EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6979
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Applications of Fuzzy Differential Subordination on
Analytic Functions Connected with Lommel Function

Ekram E. Ali1,*, Rabha M. El-Ashwah2, Altaf Alshuhail1, Maryam F. Alshammari1

1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 81451,
Saudi Arabia
2 Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

Abstract. The findings of this study are connected with geometric function theory and were acquired
by using Fuzzy subordination-based techniques in conjunction with the convolution concept and Lommel
Function LMn,v. The first class introduced and investigated here is a generalized class of analytic func-
tions. It is also shown that for particular choice of parameters for the new generalized class, the class
of close-to-convex functions emerges. Using the properties of the convolution and subordination, certain
characterization properties of this class are proved involving combinations of the functions from the class.
Further, three more classes are defined in connection to this first class, developing new applications of
Lommel function by using the fuzzy subordination technique and convolutions.

2020 Mathematics Subject Classifications: 30C45, 30C80
Key Words and Phrases: Analytic function, fuzzy set, differential subordination, Lommel function

1. introduction

Lotfi A. Zadeh [1] established the foundations of fuzzy set theory, which has since become a
central tool for handling uncertainty in mathematical analysis. The study of geometric function
theory (GFT) has benefited from the contributions of fuzzy set theory and complex analysis since
the first work introducing the idea of subordination in fuzzy set theory was published in 2011 [2].
Miller and Mocanu’s traditional qualities of subordination [3, 4] served as the inspiration for this
concept. Later papers that studied fuzzy differential subordination, which included components
from the previously established theory of differential subordination [5–7], followed the study path
laid forth by Miller and Mocanu. The idea was immediately embraced by GFT researchers, and
all of the conventional research paths in this area were changed to account for the novel fuzzy
properties. An essential area of research in GFT is operator-related research. Shortly after the no-
tion was launched, in 2013, such experiments were published to acquire new fuzzy subordination
results [8]. We only highlight a few of the numerous publications that have been published in the

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6979

Email addresses: e.ahmad@uoh.edu.sa (E. E. Ali),
r elashwah@yahoo.com (R. M. El-Ashwah)

https://www.ejpam.com 1 Copyright:© 2025 The Author(s). (CC BY-NC 4.0)



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 2 of 18

past few years to demonstrate how the body of knowledge on this subject is constantly growing
[9–16].

In a related study, Haydar [17] extended these ideas and derived new results for fuzzy differen-
tial subordinations, highlighting how the use of diverse operators enriches the subject. A number
of researchers have since examined linear operators in this setting, producing a wide body of work
on fuzzy third order differential subordination [18, 19]. These contributions represent the first
systematic attempts to employ fuzzy sets in the geometric theory of analytic functions.

Let A denote the class of function satisfying f(0) =f
′

(0) − 1 = 0 written as:

f (ξ) = ξ +
∞∑
κ=2

aκξκ. (1)

which are analytic and univalent in the open unit disc U = {ξ : |ξ| < 1}. If f and g are
analytic in U, f is subordinate to g, denoted f(ξ) ≺ g(ξ), if there exists an analytic function ϖ,
with ϖ(0) = 0 and |ϖ(ξ)| < 1 for all ξ ∈ U, such that f(ξ) =g(ϖ(ξ)), ξ ∈ U. If the function g is
univalent inU, f(ξ) ≺g(ξ) is given as (see [4, 20, 21]):

f (0) = g(0) and f (U) ⊂ g(U).

For two functions fι(ξ) ∈ A(ι = 1, 2) are given by

fι(ξ) = ξ +
∞∑
κ=2

aκ,ιξκ,

we define the convolution of f1(ξ) and f2(ξ) as

( f1 ∗ f2)(ξ) = ξ +
∞∑
κ=2

aκ,1aκ,2ξκ = ( f2 ∗ f1)(ξ).

The Lommel function is a special type of mathematical function that arises in various areas
of applied mathematics and physics. It is often encountered in problems involving wave propa-
gation, optics, and acoustics, particularly when dealing with cylindrical geometries. The Lommel
functions are solutions to a specific type of differential equation, known as Bessel’s differential
equation, which describes the behavior of waves in cylindrical coordinates. These functions are
particularly valuable because they allow for the representation of waveforms that are not easily
handled by simpler functions, thus providing more accurate models in physical applications.

