EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6745
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Geometric Characterizations of Imaginary Error
Functions in Subclasses of Spirallike Analytic Functions

Feras Yousef1, Maryam M. Alholi2, Tariq Al-Hawary3,∗

1 Department of Mathematics, The University of Jordan, Amman 11942, Jordan
2 Applied College, Taibah University, Saudi Arabia
3 Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun
26816, Jordan

Abstract. In this paper, we investigate the geometric behavior of the generalized normalized
imaginary error function Υik (z) and the associated convolution operator Iik(z) within the frame-
work of analytic function theory. Specifically, we establish necessary and sufficient conditions under
which these functions belong to the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ) of spirallike and convex
spirallike analytic functions, respectively. Additionally, we derive sharp criteria for an integral
operator involving Υik (z) to be a member of these subclasses. These results extend and generalize
several known findings and may inspire further applications of the imaginary error function in
geometric function theory.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic, univalent, spirallike, convex spirallike, error function

1. Introduction and Preliminaries

Let E symbolize for the class of analytic functions of the form:

q(z) = z +
∞∑
ϵ=2

βϵz
ϵ , z ∈ Γ = {z ∈ C : |z| < 1}. (1)

Further, let NE be a subclass of E consisting of functions of the form:

q(z) = z −
∞∑
ϵ=2

βϵz
ϵ , βϵ ≥ 0, z ∈ Γ. (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6745

Email addresses: fyousef@ju.edu.jo (F. Yousef),
mholi@taibahu.edu.sa (M. M. Alholi), tariq amh@bau.edu.jo (T. Al-Hawary)

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



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 2 of 13

A function q ∈ E is spirallike if

R

(
e−iϑ zq

′(z)

q(z)

)
> 0, |ϑ| < π/2, z ∈ Γ.

Also, q(z) is convex spirallike if zq′(z) is spirallike.
Selvaraj and Geetha [1] introduced the subclasses of uniformly spirallike functions

SP(ϑ, ζ) and CSP(ϑ, ζ), as given in the following definition.

Definition 1. A function q of the form (1) is said to be in the subclass SP(ϑ, ζ), if it
satisfies the following condition:

R

{
e−iϑ

(
zq′(z)

q(z)

)}
≥
∣∣∣∣zq′(z)q′(z)

− 1

∣∣∣∣+ ζ (z ∈ Γ; |ϑ| < π/2 ; 0 ≤ ζ < 1)

and q ∈ CSP(ϑ, ζ) iff zq′(z) ∈ SP(ϑ, ζ), which is equivalent the following condition:

R

{
e−iϑ

(
1 +

zq′′(z)

q′(z)

)}
≥
∣∣∣∣zq′′(z)q′(z)

∣∣∣∣+ ζ (z ∈ Γ; |ϑ| < π/2 ; 0 ≤ ζ < 1).

We write
SPE(ϑ, ζ) = SP(ϑ, ζ) ∩NE

and
CSPE(ϑ, ζ) = CSP(ϑ, ζ) ∩NE .

We note that, for ζ = 0, the subclasses of uniformly spirallike SP(ϑ, 0) = SP(ϑ) and
uniformly convex spirallike CSP(ϑ, 0) = CSP(ϑ) introduced by Ravichandran et al. [2].
For ϑ = 0, the subclasses SP(ϑ) = SP and CSP(ϑ) = CSP introduced and studied by

Rønning [3]. For more intriguing discoveries of some related subclasses of consistently
uniformly spirallike and uniformly convex spirallike, see the works of Al-Hawary et al.
[4, 5], Bharati et al. [6], Frasin et al. [7], Goodman [8], Kanas and Wisniowska [9].

Definition 2. [10] A function h ∈ E is said to be in the class Gτ (C1, C2), τ ∈ C\{0},
−1 ≤ C2 < C1 ≤ 1, if it satisfies the condition∣∣∣∣ q′(z)− 1

(C1 − C2)τ − C2[q′(z)− 1]

∣∣∣∣ < 1, z ∈ Γ.

