EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 7071
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analytic Subclass Conditions for Operators Involving
the Generalized Imaginary Error Function

A. Alameer1,∗, Tariq Al-Hawary2,∗, Basem Aref Frasin3, Feras Yousef4

1 Department of Mathematics, University of Hafr Al-Batin, Hafr Al Batin 31991,
Saudi Arabia
2 Department of Applied Science, Ajloun College, Al-Balqa Applied University,
Ajloun 26816, Jordan
3 Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
4 Department of Mathematics, The University of Jordan, Amman 11942, Jordan

Abstract. In this paper, we establish necessary and sufficient criteria for the generalized imaginary
error function ϖin (z) to belong to the class F(ℏ1, ℏ2). We also derive an inclusion relation between
the classes Gτ (A1, A2) and F(ℏ1, ℏ2). Furthermore, we provide a necessary and sufficient condition

for the integral operator Gim(z) :=
∫ z

0
ϖin(z)

t dt to lie in the class F(ℏ1, ℏ2).
2020 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Analytic, univalent, error function

1. Introduction and preliminaries

Let AN indicate the class of analytic functions of the form

g(z) = z +
∞∑
ρ=2

aρz
ρ; z ∈ ∆ = {z ∈ C : |z| < 1}. (1)

Also, consider S to be the subclass of AN consisting of functions which are univalent
in ∆.

According to Kamali et al. [1], let F(ℏ1, ℏ2) be a subclass of AN , containing the
functions of the form (1) that satisfy the following analytical requirement:

Re

(
ℏ2z3g′′′(z) + (2ℏ2 + 1)z2g′′(z) + zg′(z)

ℏ2z2g′′(z) + zg′(z)

)
> ℏ1; (ℏ1, ℏ2 ∈ [0, 1), |z| < 1).

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7071

Email addresses: aamalameer@uhb.edu.sa (A. Alameer),
tariq amh@bau.edu.jo (T. Al-Hawary), bafrasin@yahoo.com (B. A. Frasin),
fyousef@ju.edu.jo (F. Yousef)

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



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 2 of 11

Example 1. [2] For some ℏ1 ∈ [0, 1), by choosing ℏ2 = 0 and taking g(z) of the form (1),
let the subclass F(ℏ1, 0) ≡ K(ℏ1) consists of functions in AN that satisfy the inequality:

Re

(
zg′′(z)

g′(z)
+ 1

)
> ℏ1; |z| < 1.

Also, if ℏ1 = ℏ2 = 0, the subclass F(0, 0) ≡ K(0) ≡ K consists of functions in AN
satisfying the inequality:

Re

(
zg′′(z)

g′(z)
+ 1

)
> 0; |z| < 1.

Both subclasses K(ℏ1) and K are well known subclasses of convex functions of order
ℏ1 and convex functions (convex functions of order zero), respectively (see [2]).

For functions of the previously described subclass, we require the following sufficient
and necessary conditions.

Lemma 1. [1] A function g ∈ F(ℏ1, ℏ2) if and only if

∞∑
ρ=2

ρ(ρ− ℏ1) (ℏ2(ρ− 1) + 1) |aρ| ≤ 1− ℏ1. (2)

Definition 1. [3] A function g ∈ AN is said to be in the class Gτ (A1, A2), τ ∈ C\{0},
−1 ≤ A2 < A1 ≤ 1, if it satisfies the inequality:∣∣∣∣ g′(z)− 1

(A1 −A2)τ −A2[g′(z)− 1]

∣∣∣∣ < 1; z ∈ ∆.

Special functions are widely acknowledged as being crucial to geometric function theory
and are employed in numerous applications in applied mathematics, statistics, engineering,
and physics [4–8]. Because these functions are widely used, many academics are interested
in obtaining the geometric features of special functions; see [9–15].

Abramowitz and Stegun [16] defined the following error function

erg (z) =
2√
π

∫ z

0
e−t2dt =

2√
π

∞∑
ρ=0

(−1)ρ z2ρ+1

(2ρ+ 1) ρ!
, (z ∈ C), (3)

whereas the imaginary error function

ergi (z) =
2√
π

∫ z

0
et

2
dt =

2√
π

∞∑
ρ=0

z2ρ+1

(2ρ+ 1) ρ!
, (z ∈ C). (4)

The physics of partial differential equations, probability theory, statistics, and applied
mathematics make extensive use of the error function. The error function is crucial to
estimating the likelihood of seeing a particle in a given area in quantum mechanics. Alzer
[17] and Coman [18] demonstrated several characteristics and inequalities of the error
function, whereas Elbert et. al [19] studied the properties of complementary error function.



