EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5678
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Subfamilies of Bi-Univalent Functions Defined by
Imaginary Error Functions Subordinate to Horadam

Polynomials

Tariq Al-Hawary1,∗, Basem Aref Frasin2, Ala Amourah3,4, Jamal Salah5,∗

1 Department of Applied Science, Ajloun College, Al Balqa Applied University,
Ajloun 26816. Jordan
2 Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
3 Mathematics Education Program, Faculty of Education and Arts, Sohar University,
Sohar 3111, Oman
4 Applied Science Research Center. Applied Science Private University, Amman, Jordan
5 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42,
Post Code No. 400 Ibra, Sultanate of Oman

Abstract. Several different subfamilies of the bi-univalent function family Ω were introduced and
studied by numerous researchers using special functions. In the present paper, utilizing the imag-
inary error function, we introduce and study a new subfamily FΩ(s, r, u, y, t, λ, τ) of bi-univalent
functions in the open unit disk Θ, which are connected to the Horadam polynomials, and determine
initial coefficients in the Maclaurin series of functions in this subfamily. Moreover, we determine
the Fekete-Szegö inequality for functions in this subfamily. The parameters employed in our major
results are specialized, and several fresh outcomes are shown to follow.

2020 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Analytic, bi-univalent, Fekete-Szegö, Horadam, imaginary error
function

1. Introduction and Preliminaries

Ordinary differential equations that meet model constraints are frequently solved using
orthogonal polynomials [19] in mathematical model solving. Orthogonal polynomials are
useful in physics and engineering and are significant in modern mathematics. The impor-
tance of these polynomials in issues pertaining to approximation theory is well known.
They are present in quantum physics, approximation theory, probability theory, interpo-
lation, differential equation theory, and mathematical statistics. They also model and

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5678

Email addresses: tariq amh@bau.edu.jo (T. Al-Hawary), bafrasin@yahoo.com (B. A. Frasin),
AAmourah@su.edu.om (A. Amourah), damous73@yahoo.com (J. Salah)

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



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 2 of 12

analyze complicated systems and data sets in the fields of signal processing, image pro-
cessing, and data analysis (see [5, 11]).

The pair of polynomials Jϵ and Jε, of order ϵ and ε, respectively, are orthogonal if

⟨Jϵ, Jε⟩ =
∫ σ2

σ1

Jϵ(y)Jε(y)r(y)dy = 0, for ϵ ̸= ε, (1)

The integral of all finite order polynomials Jϵ(y) is properly defined since r(y) is a
non-negative function in the interval (σ1, σ2).

Several families of orthogonal polynomials are well-known, such as the Jacobi, La-
guerre, Legendre, Hermite, and Chebyshev families. Orthogonal polynomials have many
practical qualities and applications, and each family has its own weight function and in-
terval.

The Recurrence relations define the Horadam polynomials as the family of polynomials
that is a generalization of the Fibonacci and Lucas polynomials. Murray S. Klamkin
Horadam, an Australian mathematician, is credited with their introduction in 1978, hence
its name.

Numerous intriguing characteristics of Horadam polynomials and their relationships
to other branches of mathematics, such as algebraic geometry, combinatorics, and number
theory.

Horzum and Kocer (2009) examined the Horadam polynomials hα(y), which are as-
certained by the following recurrence relation [21].

hα(y) = ryhα−1(y) + uhα−2(y), α ∈ {3, 4, · · · } , (2)

with
h1(y) = s, h2(y) = ty and h3(y) = rty2 + su, s, r, u, t ∈ R. (3)

The Horadam polynomials hα(y) have the generator

Υ(y, ℘) =
∞∑
α=1

hα(y)℘
α−1 =

s+ (t− sr)y℘

1− ry℘− u℘2
. (4)

Remark 1. Various polynomials can be obtained from the Horadam polynomials hα(y)
for specific values of s, t, r and u (see [18, 21]). For instance:

(i) When s = t = r = u = 1, we receive the Fibonacci polynomials Fα(y);

(ii) When s = 2 and t = r = u = 1, we receive the Lucas polynomials Lα(y);

(iii) When s = t = 1, r = 2 and u = −1, we receive the first kind of Chebyshev polyno-
mials Tα(y);

(iv) When s = 1, t = r = 2 and u = −1, we receive the second kind of Chebyshev
polynomials Uα(y);

(v) When s = u = 1 and t = r = 2, we receive the Pell polynomials PLα(y);



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 3 of 12

(vi) When s = t = r = 2 and u = 1, we receive the first kind of Pell-Lucas polynomials
uα(y).