If we put τ = 1, C1 = ϱ and C2 = −ϱ(0 < ϱ ≤ 1), we get the class of functions q ∈ E
satisfying the condition ∣∣∣∣q′(z)− 1

q′(z) + 1

∣∣∣∣ < ϱ, (z ∈ Γ, 0 < ϱ ≤ 1)

which was studied by (among others) Caplinger and Causey [11].
It is commonly known that special functions are crucial to the theory of geometric

functions, and that their use is not restricted to the theory of geometric functions; they



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 3 of 13

are used in a wide variety of problems and in other areas of mathematics and the applied
sciences, see [12–23].

The error function erq defined by Abramowitz and Stegun [24] as:

erq (z) =
2√
π

∫ z

0
e−t2dt =

2√
π

∞∑
ϵ=0

(−1)ϵ z2ϵ+1

(2ϵ+ 1) ϵ!
, (z ∈ C), (3)

whereas the imaginary error function

erqi (z) =
2√
π

∫ z

0
et

2
dt =

2√
π

∞∑
ϵ=0

z2ϵ+1

(2ϵ+ 1) ϵ!
, (z ∈ C). (4)

The error function is widely used in statistics, probability theory, applied mathematics,
and the physics of partial differential equations. In quantum physics, the error function is
an essential tool for calculating the probability of observing a particle in a specific location.
While Alzer [25] and Coman [26] demonstrated numerous features and inequalities of
the error function, Elbert et al. [27] examined the characteristics of the complementary
error function. Figure 1 below illustrates the behavior of the real and imaginary parts
of erq (z) in the complex plane. It reveals rich geometric structure, including symmetry
and curvature, which motivates its role in the geometric characterization of subclasses of
analytic functions.

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

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

erqk (z) =
k!√
π

∫ z

0
e−tkdt, k ∈ N0= N∪{0}

=
k!√
π

∞∑
ϵ=0

(−1)ϵ zkϵ+1

(kϵ+ 1) ϵ!
, (z ∈ C). (5)



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 4 of 13

And a generalization of the imaginary error function given by (4) is defined by

erqik (z) =
k!√
π

∫ z

0
et

k
dt, k ∈ N0

=
k!√
π

∞∑
ϵ=0

zkϵ+1

(kϵ+ 1) ϵ!
, (z ∈ C). (6)

From (5) and (6), we get

erq0 (z) =
z

e
√
π
, erq1 (z) =

1− ez√
π

= −erqi1 (z) , erq2 (z) = erq (z) and erqi2 (z) = erqi (z) .

The functions erqk (z) and erqik (z) are not in the class E . Therefore, we will consider
the following functions given by Al-Hawary et al. [28].

εk (z) =

√
π

k!
z
(1− 1

k )
k erqk

(
z1/k

)
= z +

∞∑
ϵ=2

(−1)ϵ−1

((ϵ− 1) k + 1) (ϵ− 1)!
zϵ, (k ∈ N), (7)

and

εik (z) =

√
π

k!
z(1−

1
k )erqik

(
z1/k

)
= z +

∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!
zϵ, (k ∈ N). (8)

From (7) and (8), we get

ε1 (z) =
√
πerq1 (z) = 1− ez, εi1 (z) =

√
πerqi1 (z) = ez − 1

and

ε2 (z) =

√
πz

2
erq2

(√
z
)
and εi1 (z) =

√
πz

2
erqi2

(√
z
)
.

Let the function Υik (z) be defined as:

Υik (z) = 2z − εik (z) = z −
∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!
zϵ, z ∈ Γ, (9)

and the linear operator
Iik : E → E

defined as:

Iik(z) = εik (z) ∗ q(z) = z +

∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!
βϵz

ϵ. (10)

Inspired by the works of several researchers who have employed a variety of special
functions to identify certain conditions to belong to subclasses of analytic functions (see,
[29–36]), we will determine some conditions for the error functions Υik (z) and Iik(z), and
an integral operator to belong to the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ).