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 3 of 11

Figure 1: Real (left) and imaginary (right) parts of the error function erq (z) over the complex plane.

The behavior of the real and imaginary components of erg (z) in the complex plane is
depicted in Figure 1 below. Its relevance in the geometric characterization of subclasses
of analytic functions is motivated by the rich geometric structure it displays, including
symmetry and curvature.

A generalization of the error function (3) is given by

ergn (z) =
n!√
π

∫ z

0
e−tndt, n ∈ N0= N∪{0}

=
n!√
π

∞∑
ρ=0

(−1)ρ znρ+1

(nρ+ 1) ρ!
, (z ∈ C). (5)

In addition, a generalization of the imaginary error function (4) is given by

ergin (z) =
n!√
π

∫ z

0
et

n
dt, n ∈ N0

=
n!√
π

∞∑
ρ=0

znρ+1

(nρ+ 1) ρ!
, (z ∈ C). (6)

From (5) and (6), we get

erg0 (z) =
z

e
√
π
, erg1 (z) =

1− ez√
π

= −ergi1 (z) , erg2 (z) = erg (z) and ergi2 (z) = ergi (z) .

It is clear that the functions ergn (z) and ergin (z) are not members of the class AN .
Thus, it is natural to consider the following functions given by Al-Hawary et. al [20]

En (z) =
√
π

n!
z
(1− 1

n)
n ergn

(
z1/n

)
= z +

∞∑
ρ=2

(−1)ρ−1

((ρ− 1)n+ 1) (ρ− 1)!
zρ, (n ∈ N), (7)



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 4 of 11

and

Ein (z) =
√
π

n!
z(1−

1
n) erf in

(
z1/n

)
= z +

∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!
zρ, (n ∈ N). (8)

Specifically, we get the normalizations E2g = Erg and Ei2g = Ergi given by Ra-
machandran et al. [21] and is the normalization given by Mohammed et al. [22].

From (7) and (8), we have

E1g (z) =
√
πerg1 (z) = 1− ez, Ei1g (z) =

√
πergi1 (z) = ez − 1

and

E2g (z) =
√
πz

2
erg2

(√
z
)
and Ei1g (z) =

√
πz

2
ergi2

(√
z
)
.

Let the function ϖin (z) be defined as

ϖin (z) = 2z − Ein (z) = z −
∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!
zρ; z ∈ ∆, (9)

and consider the linear operator

Iin : AN → AN

defined by the convolution or Hadamard product

Iin(z) = Ein (z) ∗ g(z) = z +
∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!
aρz

ρ. (10)

Recently, several academics determined the necessary and sufficient conditions using
Rabotnov functions [23], generalized Bessel functions [24, 25], Struve functions [26, 27],
hypergeometric functions [28], Pascal distribution series [29, 30], Poisson distribution series
[31], normalized Wright functions [32], ad Touchard polynomials [33].

The remainder of the document is structured as follows. In Section 2, we derive the
necessary and sufficient conditions for the function ϖin to be in the class F(ℏ1, ℏ2). In
Section 3, we will study the action of the function Iin(z) on the class F(ℏ1, ℏ2). Finally, in
Section 4, we give a necessary and sufficient condition for an integral operator Gin(z) :=∫ z
0

ϖin(t)
t dt to be in the class F(ℏ1, ℏ2).

We recall the following lemma, which will be helpful in derive our result in Section 3.

Lemma 2. [3] If g ∈ Gτ (A1, A2) is of the form (1), then

|aρ| ≤ (A1 −A2)
|τ |
ρ
; ρ ∈ N− {1}.

The result is sharp.



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 5 of 11

We employ the following well-known series sums throughout the sequel.