Geometric properties of several families of special functions are discussed in many articles,
especially the Bessel functions (see [22–25]) and the generalized hypergeometric functions (see
[26–29]). The theory of Bessel functions contains the first and second class Lommel functions
as specific solutions of certain second-order differential equations [30–32]. We now review the
Lommel function, which is represented by Lρ,ν(ξ) and provided by

Lρ,ν(ξ) =
ξρ+1

4

∞∑
κ=0

(−1)κ Γ
(
ρ−ν+1

2

)
Γ
(
ρ+ν+1

2

)
Γ
(
ρ−ν+3

2 + κ
)
Γ
(
ρ+ν+3

2 + κ
) (ξ

2

)2κ
(2)



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 3 of 18

which is a particular solution of the nonhomogeneous Bessel differential equation

ξ2w′′(ξ) + ξw′(ξ) +
[
ξ2 − ν2

]
w(ξ) = ξρ+1, (3)

where ρ−ν+1
2 ,

ρ+ν+1
2 ∈ C\Z−;Z−={−1,−2, ...} and Γ stands for Euler gamma function, it is clear that

the function Lρ,ν is analytic for all ξ ∈ C.
Now, we define Lρ,ν(ξ) as follows:

Lρ,ν(ξ) = 4
(
ρ−ν+1

2

) (
ρ+ν+1

2

)
ξ

1−ρ
2 Lρ,ν(

√
ξ), (4)

and using the shifted factorial (y)n defined as

(y)n =
Γ (y + n)
Γ (y)

=

{
1, (n = 0, y ∈ C\ {0}) ,
y (y + 1) ... (y + n − 1) , (n ∈ N, y ∈ C) ,

then Lρ,ν(ξ) can be represented by the following series representation

Lρ,ν(ξ) = ξ +
∞∑
κ=2

(−1)κ−1

4κ−1
(
ρ−ν+1

2 + 1
)
κ−1

(
ρ+ν+1

2 + 1
)
κ−1

ξκ, (5)

for simplicity, let n = ρ−ν+3
2 and v = ρ+ν+3

2 . Thus the function Ln,v can be defined as following:

Ln,v(ξ) = ξ +
∞∑
κ=2

(−1)κ−1

4κ−1 (n)κ−1 (v)κ−1
ξκ, (6)

the function Ln,v(ξ) is analytic for all ξ ∈ C and n, v ∈ C\Z−0 , it is clear that Ln,v(ξ) ∈ A
Now, we define the new operator

LMn,v : A −→ A,

by means of Hadamard product as following:

LMn,vf(ξ) :=
(
Ln,v ∗ f

)
(ξ) = ξ +

∞∑
κ=2

(−1)κ−1

4κ−1 (n)κ−1 (v)κ−1
aκξκ. (7)

Remark 1. We note that by taking v = 1 in (7), then we get the operator LMn defined as following:

LMn,vf(ξ) = ξ +
∞∑
κ=2

(−1)κ−1

4κ−1 (n)κ−1 (κ − 1)!
aκξκ.

which is related to Bessel functions of the first kind (see [24]).

The operator LMn,vf(ξ), satisfying

ξ
(
LMn+1,vf(ξ)

)′
= nLMn,vf(ξ) − (n − 1)LMn+1,vf(ξ) (8)

and
ξ
(
LMn,v+1f(ξ)

)′
= vLMn,vf(ξ) − (v − 1)LMn,v+1f(ξ) (9)

.



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 4 of 18

2. Definitions and Preliminaries

Definition 1. [2] A fuzzy set is pair (S , F), where S is a set , S , ϕ and F : S → [0, 1] a
membership function.

The fuzzy subset is likewise covered by the following idea.

Definition 2. [2] A fuzzy subset of S is a pair (ℓ, Fℓ), where the support of the fuzzy set (ℓ, Fℓ) is
defined as ℓ = {x ∈ S : 0 < Fℓ(x) ≤ 1} = sup (ℓ, Fℓ) and Fℓ : S → [0, 1] is belongs to (ℓ, Fℓ) .

Definition 3. [2] Fuzzy subsets
(
Y1, FY1

)
and
(
Y2, FY2

)
of S are equal iff Y1 = Y2, whereas(

Y1, FY1

)
⊆
(
Y2, FY2

)
iff FY1 (η) ≤ FY2 (η), η ∈ S .

Definition 4. [2] LetU ⊂ C and ξ0 are a fixed point inU and let the functions f, µ ∈ H(U). f is
said to be fuzzy subordinate to µ and write f ≺F µ or f(ξ) ≺F µ(ξ) if

f(ξ0) = µ(ξ0) and Ff(U) (f (ξ)) ≤ Fµ(U) (µ (ξ)) , ξ ∈ U,

where
f(U) = sup(f(U), Ff(U)) = {f (ξ) : 0 < Ff(U)(f (ξ)) ≤ 1, ξ ∈ U

and
µ(U) = sup(µ(U), Fµ(U)) = {µ (ξ) : 0 < Fµ(U)(µ (ξ)) ≤ 1, ξ ∈ U.

The following lemma is necessary to validate our research.

Lemma 1. [6] Let β, γ ∈ C. Also let µ ∈ A be convex univalent inU with Re[βµ(ξ) + γ] > 0 (ξ ∈
U), µ(0) = 1 and p(ξ) ∈ A with p(ξ) = 1 + p1ξ + p2ξ

2 + .... is analytic inU. If

Fψ(C2×U)

[
p(ξ) +

ξp
′

(ξ)
βp(ξ) + γ

]
≤ Fµ(U)µ(ξ),

implies
Fp(U) p(ξ) ≤ Fµ(U)µ(ξ), ξ ∈ U.