The lemmas listed below will be useful in deriving our main findings.



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 5 of 13

Lemma 1. (see [1]) (i) A sufficient condition for a function q of the form (1) to be in
the subclass SP(ϑ, ζ) is

∞∑
ϵ=2

(2ϵ− ζ − cosϑ) |βϵ| ≤ cosϑ− ζ (|ϑ| < π/2 ; 0 ≤ ζ < 1), (11)

and a necessary and sufficient condition for a function q of the form (2) to be in the
subclass SPE(ϑ, ζ) is that the condition (11) is satisfied. In particular, when ζ = 0, we
obtain a sufficient condition for a function q of the form (1) to be in the subclass SP(ϑ)
is

∞∑
ϵ=2

(2ϵ− cosϑ) |βϵ| ≤ cosϑ (|ϑ| < π/2), (12)

and a necessary and sufficient condition for a function q of the form (2) to be in the
subclass SPE(ϑ)is that the condition (12) is satisfied.

(ii) A sufficient condition for a function q of the form (1) to be in the subclass
CSP(ϑ, ζ)is

∞∑
ϵ=2

ϵ(2ϵ− ζ − cosϑ) |βϵ| ≤ cosϑ− ζ (|ϑ| < π/2 ; 0 ≤ ζ < 1) (13)

and a necessary and sufficient condition for a function q of the form (2) to be in the
subclass CSPE(ϑ, ζ) is that the condition (13) is satisfied. In particular, when ζ = 0, we
obtain a sufficient condition for a function q of the form (1) to be in the subclass CSP(ϑ)
is that

∞∑
ϵ=2

ϵ(2ϵ− cosϑ) |βϵ| ≤ cosϑ (|ϑ| < π/2) (14)

and a necessary and sufficient condition for a function q of the form (2) to be in the
subclass CSPE(ϑ)is that the condition (14) is satisfied.

Lemma 2. [10] If q of the form (1) and q ∈ Gτ (C1, C2), then

|βϵ| ≤
(C1 − C2) |τ |

ϵ
, ϵ ∈ N− {1}. (15)

The result is sharp for the function q(z) given by

q(z) =

z∫
0

(
1 +

(C1 − C2)τt
ϵ−1

1 + C2tϵ−1

)
dt ( z ∈ Γ, ϵ ≥ 2). (16)

We need the following well-known series sums to prove our main findings.

∞∑
ϵ=2

1

(ϵ− 1) 2ϵ
=

1

2
ln 2 (17)



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 6 of 13

and
∞∑
ϵ=3

1

2ϵ (ϵ− 1)
=

1

2
ln 2− 1

4
. (18)

Note that
∞∑
ϵ=d

1

(ϵ− 1) 2ϵ
=

1

2
ln 2−

d−1∑
ϵ=2

1

(ϵ− 1) 2ϵ
, d = 3, 4, · · · . (19)

The following inequalities are also required

(ϵ− 1)k + 1 > (ϵ− 1)k (ϵ, k ∈ N) (20)

and
ϵ! ≥ 2ϵ−1 (ϵ ∈ N). (21)

2. Necessary and sufficient conditions for the function Υik

In this section, we find some necessary and sufficient conditions for the function Υik
to be in the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ).

Theorem 1. If k ∈ N, then Υik (z) is in the subclass SPE(ϑ, ζ) if and only if

2(6− ζ − cosϑ) ln 2 ≤ k (cosϑ− ζ) . (22)

Proof. Since

Υik (z) = z −
∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!
zϵ, (23)

by virtue of (11) it suffices to show that L1(ϑ, ζ) ≤ cosϑ− ζ, where

L1(ϑ, ζ) =

∞∑
ϵ=2

[2ϵ− ζ − cosϑ]
1

((ϵ− 1) k + 1) (ϵ− 1)!
.