∞∑
ρ=2

1

(ρ− 1) 2ρ
=

1

2
ln 2, (11)

∞∑
ρ=3

1

2ρ (ρ− 1)
=

1

2
ln 2− 1

4
, (12)

and
∞∑
ρ=4

1

2ρ (ρ− 1)
=

1

2
ln 2− 1

4
− 1

16
. (13)

Note that
∞∑
ρ=j

1

(ρ− 1) 2ρ
=

1

2
ln 2−

j−1∑
ρ=2

1

(ρ− 1) 2ρ
, j = 3, 4, · · · . (14)

In addition, we need the following inequalities.

(ρ− 1)n+ 1 > (ρ− 1)n (ρ, n ∈ N) (15)

and
ρ! ≥ 2ρ−1 (ρ ∈ N). (16)

2. Necessary and sufficient conditions

In this section, we give the necessary and sufficient conditions for the function ϖin to
be in the class F(ℏ1, ℏ2).

Theorem 1. If ℏ1, ℏ2 ∈ [0, 1) and n ∈ N, then ϖin (z) is in the class F(ℏ1, ℏ2) if and
only if

[72ℏ2 − 16ℏ1ℏ2 − 6ℏ1 + 22] ln 2− [30ℏ2 + 4(1− ℏ1ℏ2)] ≤ n (1− ℏ1) . (17)

Proof. Since

ϖin (z) = z −
∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!
zρ,

by virtue of Lemma 1 it suffices to show that L(γ, β) ≤ 1− ℏ1, where

L(ℏ1, ℏ2) :=
∞∑
ρ=2

ρ(ρ− ℏ1) (ℏ2(ρ− 1) + 1)
1

((ρ− 1)n+ 1) (ρ− 1)!
.

Writing
ρ = (ρ− 1) + 1, (18)

ρ2 = (ρ− 1)(ρ− 2) + 3(ρ− 1) + 1, (19)



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 6 of 11

ρ3 = (ρ− 1)(ρ− 2)(ρ− 3) + 6(ρ− 1)(ρ− 2) + 7(ρ− 1) + 1, (20)

we get

L(ℏ1, ℏ2) =
∞∑
ρ=2

[ℏ2ρ3 + (1− ℏ2(ℏ1 + 1))ρ2 + ℏ1(ℏ2 − 1))ρ]
1

((ρ− 1)n+ 1) (ρ− 1)!

= ℏ2
∞∑
ρ=2

(ρ− 1)(ρ− 2)(ρ− 3)
1

((ρ− 1)n+ 1) (ρ− 1)!

+ (ℏ2(5− ℏ1) + 1)

∞∑
ρ=2

(ρ− 1)(ρ− 2)
1

((ρ− 1)n+ 1) (ρ− 1)!

+ (2ℏ2(2− ℏ1) + 3− ℏ1)
∞∑
ρ=2

(ρ− 1)
1

((ρ− 1)n+ 1) (ρ− 1)!

+ (1− ℏ1)
∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!

= ℏ2
∞∑
ρ=4

1

((ρ− 1)n+ 1) (ρ− 4)!
+ (ℏ2(5− ℏ1) + 1)

∞∑
ρ=3

1

((ρ− 1)n+ 1) (ρ− 3)!

+ (2ℏ2(2− ℏ1) + 3− ℏ1)
∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 2)!
+ (1− ℏ1)

∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!
.

By (15), we get

L(ℏ1, ℏ2) ≤
ℏ2
n

∞∑
ρ=4

1

(ρ− 1) (ρ− 4)!
+

ℏ2(5− ℏ1) + 1

n

∞∑
ρ=3

1

(ρ− 1) (ρ− 3)!

+
2ℏ2(2− ℏ1) + 3− ℏ1

n

∞∑
ρ=2

1

(ρ− 1) (ρ− 2)!
+

1− ℏ1
n

∞∑
ρ=2

1

(ρ− 1) (ρ− 1)!
.

By (16), we get

L(ℏ1, ℏ2) ≤
32ℏ2
n

∞∑
ρ=4

1

(ρ− 1) 2ρ
+

16 (ℏ2(5− ℏ1) + 1)

n

∞∑
ρ=3

1

(ρ− 1) 2ρ

+
8 (2ℏ2(2− ℏ1) + 3− ℏ1)

n

∞∑
ρ=2

1

(ρ− 1) 2ρ
+

4 (1− ℏ1)
n

∞∑
ρ=2

1

(ρ− 1) 2ρ
.