Lemma 2. [6] Let β, y ∈ C. Also let µ ∈ A be convex univalent inU with µ(0) = 1 and Re[βµ(ξ)+
y] > 0 (ξ ∈ U), and q(ξ) ∈ A with q(0) = 1 and q(ξ) ≺ µ(ξ) (ξ ∈ U). If p(ξ) = 1 + p1ξ + p2ξ

2 + ...

is analytic inU,

Fψ(C2×U)

[
p(ξ) +

ξp
′

(ξ)
βq(ξ) + y

]
≤ Fµ(U)µ(ξ) (ξ ∈ U),

then
Fp(U) p(ξ) ≤ Fµ(U)µ(ξ).

where F : U→ [0, 1].



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 5 of 18

Using the operator LMn,vf(ξ), we introduce the class Fℜn,v(ϑ; µ), of analytic functions f =

{f1, f2, ..., fϑ} on open unit discU satisfying ξ(LMn+1,vfi(ξ))
′

1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

≺F µ(ξ)(fi ∈ A, i = 1, 2, ..., ϑ, ξ ∈ U),

where ξ−1
ϑ∑

j=1
LMn+1,vf j(ξ) , 0 and µ is convex univalent in U with µ(0) = 1. Also we describe

F = {F1,F2, ...,Fϑ} where Fi(ξ) =
ς+1
ξς

ξ∫
0

tς−1fi(t)dt (ς ∈ C; Re(ς) > 0; i = 1, 2, ..., ϑ). and

proved that F ∈ Fℜn,v(ϑ; µ), whenever f ∈ Fℜn,v(ϑ; µ). There more such classes denoted by
Fℵn,v(ϑ; µ), F℘n,v(ϑ;α, µ) and Fℜn,v(ϑ;α, µ) are introduced and studied here by Fuzzy subordi-
nation method and convolutions.

3. Main Results

Throughout this paper, unless otherwise mentioned, we set n, v > 1.

3.1. The class Fℜn,v(ϑ; µ)

Definition 5. Let f = {f1,f2, ...,fϑ}, fi ∈ A, 1 ≤ i ≤ ϑ be such that

Fψ(C2×U)


ξ
(
LMn+1,vfi(ξ)

)′
1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

 ≤ Fµ(U)µ(ξ) (ξ ∈ U; i = 1, 2, ..., ϑ),

where ξ−1
ϑ∑

j=1
LMn+1,vf j(ξ) , 0 in U, µ is convex univalent in U with µ(0) = 1. Then we say that

f = {f1, f2, ...., fϑ} ∈ Fℜn,v(ϑ; µ).

Theorem 1. Let f = {f1, f2, ..., fϑ} ∈ Fℜn,v(ϑ; µ) and F (ξ) = 1
ϑ

ϑ∑
i=1

fi(ξ). Then F (ξ) satisfies:

Fψ(C2×U)

ξ (LMn+1,vF (ξ)
)′

LMn+1,vF (ξ)

 ≤ Fµ(U)µ(ξ) (ξ ∈ U) (10)

Proof. Let f = {f1, f2, ..., fϑ} ∈ Fℜn,v(ϑ; µ).Then for any ξ0 ∈ U, we have

ξ0
(
LMn+1,vfi(ξ0)

)′
1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ0)

≺F µ(ξ)

and hence equals to µ(wi)(say) for some wi ∈ U, i = 1, 2, ..., ϑ. Then
ϑ∑

i=1
ξ0
(
LMn+1,vfi(ξ0

)
)
′

1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ0)

=

ϑ∑
i=1

µ(wi).



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 6 of 18

Let f(ξ) = ξ +
∞∑
κ=2

aκξκ. Then, from (7), we see that

LMn,vf(ξ) = f(ξ) ∗
{
ξ +∞κ=2

(−1)κ−1

4κ−1 (n)κ−1 (v)κ−1
ξk
}

=
(
Ln,v ∗ f

)
(ξ),

where

Ln,v(ξ) = ξ +
∞∑
κ=2

(−1)κ−1

4κ−1 (n)κ−1 (v)κ−1
ξk. (11)

Hence

ξ0
(
LMn+1,vF (ξ0)

)′
LMn+1,vF (ξ0)

=

ξ0[Ln,v(ξ) ∗
ϑ∑

i=1
fi(ξ0)]

′

Ln,v(ξ) ∗
ϑ∑

j=1
f j(ξ0)

.