Writing ϵ = (ϵ− 1) + 1, we get

L1(ϑ, ζ) =
∞∑
ϵ=2

2(ϵ− 1)

((ϵ− 1) k + 1) (ϵ− 1)!
+

∞∑
ϵ=2

2− ζ − cosϑ

((ϵ− 1) k + 1) (ϵ− 1)!

=

∞∑
ϵ=2

2

((ϵ− 1) k + 1) (ϵ− 2)!
+

∞∑
ϵ=2

2− ζ − cosϑ

((ϵ− 1) k + 1) (ϵ− 1)!
.

By (20), we get

L1(ϑ, ζ) ≤
2

k

∞∑
ϵ=2

1

(ϵ− 1) (ϵ− 2)!
+

2− ζ − cosϑ

k

∞∑
ϵ=2

1

(ϵ− 1) (ϵ− 1)!
.



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 7 of 13

By (21), we get

L1(ϑ, ζ) ≤
16

k

∞∑
ϵ=3

1

(ϵ− 1) 2ϵ
+

4 (2− ζ − cosϑ)

k

∞∑
ϵ=2

1

(ϵ− 1) 2ϵ
.

Using the series sums (17), we get

L1(ϑ, ζ) ≤
8

k
(ln 2) +

2 (2− ζ − cosϑ)

k
(ln 2) =

2(6− ζ − cosϑ)

k
ln 2.

However, if and only if (22) holds, the last expression is bounded above by cosϑ− ζ.

Theorem 2. If k ∈ N, then Υik (z) is in the subclass CSPE(ϑ, ζ) if and only if

2(22− 3ζ − 3 cosϑ) ln 2 ≤ 8 + k(cosϑ− ζ). (24)

Proof. Since Υik (z) is given by (23) and by virtue (13), it suffices to show that
L2(λ1, λ2) ≤ cosϑ− ζ, where

L2(ϑ, ζ) =
∞∑
ϵ=2

ϵ [2ϵ− ζ − cosϑ]
1

((ϵ− 1) k + 1) (ϵ− 1)!

=

∞∑
ϵ=2

[
2ϵ2 − (ζ + cosϑ) ϵ

] 1

((ϵ− 1) k + 1) (ϵ− 1)!
.

Writing
ϵ = (ϵ− 1) + 1, (25)

and
ϵ2 = (ϵ− 1)(ϵ− 2) + 3(ϵ− 1) + 1, (26)

we get

L2(ϑ, ζ) = 2
∞∑
ϵ=2

(ϵ− 1)(ϵ− 2)

((ϵ− 1) k + 1) (ϵ− 1)!

+ (6− ζ − cosϑ)

∞∑
ϵ=2

ϵ− 1

((ϵ− 1) k + 1) (ϵ− 1)!

+ (2− ζ − cosϑ)
∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!

= 2

∞∑
ϵ=3

1

((ϵ− 1) k + 1) (ϵ− 3)!
+ (6− ζ − cosϑ)

∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 2)!

+ (2− ζ − cosϑ)
∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!
.



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 8 of 13

By (20), we get

L2(ϑ, ζ) ≤
2

k

∞∑
ϵ=3

1

(ϵ− 1) (ϵ− 3)!
+

(6− ζ − cosϑ)

k

∞∑
ϵ=2

1

(ϵ− 1) (ϵ− 2)!

+
(2− ζ − cosϑ)

k

∞∑
ϵ=2

1

(ϵ− 1) (ϵ− 1)!
.

By (21), we get

L2(ϑ, ζ) ≤
32

k

∞∑
ϵ=3

1

(ϵ− 1) 2ϵ
+

8(6− ζ − cosϑ)

k

∞∑
ϵ=2

1

(ϵ− 1) 2ϵ

+
4(2− ζ − cosϑ)

k

∞∑
ϵ=2

1

(ϵ− 1) 2ϵ
.