Using the series sums (11), (12) and (13), we get

L(ℏ1, ℏ2) =
ℏ2
n

(16 ln 2− 8− 2) +
ℏ2(5− ℏ1) + 1

n
(8 ln 2− 4)



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 7 of 11

+
2ℏ2(2− ℏ1) + 3− ℏ1

n
(4 ln 2) +

1− ℏ1
n

(2 ln 2)

=

(
72ℏ2 − 16ℏ1ℏ2 − 6ℏ1 + 22

n

)
ln 2− 30ℏ2 + 4(1− ℏ1ℏ2)

n
.

But this expression is bounded above by 1− ℏ1 if and only if (17) holds.

3. An inclusion property

Making use of Lemma 2, we will study the action of the function Iin(z) on the class
F(ℏ1, ℏ2).

Theorem 2. Let ℏ1, ℏ2 ∈ [0, 1) and n ∈ N. If g ∈ Gτ (A1, A2), then Iin(z) is in the
class F(ℏ1, ℏ2) if

(8ℏ2 − 2ℏ1ℏ2 − ℏ1 + 3) ln 2− 2ℏ2 ≤
n (1− γ)

2(A1 −A2)|τ |
. (21)

Proof. In view of Lemma 1, it suffices to show that

M(ℏ1, ℏ2) :=
∞∑
ρ=2

ρ(ρ− ℏ1) (ℏ2(ρ− 1) + 1)
1

((ρ− 1)n+ 1) (ρ− 1)!
|aρ| ≤ 1− γ.

Since g ∈ Gτ (A1, A2), then by Lemma 2, we have

|aρ| ≤
(A1 −A2) |τ |

ρ
. (22)

Thus, we have

M(ℏ1, ℏ2) =
∞∑
ρ=2

[ℏ2ρ3 + (1− ℏ2(ℏ1 + 1))ρ2 + ℏ1(ℏ2 − 1))ρ]
1

((ρ− 1)n+ 1) (ρ− 1)!
|aρ|

≤ (A1 −A2)|τ |
∞∑
ρ=2

[ℏ2ρ2 + (1− ℏ2(ℏ1 + 1))ρ + ℏ1(ℏ2 − 1))]
1

((ρ− 1)n+ 1) (ρ− 1)!
.

By (18) and (19), we get

M(ℏ1, ℏ2) ≤ (A1 −A2)|τ |
∞∑
ρ=2

[ℏ2(ρ− 1)(ρ− 2) + (ℏ2(2− ℏ1) + 1)(ρ− 1) + 1− ℏ1]

× 1

((ρ− 1)n+ 1) (ρ− 1)!

= (A1 −A2)|τ |

 ∞∑
ρ=3

ℏ2
((ρ− 1)n+ 1) (ρ− 3)!

+

∞∑
ρ=2

ℏ2(2− ℏ1) + 1

((ρ− 1)n+ 1) (ρ− 2)!



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 8 of 11

+
∞∑
ρ=2

1− ℏ1
((ρ− 1)n+ 1)ρ!

 .

By (15) and (16), we get

M(ℏ1, ℏ2) ≤
(A1 −A2)|τ |

n

16

∞∑
ρ=3

ℏ2
(ρ− 1) 2ρ

+ 8

∞∑
ρ=2

ℏ2(2− ℏ1) + 1

(ρ− 1) 2ρ
+ 4

∞∑
ρ=2

1− ℏ1
(ρ− 1) 2ρ

 .

By (11) and (12), we get

M(ℏ1, ℏ2) ≤
(A1 −A2)|τ |

n
[2 (8ℏ2 − 2ℏ1ℏ2 − ℏ1 + 3) ln 2− 4ℏ2] .

But this last expression is bounded by 1− ℏ1, if (21) holds.

4. An integral operator

Theorem 3. Let ℏ1, ℏ2 ∈ [0, 1) and n ∈ N.The integral operator

Gin(z) :=
∫ z

0

ϖin(t)

t
dt, z ∈ ∆, (23)

is in the class F(ℏ1, ℏ2) if and only if the inequality

(8ℏ2 − 2ℏ1ℏ2 − ℏ1 + 3) ln 2− 2ℏ2 ≤
n (1− γ)

2
(24)

holds.

Proof. According to (9) it follows that

Gin(z) = z −
∞∑
ρ=2

1

((ρ− 1)n+ 1) (ρ− 1)!

zρ

ρ
, z ∈ ∆.