Let AU be the family of analytic and univalent functions L in the open disk Θ = {℘ :
|℘| < 1}, of the form:

L(℘) = ℘+ c2℘
2 + c3℘

3 + · · · . (5)

For analytic functions L and V , L subordination to V (denoted by L ≺ V ) for all
℘ ∈ Θ, if there exists a function ϖ via ϖ(0) = 0 and |ϖ(℘)| < 1, such that

L(℘) = V (ϖ(℘)).

In addition, if V is univalent in Θ, then

L(℘) ≺ V (℘), iff, L(0) = V (0)

and
L(Θ) ⊂ V (Θ).

Every function L ∈ AU has an inverse L−1 defined by (see [12, 20]):

L−1(L(℘)) = ℘ (℘ ∈ Θ)

and

ϖ = L(L−1(ϖ)) (|ϖ| < r0(L); r0(V ) ≥ 1

4
),

where

V (ϖ) = L−1(ϖ) = ϖ − c2ϖ
2 + (2c22 − c3)ϖ

3 − (c4 + 5c32 − 5c3c2)ϖ
4 + · · · . (6)

A function L ∈ AU is said to be bi-univalent in Θ (the family of bi-univalent functions in
Θ denoted by Ω) if both L(℘) and L−1(℘) are univalent in Θ (see [15, 22]).

The error function is important in many scientific domains, such as probability, statis-
tics, partial differential equations, and numerous engineering issues. As a result, math-
ematics has given it a lot of attention. For the error function, a number of noteworthy
inequalities and associated subjects were reported; for examples, see [9, 13, 16]. When
predicting events that hold with high or low probability, the error function and its ap-
proximations are typically utilized.

erf(℘) =
2√
π

℘∫
0

e−y2dy =
2√
π

∞∑
ν=0

(−1)ν℘2ν+1

(2ν + 1)ν!
, ℘ ∈ C. (7)

The imaginary error functions Maclaurin series can be obtained as shown above by wring-
ing the integrand e−y2 as Maclaurin series and integrating term by term; additionally, the



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 4 of 12

imaginary error function, represented by the symbol erfi, has a very similar Maclaurin
series, which is explained by (see [2, 10]):

erf i(℘) =
2√
π

℘∫
0

e−y2dy =
2√
π

∞∑
ν=0

℘2ν+1

(2ν + 1)ν!
, ℘ ∈ C. (8)

Using (7), Ramachandran et al. [8] investigated the normalized analytic error function
regarding the form:

Erf(℘) =

√
π℘

2
erf(

√
℘) = ℘+

∞∑
ν=2

(−1)ν−1℘ν

(2ν − 1)(ν − 1)!
, (9)

and utilizing the convolution product “∗” defined the following family

Erf ∗AU =

{
R : R(℘) = (Erf ∗ L)(℘) = ℘+

∞∑
ν=2

(−1)ν−1cν
(2ν − 1)(ν − 1)!

℘ν , L ∈ AU

}
. (10)

From (8), the normalized form of the error function Erfi defined by:

Erfi(℘) =

√
π℘

2
erf i(

√
℘) = ℘+

∞∑
ν=2

℘ν

(2ν − 1)(ν − 1)!

and by convolution product, we define

EL(℘) = (Erfi ∗ L)(℘) = ℘+
∞∑
ν=2

cν
(2ν − 1)(ν − 1)!

℘ν .

After we introduced the Horadam polynomials and the normalized form of the error func-
tion, we will define the following definition.

Definition 1. A function L ∈ Ω given by (5) is said to be in the family FΩ(s, r, u, y, t, λ, τ)
if satisfying the below two conditions

(1− τ)
EL(℘)

℘
+ τ (EL(℘))′ + λ℘ (EL(℘))′′ ≺ Υ(y, ℘) + 1− s (11)

and

(1− τ)
EV (ϖ)

ϖ
+ τ (EV (ϖ))′ + λϖ (EV (ϖ))′′ ≺ Υ(y,ϖ) + 1− s, (12)

where ℘,ϖ ∈ Θ , τ, λ ≥ 0, y ∈ R, and the function V = L−1 is given by (6).