Using the series sums (17) and (18), we get

L2(ϑ, ζ) ≤
1

k
(16 ln 2− 8) +

6− ζ − cosϑ

k
(4 ln 2) +

2− ζ − cosϑ

k
(2 ln 2)

=
2(22− 3ζ − 3 cosϑ)

k
ln 2− 8

k
.

However, if and only if (24) holds, the last expression is bounded above by cosϑ− ζ.

3. Necessary and sufficient conditions for the convolution operator Iik(z)

In this section, we find sufficient conditions for the convolution operator Iik(z) to be
in the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ).

Theorem 3. Let k ∈ N. If q ∈ Gτ (C1, C2), then Iik(z) is in the subclass SPE(ϑ, ζ) if

(C1 − C2)|τ |(4− ζ − cosϑ) ln 2 ≤ k (cosϑ− ζ) . (27)

Proof. In view of (11), it suffices to show that

M1(ϑ, ζ) =

∞∑
ϵ=2

[2ϵ− ζ − cosϑ]
1

((ϵ− 1) k + 1) (ϵ− 1)!
|βϵ| ≤ cosϑ− ζ.

Since q ∈ Gτ (C1, C2), then by virtue (15), we have

M1(ϑ, ζ) ≤ (C1 − C2)|τ |

( ∞∑
ϵ=2

2

((ϵ− 1) k + 1) (ϵ− 1)!
−

∞∑
ϵ=2

ζ + cosϑ

((ϵ− 1) k + 1)ϵ!

)
.

By (20) and (21), we get

M1(ϑ, ζ) ≤
(C1 − C2)|τ |

k

(
8

∞∑
ϵ=2

1

(ϵ− 1) 2ϵ
− 2

∞∑
ϵ=2

ζ + cosϑ

(ϵ− 1) 2ϵ

)
.



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 9 of 13

By (17), we get

M1(ϑ, ζ) ≤
(C1 − C2)|τ |

k
(4− ζ − cosϑ) ln 2.

However, the last expression is bounded above by cosϑ− ζ if (27) holds.

Theorem 4. Let k ∈ N. If q ∈ Gτ (C1, C2), then Iik(z) is in the subclass CSPE(ϑ, ζ) if

2(C1 − C2)|τ | (6− ζ − cosϑ) ln 2 ≤ k (cosϑ− ζ) . (28)

Proof. By view of (13), it suffices to show that

M2(ϑ, ζ) =
∞∑
ϵ=2

ϵ [2ϵ− ζ − cosϑ]
1

((ϵ− 1) k + 1) (ϵ− 1)!
|βϵ| ≤ cosϑ− ζ.

Since q ∈ Gτ (C1, C2), then by virtue (15), we have

M2(ϑ, ζ) ≤ (C1 − C2)|τ |

( ∞∑
ϵ=2

2ϵ

((ϵ− 1) k + 1) (ϵ− 1)!
−

∞∑
ϵ=2

ζ + cosϑ

((ϵ− 1) k + 1) (ϵ− 1)!

)
.

By (25), we get

M2(ϑ, ζ) ≤ (C1 − C2)|τ |

( ∞∑
ϵ=2

2(ϵ− 1)

((ϵ− 1) k + 1) (ϵ− 1)!
+

∞∑
ϵ=2

2− ζ − cosϑ

((ϵ− 1) k + 1) (ϵ− 1)!

)

= (C1 − C2)|τ |

( ∞∑
ϵ=2

2

((ϵ− 1) k + 1) (ϵ− 2)!
+

∞∑
ϵ=2

2− ζ − cosϑ

((ϵ− 1) k + 1) (ϵ− 1)!

)
.