Using Lemma 1, the function Gin(z) belongs to F(ℏ1, ℏ2) if and only if

∞∑
ρ=2

[ρ(ρ− ℏ1) (ℏ2(ρ− 1) + 1)]
1

ρ((ρ− 1)n+ 1) (ρ− 1)!
≤ 1− ℏ1.

By a similar proof of Theorem 2 we get that Gin ∈ F(ℏ1, ℏ2) if and only if (24) holds.

We obtain many corollaries by specializing the parameters ℏ1 and ℏ2 in our theorems,
for example, if ℏ2 = 0, we get the following corollary.

Corollary 1. If n ∈ N and ℏ1 ∈ [0, 1), then



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 9 of 11

(i) ϖin (z) ∈ K(ℏ1) if and only if

[22− 6ℏ1] ln 2− 4 ≤ n (1− ℏ1) .

(ii) For g ∈ Gτ (A1, A2), Iin(z) ∈ K(ℏ1) if and only if

(3− ℏ1) ln 2 ≤ n (1− ℏ1)
2(A1 −A2)|τ |

.

(iii) Gin(z) ∈ K(ℏ1) if and only if

(3− ℏ1) ln 2 ≤ n (1− ℏ1)
2

.

5. Conclusion

For the generalized normalized imaginary error function ϖin to belong to the class
F(ℏ1, ℏ2) of analytic functions defined on the open unit disk ∆, we establish necessary
and sufficient criteria. We also investigate the action of the function Iin(z) on the class
F(ℏ1, ℏ2). Furthermore, we derive a necessary and sufficient condition for the integral
operator Gin(z) to lie in the same class. This study may further motivate researchers to
derive new criteria under which the generalized normalized imaginary error function ϖin
belongs to other families of analytic functions defined on ∆.

References

[1] M. Kamali and S. Akbulut. On a subclass of certain convex functions with negative
coefficients. Applied mathematics and computation, 145(2-3):341–350, 2003.

[2] H. Silverman. Univalent functions with negative coefficients. Proceedings of the Amer-
ican mathematical society, 51(1):109–116, 1975.

[3] K. K. Dixit and S. K. Pal. On a class of univalent functions related to complex order.
Indian Journal of Pure and Applied Mathematics, 26(9):889–896, 1995.

[4] A. A. Amourah, F. Yousef, T. Al-Hawary, and M. Darus. On a class of p valent non-
bazilevič functions of order µ+ iβ. International Journal of Mathematical Analysis,
10(15):701–710, 2016.

[5] A. A. Attiya. Some applications of Mittag-Leffler function in the unit disk. Filomat,
30(7):2075–2081, 2016.

[6] F. Yousef, A. A. Amourah, and M. Darus. Differential sandwich theorems for p-
valent functions associated with a certain generalized differential operator and integral
operator. Italian Journal of Pure and Applied Mathematics, 36:543–556, 2016.

[7] M. El-Ityan, M. A. Sabri, H. Suha, B. Basem, T. Al-Hawary, and F. Yousef. Third-
order Hankel determinant for a class of bi-univalent functions associated with sine
function. Mathematics, 13(17):2887, 2025.

[8] T. Al-Hawary, B. A. Frasin, and F. Yousef. Coefficients estimates for certain classes
of analytic functions of complex order. Afrika Matematika, 29(7):1265–1271, 2018.



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 10 of 11

[9] A. O. Mostafa. A study on starlike and convex properties for hypergeometric func-
tions. J. Inequal. Pure Appl. Math., 10(3):1–16, 2009.

[10] E. Merkes and B. T. Scott. Starlike hypergeometric functions. Proceedings of the
American Mathematical Society, 12(6):885–888, 1961.

[11] M. Illafe, A. Hussen, M. H. Mohd, and F. Yousef. On a subclass of bi-univalent
functions affiliated with Bell and Gegenbauer polynomials. Boletim da Sociedade
Paranaense de Matematica, 43:1–10, 2025.

[12] M. Illafe, M. H. Mohd, F. Yousef, and S. Supramaniam. A subclass of bi-univalent
functions defined by asymmetric q-derivative operator and Gegenbauer polynomials.
European Journal of Pure and Applied Mathematics, 17(4):2467–2480, 2024.