Since

LMn+1,v

ϑ∑
j=1

f j(ξ) =
ϑ∑

j=1

LMn+1,vf j(ξ),

we have

ξ0
(
LMn+1,vF (ξ0)

)′
LMn+1,vF (ξ0)

=
1
ϑ


ξ0

ϑ∑
i=1

(
LMn+1,vfi(ξ0)

)′
1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ0)


=

1
ϑ

ϑ∑
i=1

µ(wi) = µ(w0),

for some w0 ∈ U, since µ is convex inU.

Theorem 2. Suppose f = {f1, f2, ..., fϑ} ∈ Fℜn,v(ϑ; µ). Define

Fi(ξ) =
ς + 1
ξς

ξ∫
0

tς−1
fi(t)dt (ς ∈ C; Re(ς) > 0; i = 1, 2, ..., ϑ).

If µ is bounded inU and Re{µ(ξ) + ς} > 0, then F = {F1,F2, ...,Fϑ} ∈ Fℜn,v(ϑ; µ).

Proof. From the definition of Fi(ξ), it follows that

ξF
′

i (ξ) + ςFi(ξ) = (ς + 1)fi(ξ),

and on taking convolution with Ln,v given by (11), we obtain

ξ[LMn,vFi(ξ)]
′

+ ςLMn,vFi(ξ) = (ς + 1)LMn,vfi(ξ), i = 1, 2, ..., ϑ. (12)



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 7 of 18

Let

pi(ξ) =
ϑξ[LMn,vFi(ξ)]

′

ϑ∑
j=1
LMn,vF j(ξ)

. (13)

From (12), we have

pi(ξ)
ϑ

ϑ∑
j=1

LMn,vF j(ξ) + ςLMn,vFi(ξ) = (ς + 1)LMn,vfi(ξ). (14)

Differentiating (14) with respect to ξ, we obtain

p
′

i(ξ)
ϑ

ϑ∑
j=1

LMn,vF j(ξ) +
pi(ξ)
ϑ

ϑ∑
j=1

[LMn,vF j(ξ)]
′

+ ς[LMn,vFi(ξ)]
′

= (ς + 1)[LMn,vfi(ξ)]
′

.

From (13), we have

p
′

i(ξ)

ϑ∑
j=1
LMn,vF j(ξ)

ϑ
+

pi(ξ)
ϑ

ϑ∑
i=1

pi(ξ)
ϑ∑

j=1
LMn,vF j(ξ)

ϑξ
+ ς

pi(ξ)
ϑ∑

j=1
LMn,vF j(ξ)

ϑξ

= (ς + 1)[LMn,vfi(ξ)]
′

.

Hence

p
′

i(ξ) +
pi(ξ)
ϑξ

ϑ∑
i=1

pi(ξ) + ς
pi(ξ)
ξ
=

(ς + 1)[LMn,vfi(ξ)]
′

1
ϑ

ϑ∑
j=1
LMn,vF j(ξ)

.

Then
ξp
′

i(ξ)

1
ϑ

ϑ∑
i=1

pi(ξ) + ς
+ pi(ξ) =

(ς + 1)ξ[LMn,vfi(ξ)]
′

1
ϑ

ϑ∑
j=1
LMn,vF j(ξ)

.
1

1
ϑ

ϑ∑
i=1

pi(ξ) + ς

=
(ς + 1)ξ[LMn,vfi(ξ)]

′

1
ϑ

{
1
ϑ

ϑ∑
j=1
LMn,vF j(ξ).

ϑ∑
i=1

pi(ξ) + ς
ϑ∑

j=1
LMn,vF j(ξ)

} .
From (14), we have

Fψ(C2×U)


ξp
′

i(ξ)

1
ϑ

ϑ∑
i=1

pi(ξ) + ς
+ pi(ξ)

 = Fψ(C2×U)


(ς + 1)ξ[LMn,vfi(ξ)]

′

1
ϑ (ς + 1)

ϑ∑
i=1
LMn,vfi(ξ)

 ≤ Fµ(U)µ(ξ), (15)

since f = {f1, f2, ...., fϑ} ∈ Fℜn,v(ϑ; µ). Now we can write for any ξ0 ∈ U,

1
ϑξ0 p

′

i(ξ0)

1
ϑ

ϑ∑
j=1

p j(ξ0) + ς
+

1
ϑ

pi(ξ0) =
1
ϑ
µ(wi),



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 8 of 18

for some wi ∈ U. This is true for i = 1, 2, ..., ϑ. Since µ is convex, there exists a w0 ∈ U such that

ξ0Q
′

(ξ0)
Q(ξ0) + ς

+ Q(ξ0) = µ(w0),

where Q(ξ) = 1
ϑ

ϑ∑
i=1

pi(ξ). Hence

Fψ(C2×U)

[
ξQ

′

(ξ)
Q(ξ) + ς

+ Q(ξ)
]
≤ Fµ(U)µ(ξ).

