order fuzzy differential superordination presenting dual results to those reported in
Section 3. Using the previously established new outcome in
Section 3 and
Section 4, sandwich-type results are stated in
Section 5.
The present endeavor provides important knowledge to improve the newly started lines of research because there are currently no published papers on the idea of using the two new dual theories to develop third-order fuzzy sandwich-type outcomes. Also, new results that would enhance our knowledge regarding the topic of third-order fuzzy differential subordination theory could be further developed in view of the statement of Remark 2.
Author Contributions
Conceptualization, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Methodology, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Software, E.E.A. and G.I.O.; Validation, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Formal analysis, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Investigation, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Resources, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Data curation, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Writing—original draft, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Writing—review & editing, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Visualization, E.E.A., G.I.O., R.M.E.-A. and A.M.A.; Supervision, G.I.O.; Project administration, E.E.A.; Funding acquisition, G.I.O. All authors have read and agreed to the published version of the manuscript.
Funding
The publication of this paper was funded by the University of Oradea, Romania.
Data Availability Statement
The original contributions presented in this study are included in the article; further inquiries can be directed to the corresponding author.
Conflicts of Interest
The authors declare no conflicts of interest.
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