[13] M. Illafe, M. Haji Mohd, F. Yousef, and S. Supramaniam. Bounds for the second
Hankel determinant of a general subclass of bi-univalent functions. International
Journal of Mathematics, Engineering, and Management Sciences, 9(5):1226–1239,
2024.

[14] N. E. Cho, S. Y. Woo, and S. Owa. Uniform convexity properties for hypergeometric
functions. Fractional Calculus and Applied Analysis, 5(3):303–314, 2002.

[15] T. Al-Hawary, A. Amourah, and B. A. Frasin. Fekete–Szegö inequality for bi-univalent
functions by means of Horadam polynomials. Boletin de la Sociedad Matematica
Mexicana, 27(3):79, 2021.

[16] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas,
graphs and matematical tables. Dorer Publications Inc., New York, 1965.

[17] H. Alzer. Error functions inequalities. Advances in Computational Mathematics,
33(3):349–379, 2010.

[18] D. Coman. The radius of starlikeness for error function. Stud. Univ. Babes-Bolyai
Math, 36(2):13–16, 1991.

[19] A. Elbert and A. Laforgia. The zeros of the complementary error function. Numerical
Algorithms, 49(1):153–157, 2008.

[20] T. Al-Hawary, B.A. Frasin and J. Salah. Comprehensive subfamilies of bi-univalent
functions defined by error function subordinate to Euler polynomials. Symmetry,
17(2):256, 2025.

[21] C. Ramachandran, L. Vanitha, and S. Kanas. Certain results on q-starlike and q-
convex error functions. Mathematica Slovaca, 68(2):361–368, 2018.

[22] N. H. Mohammed, N. E. Cho, E. A. Adegani, and T. Bulboaca. Geometric properties
of normalized imaginary error function. Studia Universitatis Babes-Bolyai Matemat-
ica, 67(2):455–462, 2022.

[23] A. Fallatah, T. Al-Hawary, M. O. Massa’deh, and F. Yousef. Subfamilies of analytic
functions associated with Rabotnov function. International Journal of Neutrosophic
Science, 26(1):33–39, 2025.

[24] B. A. Frasin, F. Yousef, T. Al-Hawary, and I. Aldawish. Application of generalized
Bessel functions to classes of analytic functions. Afrika Matematika, 32(3):431–439,
2021.

[25] S. R. Mondal and A. Swaminathan. Geometric properties of generalized Bessel func-
tions. Bull. Malays. Math. Sci. Soc., 35(1):179–194, 2012.



A. Alameer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7071 11 of 11

[26] B. A. Frasin, T. Al-Hawary, F. Yousef, and I. Aldawish. On subclasses of analytic
functions associated with Struve functions. Nonlinear Functional Analysis and Ap-
plications, 27(1):99–110, 2022.

[27] T. Janani and G. Murugusundaramoorthy. Inclusion results on subclasses of starlike
and convex functions associated with Struve functions. Italian Journal of Pure and
Applied Mathematics, 32:467–476, 2014.

[28] B.A. Frasin, T. Al-Hawary, and F. Yousef. Necessary and sufficient conditions for
hypergeometric functions to be in a subclass of analytic functions. Afrika Matematika,
30(1):223–230, 2019.

[29] G. Murugusundaramoorthy, B. A. Frasin, and T. Al-Hawary. Uniformly convex spiral
functions and uniformly spirallike function associated with Pascal distribution series.
Mat. Bohem., 23:1–11, 2021.

[30] S. M. El-Deeb, T. Bulboacă and J. Dziok. Pascal distribution series connected
with certain subclasses of univalent functions. Kyungpook Mathematical Journal,
59(2):301–314, 2019.

[31] R. M. El-Ashwah and W. Y Kota. Some condition on a poisson distribution series to
be in subclasses of univalent functions. Acta Univ. Apulensis, 51:89–103, 2017.

[32] T. Al-Hawary, I. Aldawish, B. A. Frasin, O. Alkam, and F. Yousef. Necessary and
sufficient conditions for normalized Wright functions to be in certain classes of analytic
functions. Mathematics, 10(24):4693, 2022.

[33] T. Al-Hawary, M. Illafe, and F. Yousef. Certain constraints for functions provided
by Touchard polynomials. International Journal of Mathematics and Mathematical
Sciences, 2025(1):2581058, 2025.