Since Re{µ} is bounded and Re{µ(ξ) + ς} > 0, it follows from Lemma 1 that Q(ξ) ≺F µ(ξ) (ξ ∈
U). From (15), we have

Fψ(C2×U)

 ξp
′

i(ξ)
Q(ξ) + ς

+ pi(ξ)

 ≤ Fµ(U)µ(ξ),

where Q(ξ) ≺F µ(ξ). Lemma 2 gives pi(ξ) ≺F µ(ξ) (ξ ∈ U), i = 1, 2, ..., ϑ, that is

Fψ(C2×U)


ξ[LMn,vFi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vF j(ξ)

 ≤ Fµ(U)µ(ξ).

Now

Fi(ξ) =
ς + 1
ξς

ξ∫
0

tς−1
fi(t)dt, ς ∈ C,Reς > 0.

It can be proved, easily, that , for every i, 1 ≤ i ≤ ϑ,

LMn,vFi(ξ) =
ς + 1
ξς

ξ∫
0

tς−1
LMn,vfi(t)dt,

and hence
ϑ∑

i=1

LMn,vFi(ξ) =
ς + 1
ξς

ξ∫
0

tς−1
ϑ∑

i=1

LMn,vfi(t)dt

=
ς + 1
ξς

ϑ∫
0

tςg(t)dt,

where g(t) = t−1
ϑ∑

i=1
LMn,vfi(t) , 0, for ξ ∈ U. Now define

Ω(ξ) =
∞∑

k=1

ς + 1
ς + k

ξk−1, Re(ς) > 0.



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 9 of 18

Then an easy calculations show that

ξ−1
ϑ∑

i=1

LMn,vFi(ξ) = (Ω ∗ g)(ξ) , 0.

Thus F = {F1,F2, ...,Fϑ} ∈ Fℜn,v(ϑ; µ).

Theorem 3. If f = {f1,f2, ...,fϑ} ∈ Fℜn,v(ϑ; µ), and Re{µ} is bounded in U, then f = {f1,f2, ...,fϑ} ∈
Fℜn+1,v(ϑ; µ) holds for Re{µ(ξ) + (n − 1)} > 0 inU.

Proof. Let

pi(ξ) =
ϑξ[LMn+1,vfi(ξ)]

′

ϑ∑
j=1
LMn+1,vf j(ξ)

(ξ ∈ U; i = 1, 2, ..., ϑ). (16)

From (8) and (16), we have

1
ϑ

pi(ξ)
ϑ∑

j=1

LMn+1,vf j(ξ) = nLMn,vfi(ξ) − (n − 1)LMn+1,vfi(ξ). (17)

Differentiating (17) with respect to ξ, we get

ξ

ϑ
p
′

i(ξ)
ϑ∑

j=1

LMn+1,vf j(ξ) +
ξ

ϑ
pi(ξ)

ϑ∑
j=1

[LMn+1,vf j(ξ)]
′

= nξ[LMn,vfi(ξ)]
′

− (n − 1)ξ[LMn+1,vfi(ξ)]
′

.

Using (16), we obtain

ξ

ϑ
p
′

i(ξ)
ϑ∑

j=1

LMn+1,vf j(ξ) + pi(ξ)

 ξϑ
ϑ∑

j=1

[LMn+1,vf j(ξ)]
′

+
(n − 1)
ϑ

ϑ∑
j=1

LMn+1,vf j(ξ)


= nξ[LMn,vfi(ξ)]

′

.

Then

ξ

ϑ
p
′

i(ξ)
ϑ∑

j=1
LMn+1,vf j(ξ)

ξ

ϑ

ϑ∑
j=1

[LMn+1,vf j(ξ)]
′
+

(n−1)
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

+ pi(ξ)

=
nξ[LMn,vfi(ξ)]

′

ξ

ϑ

ϑ∑
j=1

[LMn+1,vf j(ξ)]
′
+

(n − 1)
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

.

Using (8), we have

ξ

ϑ
p
′

i(ξ)
ϑ∑

j=1
LMn+1,vf j(ξ)

ξ

ϑ

ϑ∑
j=1

[LMn+1,vf j(ξ)]
′
+

(n−1)
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

+ pi(ξ) =
ξ[LMn,vfi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vf j(ξ)

. (18)



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 10 of 18

On the left side of (18), by using (16), we’ve

ξp
′

i(ξ)

1
ϑ

ϑ∑
j=1

p j(ξ) + (n − 1)
+ pi(ξ) =

ξ[LMn,vfi(ξ)]
′

1
ϑ

ϑ∑
j=1
LMn,vf j(ξ)

.