By (20) and (21), we get

M2(ϑ, ζ) ≤
(C1 − C2)|τ |

k

(
16

∞∑
ϵ=2

1

(ϵ− 1) 2ϵ
+ 4

∞∑
ϵ=2

2− ζ − cosϑ

(ϵ− 1) 2ϵ

)
.

By (17), we get

M2(ϑ, ζ) ≤
2(C1 − C2)|τ |

k
(6− ζ − cosϑ) ln 2.

However, the last expression is bounded above by cosϑ− ζ if (28) holds.



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 10 of 13

4. Necessary and sufficient conditions for the integral operator Lik(z)

In this section, we find necessary and sufficient conditions of the integral operator

Lik(z) :=

∫ z

0

Υik(t)

t
dt, z ∈ Γ, (29)

to be in the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ).

Theorem 5. Let k ∈ N. The integral operator Lik(z) is in the subclass SPE(ϑ, ζ) if and
only if the inequality

(4− ζ − cosϑ) ln 2 ≤ k(cosϑ− ζ) (30)

holds.

Proof. According to (9) it follows that

Lik(z) = z −
∞∑
ϵ=2

1

((ϵ− 1) k + 1) (ϵ− 1)!

zϵ

ϵ
, z ∈ Γ. (31)

From (11), the integral operator Lik(z) belongs to SPE(ϑ, ζ) if and only if

∞∑
ϵ=2

[2ϵ− ζ − cosϑ]
1

ϵ((ϵ− 1) k + 1) (ϵ− 1)!

=
∞∑
ϵ=2

2

((ϵ− 1) k + 1) (ϵ− 1)!
−

∞∑
ϵ=2

ζ + cosϑ

((ϵ− 1) k + 1)ϵ!
≤ cosϑ− ζ.

By a similar proof of Theorem 3, we get that Lik(z) ∈ SPE(ϑ, ζ) if and only if (30)
holds.

Theorem 6. Let k ∈ N. The integral operator Lik(z) is in the subclass CSPE(ϑ, ζ) if and
only if the inequality (22) holds.

Proof. Since Lik(z) is given by (31) and in view (13), the integral operator Lik(z)
belongs to CSPE(ϑ, ζ) if and only if

∞∑
ϵ=2

ϵ [2ϵ− ζ − cosϑ]
1

ϵ((ϵ− 1) k + 1) (ϵ− 1)!

=
∞∑
ϵ=2

2ϵ

((ϵ− 1) k + 1) (ϵ− 1)!
−

∞∑
ϵ=2

ζ + cosϑ

((ϵ− 1) k + 1) (ϵ− 1)!
≤ cosϑ− ζ.

By a similar proof of Theorem 4, we get that Lik(z) ∈ CSPE(ϑ, ζ) if and only if (22)
holds.

Remark 1. Particularization of the parameters ϑ and ζ in our theorems, we get several
subresults related to the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ). For example, if ζ = 0 or
ϑ = 0, we get many subresults for the subclasses SPE(ϑ), CSPE(ϑ), SPE(ζ) and CSPE(ϑ).



F. Yousef, M. M. Alholi, T. Al-Hawary / Eur. J. Pure Appl. Math, 18 (4) (2025), 6745 11 of 13

5. Conclusion

In this paper, we have established several geometric criteria for the generalized nor-
malized imaginary error function Υik, its associated convolution operator Iik(z), and an
integral operator involving Υik to belong to the subclasses SPE(ϑ, ζ) and CSPE(ϑ, ζ) of
spirallike and convex spirallike analytic functions defined in the open unit disk Γ.

By employing a combination of analytic techniques and coefficient-based inequalities,
we derived sharp necessary and sufficient conditions in terms of the parameters k, ϑ, and
ζ, thereby extending known results in the theory of geometric function classes. Our
findings highlight the analytical richness of the imaginary error function and its potential
for characterizing function spaces through generalized transformations. This study may
encourage researchers to include the generalized normalized imaginary error function in
other classes of analytic functions defined on Γ and creating new necessary and sufficient
conditions.

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