Since f = {f1,f2, ...,fϑ} ∈ Fℜn,v(ϑ; µ), then we’ve

Fψ(C2×U)


ξp′i(ξ)

1
ϑ

ϑ∑
j=1

p j(ξ) + (n − 1)
+ pi(ξ)


= Fψ(C2×U)


ξ [LMn,vfi(ξ)]′

1
ϑ

ϑ∑
j=1
LMn,vf j(ξ)

 ≤ Fµ(U)µ(ξ),

i = 1, 2, . . . , ϑ. (19)

Therefore for any ξ0 ∈ U, we have

ξ0 p
′

i(ξ0)

1
ϑ

ϑ∑
j=1

p j(ξ0) + (n − 1)
+ pi(ξ0) =

1
ϑ
µ(wi)

for some w0 ∈ U. Since µ is convex, there exists a wi ∈ U, such that

ξ0

ϑ

n∑
i=1

p
′

i(ξ0)

1
ϑ

ϑ∑
j=1

p j(ξ0) + (n − 1)
+

1
ϑ

ϑ∑
j=1

p j(ξ0) =
1
ϑ

ϑ∑
i=1

µ(wi) = µ(w0).

Setting Q(ξ) = 1
ϑ

ϑ∑
i=1

pi(ξ), we have

Fψ(C2×U)

[
ξQ

′

(ξ)
Q(ξ) + (n − 1)

+ Q(ξ)
]
≤ Fµ(U)µ(ξ),

which by Lemma 1, implies that Q(ξ) ≺ µ(ξ). From (19), we’ve

Fψ(C2×U)

 ξp
′

i(ξ)
Q(ξ) + (n − 1)

+ pi(ξ)

 ≤ Fµ(U)µ(ξ),

where Q(ξ) ≺F µ(ξ). The using of Lemma 2 gives pi(ξ) ≺F µ(ξ),which implies that f = {f1, f2, ..., fϑ} ∈
Fℜn+1,v(ϑ; µ).



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 11 of 18

3.2. The class Fℵn,v(ϑ; µ)

Definition 6. Let Fℵn,v(ϑ; µ) denote the class of functions f ∈ A which satisfies

Fψ(C2×U)


ξ[LMn+1,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ)

 ≤ Fµ(U)µ(ξ) (ξ ∈ U),

where g = {g1, g2, ..., gϑ} ∈ Fℜn,v(ϑ; µ), µ is convex univalent inU with µ(0) = 1.

Theorem 4. Let f ∈ Fℵn,v(ϑ; µ). If Re(µ) is bounded inU and Re{µ(ξ) + τ} > 0, then

F (ξ) =
τ + 1
ξτ

ξ∫
0

tτ−1
f(t)dt (ξ ∈ U; τ ∈ C,Re(τ) > 0),

also belongs to Fℵn,v(ϑ; µ).

Proof. Since f ∈ Fℵn,v(ϑ; µ), then there exists g = {g1, g2, ..., gϑ} ∈ Fℜn,v(ϑ; µ), such that

Fψ(C2×U)


ξ[LMn,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vg j(ξ)

 ≤ Fµ(U)µ(ξ) (ξ ∈ U).

Let

Gi(ξ) =
τ + 1
ξτ

ξ∫
0

tτ−1
gi(t)dt (Reτ > 0).

Then by Theorem 2, we have G = {G1,G2, ...,Gϑ} ∈ Fℜn,v(ϑ; µ). Also let

p(ξ) =
ξ[LMn,vF (ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vG j(ξ)

(ξ ∈ U). (20)

Now, from the definitions of Gi and F , we have

ξ[LMn,vGi(ξ)]
′

+ τLMn,vGi(ξ) = (τ + 1)LMn,vgi(ξ), (21)

and
ξ[LMn,vFi(ξ)]

′

+ τLMn,vF (ξ) = (τ + 1)LMn,vf(ξ). (22)

From (20), (21) and (22), we have

1
ϑ

p(ξ)
ϑ∑

j=1

LMn,vG j(ξ) + τLMn,vF (ξ) = (τ + 1)LMn,vf(ξ). (23)



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 12 of 18

Differentiating (23) with respect to ξ, and multiplying the resulting equation by ξ, we have

ξ

ϑ
p
′

(ξ)
ϑ∑

j=1

LMn,vG j(ξ)+
ξ

ϑ
p(ξ)

ϑ∑
j=1

[LMn,vG j(ξ)]
′

+τξ[LMn,vF (ξ)]
′

= (τ+1)ξ[LMn,vf(ξ)]
′

. (24)

From (20) into (24), we have

ξ

ϑ
p
′

(ξ)
ϑ∑

j=1

LMn,vG j(ξ) +
ξ

ϑ
p(ξ)

ϑ∑
j=1

[LMn,vG j(ξ)]
′

+ τ
p(ξ)
ϑ

ϑ∑
j=1

LMn,vG j(ξ) = (τ + 1)ξ[LMn,vf(ξ)]
′

.

Hence, we get
ξ

ϑ
p
′

(ξ)
ϑ∑

j=1
LMn,vG j(ξ)

ξ

ϑ

ϑ∑
j=1

[LMn,vG j(ξ)]
′
+
τ

ϑ

ϑ∑
j=1
LMn,vG j(ξ)

+ p(ξ)

=
(τ + 1)ξ[LMn,vf(ξ)]

′

ξ

ϑ

ϑ∑
j=1

[LMn,vG j(ξ)]
′
+
τ

ϑ

ϑ∑
j=1
LMn,vG j(ξ)

=
ξ[LMn,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vg j(ξ)

(by using (21)).

From the above, we have

Fψ(C2×U)


ξp
′

(ξ)
1
ϑ

ϑ∑
j=1

Q j(ξ) + τ
+ p(ξ)

 = Fψ(C2×U)


ξ[LMn,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vg j(ξ)

 ≤ Fµ(U)µ(ξ),

where Q j(ξ) =
ξ[LMn,vG j(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vG j(ξ)

.Now Q j(ξ) ≺F µ(ξ), j = 1, 2, ..., ϑ, since G = {G1,G2, ...,Gϑ} ∈

Fℜn,v(ϑ; µ) and µ is a convex univalent. Since Re{µ(ξ)+τ} > 0, an application of Lemma 2 implies
that p(ξ) ≺F µ(ξ), hence F ∈ Fℵn,v(ϑ; µ). This complete the proof.

Theorem 5. If f ∈ Fℵn,v(ϑ; µ) and Re(µ) is bounded inU, then f ∈ Fℵn,v(ϑ; µ) holds for Re(µ(ξ)+
(n − 1)) > 0 inU.

Proof. This theorem’s proof is removed since it is similar to that of Theorem 3.

3.3. The class F℘n,v(ϑ;α, µ)

Definition 7. Let F℘n,v(ϑ;α, µ), α ≥ 0, denote the class of functions f ∈ A satisfying the condition



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 13 of 18

Fψ(C2×U)
[
J(α; f; g1, g2, ..., gϑ)(ξ)

]
= Fψ(C2×U)

α
ξ[LMn,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vg j(ξ)

+ (1 − α)
ξ[LMn+1,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ)


≤ Fµ(U)µ(ξ) (ξ ∈ U),

where g = {g1, g2,...,gϑ} ∈ Fℜn,v(ϑ; µ), ξ−1
ϑ∑

j=1
LMn,vg j(ξ) , 0 in U, µ is convex univalent in

U with µ(0) = 1.

Remark 2. We note that F℘n,v(ϑ; 0, µ) = Fℵn,v(ϑ; µ).

Theorem 6. If f ∈ F℘n,v(ϑ;α, µ) and Re{µ} is bounded inU, then f ∈ F℘n,v(ϑ; 0, µ) = Fℵn,v(ϑ; µ)
hold for Re{µ(ξ) + (n − 1)} ≥ 0.

Proof. For α = 0, the theorem is trivial and hence we can assume that α , 0. Let

p(ξ) =
ξ[LMn+1,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ)

(ξ ∈ U).

Then, a simple calculation reveals that

ξp
′

(ξ)

1
ϑ

ϑ∑
j=1

q j(ξ) + (n − 1)
+ p(ξ) =

ξ[LMn,vf(ξ)]
′

1
ϑ

ϑ∑
j=1
LMn,vg j(ξ)

,

where q j(ξ) =
ξ[LMn+1,vgi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ)

. Also 1
ϑ

ϑ∑
j=1

q j(ξ) ≺F µ(ξ). Since f ∈ F℘n,v(ϑ;α, µ), we have

Fψ(C2×U)
[
J(α; f; g1, g2, ..., gϑ)(ξ)

]
= Fψ(C2×U)


αξp

′

(ξ)

1
ϑ

ϑ∑
j=1

q j(ξ) + (n − 1)
+ p(ξ)

 ≤ Fµ(U)µ(ξ).

Now an application of Lemma 2 gives p(ξ) ≺F µ(ξ) which implies f ∈ Fℵn,v(ϑ; µ). This completes
the proof.

Theorem 7. Forα > β ≥ 0, and Reµ(ξ) is bounded inU, then F℘n,v(ϑ;α, µ) ⊂ F℘n,v(ϑ; β, µ).



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 14 of 18

Proof. The case β = 0 was treated in the previous theorem. Hence we assume that β , 0.
Suppose that f ∈ F℘n,v(ϑ;α, µ). Then

Fψ(C2×U)
[
J(α; f; g1, g2, ..., gϑ)(ξ)

]
≤ Fµ(U)µ(ξ). (25)

Let ξ1 be any arbitrary point inU. Then

J(α; f; g1, g2, ..., gϑ)(ξ1) ∈ µ(U).

From Theorem 6, we have

Fψ(C2×U)


ξ[LMn+1,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ)

 ≤ Fµ(U)µ(ξ). (26)

Now

J(β; f; g1, g2, ..., gϑ)(ξ) = (1 −
β

α
)
ξ[LMn+1,vf(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ)

+

β

α
J(α; f; g1, g2, ..., gϑ)(ξ).

From (25) and (26) it follows that

Fψ(C2×U)


ξ1[LMn+1,vf(ξ1)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ1)

 ≤ Fµ(U)µ(ξ)

and

Fψ(C2×U)

α
ξ1[LMn,vf(ξ1)]

′

1
ϑ

ϑ∑
j=1
LMn,vg j(ξ1)

+ (1 − α)
ξ1[LMn+1,vf(ξ1)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vg j(ξ1)

 ≤ Fµ(U)µ(ξ).

Now µ(U) is convex and
β

α
< 1, hence we have

Fψ(C2×U)
[
J(β; f; g1, g2, ..., gϑ)(ξ)

]
≤ Fµ(U)µ(ξ).

showing that f ∈ F℘n,v(ϑ; β, µ). This completes the proof.



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 15 of 18

3.4. The class Fℜn,v(ϑ;α, µ)

Definition 8. Let Fℜn,v(ϑ;α, µ), α ≥ 0, denote the class of functions f ∈ A satisfying

Fψ(C2×U)
[
J(α; f; f1, f2, ..., fϑ)(ξ)

]
= Fψ(C2×U)

α
ξ[LMn,vfi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn,vf j(ξ)

+ (1 − α)
ξ[LMn+1,vfi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)


≤ Fµ(U)µ(ξ). (ξ ∈ U),

where f = {f1,f2,...,fϑ} ∈ Fℜn,v(ϑ; µ) and ξ−1
ϑ∑

j=1
LMn,vf j(ξ) , 0 in U, µ is convex univalent in U

with µ(0) = 1.

Remark 3. We note that Fℜn,v(ϑ; 0; µ) = Fℜn,v(ϑ; µ).

Theorem 8. If f ∈ Fℜn,v(ϑ;α; µ) and Re{µ} is bounded inU, then f ∈ Fℜn,v(ϑ; 0; µ) = Fℜn,v(ϑ; µ)
hold for Re{µ(ξ) + (n − 1)} ≥ 0.

Proof. For α = 0, the theorem is trivial and hence we can assume that α , 0. Let

p(ξ) =
ξ[LMn+1,vfi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

(ξ ∈ U).

A quick calculation then reveals that

ξp
′

(ξ)

1
ϑ

ϑ∑
j=1

q j(ξ) + (n − 1)
+ p(ξ) =

ξ[LMn,vfi(ξ)]
′

1
ϑ

ϑ∑
j=1
LMn,vf j(ξ)

,

where q j(ξ) =
ξ[LMn+1,vfi(ξ)]

′

1
ϑ

ϑ∑
j=1
LMn+1,vf j(ξ)

. Also 1
ϑ

ϑ∑
j=1

q j(ξ) ≺F µ(ξ). Since f(ξ) ∈ Fℜn,v(ϑ;α; µ), we have

Fψ(C2×U)
[
J(α; f; f1, f2, ..., fϑ)(ξ)

]
= Fψ(C2×U)


αξp

′

(ξ)

1
ϑ

ϑ∑
j=1

q j(ξ) + (n − 1)
+ p(ξ)

 ≤ Fµ(U)µ(ξ).

Now an application of Lemma 2 gives p(ξ) ≺F µ(ξ) which implies f ∈ Fℜn,v(ϑ; µ). This completes
the proof.

Theorem 9. For α > β ≥ 0, and Re{µ} is bounded inU, then Fℜn,v(ϑ;α; µ) ⊂ Fℜn,v(ϑ; β; µ).

Proof. This theorem’s proof is removed since it is similar to that of Theorem 7.

Remark 4. We can get the same results if we used equation (9).



E. E. Ali et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6979 16 of 18

4. Conclusion

The new findings of this work are related to new classes of analytic normalized functions inU.
The novel results from the investigation reported in this work lead to an advancement in the theory
of fuzzy differential subordination to introduce some classes of univalent functions. The Introduc-
tion in Section 1 covers the Lommel function LMn,vf(ξ), the fundamental ideas required for the
study, and the rationale behind the topic’s investigation. The main finding is presented in Section
2. The current effort offers valuable information to advance the recently initiated research avenues.
The outcome of the present investigation could inspire the use of this operator for introducing other
new classes of analytic functions . In addition to their theoretical significance, Lommel functions
are computationally relevant. Numerical methods for evaluating these functions have been devel-
oped, allowing researchers and engineers to apply them effectively in simulations and predictive
models. The development of algorithms and software libraries that incorporate the computation of
Lommel functions has greatly enhanced our ability to analyze complex systems featuring cylindri-
cal symmetry. Consequently, the Lommel function stands out as a rich area of study, interlinking
pure mathematics with practical technological applications, offering insights that could lead to
advancements in fields ranging from telecommunications to structural engineering.

Acknowledgements

This research has been funded by Scientific Research Deanship at University of Hail-Saudi
Arabia through project number RG-25006.

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