Example 1. For λ = 0, we have, FΩ(s, r, u, y, t, 0, τ) = FΩ(s, r, u, y, t, τ), in which
FΩ(s, r, u, y, t, τ) the family of functions L ∈ Ω and satisfying the below conditions

(1− τ)
EL(℘)

℘
+ τ (EL(℘))′ ≺ Υ(y, ℘) + 1− s

and

(1− τ)
EV (ϖ)

ϖ
+ τ (EV (ϖ))′ ≺ Υ(y,ϖ) + 1− s,

where ℘,ϖ ∈ Θ, τ ≥ 0, y ∈ R.



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 5 of 12

Example 2. For λ = 0 and τ = 1, we have, FΩ(s, r, u, y, t, 1) = FΩ(s, r, u, y, t), in which
FΩ(s, r, u, y, t) the family of functions L ∈ Ω and satisfying the below conditions
and

(EV (ϖ))′ ≺ Υ(y,ϖ) + 1− s,

where ℘,ϖ ∈ Θ, y ∈ R.

Example 3. For λ = 0 and τ = 0, we have, FΩ(s, r, u, y, t, 0, 0) = FΩ(s, r, u, y, t, 0), in
which FΩ(s, r, u, y, t, 0) the family of functions L ∈ Ω and satisfying the below conditions

EL(℘)

℘
≺ Υ(y, ℘) + 1− s

and
EV (ϖ)

ϖ
≺ Υ(y,ϖ) + 1− s,

where ℘,ϖ ∈ Θ, y ∈ R.

Example 4. For s = u = 1 and t = r = 2, we have, FΩ(1, 2, 1, y, 2, λ, τ) the family of
functions L ∈ Ω and satisfying the below conditions

(1− τ)
PLα(℘)

℘
+ τ (PLα(℘))

′ + λ℘ (PLα(℘))
′′ ≺ Υ(y, ℘) + 1− s

and

(1− τ)
PVα(ϖ)

ϖ
+ τ (PVα(ϖ))′ + λϖ (PVα(ϖ))′′ ≺ Υ(y,ϖ) + 1− s,

where ℘,ϖ ∈ Θ, τ, λ ≥ 0, y ∈ R.
Recently, many researchers have examined bi-univalent functions associated with or-

thogonal polynomials and found non-sharp estimates on Maclaurin coefficients |c2| and
|c3| (for details, see [4]-[23]).

In [14], Fekete and Szegö validated the following inequality

∣∣c3 − φc22
∣∣ ≤ 1 + 2e

(
−2φ
1−φ

)

for all normalized univalent function L and φ ∈ [0, 1] . This inequality is sharp for each φ
(see [17]-[24]).

Recent years have seen a number of studies use many special functions, such as the
Borel, Poisson, Rabotnov, Pascal, Wright and Bessel, to examine important aspects of ge-
ometric function theory, such as coefficient estimates, inclusion relations, and requirements
for belonging to particular families (see, [1]-[3], [6]-[7]).

This article content is arranged as follows. In section 2 we giving bounds for the coeffi-
cients |c2| and |c3| in the Maclaurin expansions and estimation of Fekete–Szegö inequality
for functions in the family FΩ(s, r, u, y, t, λ, τ). Section 3 pertinent links between some of
the particular cases of the main results are highlighted. Section 4 concludes the study
with a few observations.



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 6 of 12

2. Bounds of the family FΩ(s, r, u, y, t, λ, τ)

Section 2 begins with bounds for the coefficients |c2| and |c3| in the Maclaurin expan-
sions for functions in the family FΩ(s, r, u, y, t, λ, τ).

Theorem 1. Let L ∈ Ω given by (5) belongs to the family FΩ(s, r, u, y, t, λ, τ). Then

|c2| ≤
ty
√
2ty√∣∣∣15 (6λ+ 2τ + 1) t2y2 − 2

9 (2λ+ τ + 1)2 (rty2 + su)
∣∣∣

and

|c3| ≤
9t2y2

(2λ+ τ + 1)2
+

10ty

6λ+ 2τ + 1
.

Proof. Let L ∈ FΩ(s, r, u, y, t, λ, τ). From Definition 1, we can write

(1− τ)
EL(℘)

℘
+ τEL′(℘) + λ℘EL′′(℘) = Υ(y,κ(℘)) + 1− s (13)

and

(1− τ)
EV (ϖ)

ϖ
+ τEV ′(ϖ) + λϖEV ′′(ϖ) = Υ(y, τ(ϖ)) + 1− s, (14)

where κ and τ are analytic and have the form:

κ(℘) = j1℘+ j2℘
2 + j3℘

3 + · · · , (℘ ∈ Θ)

and
τ(ϖ) = d1ϖ + d2ϖ

2 + d3ϖ
3 + · · · , (ϖ ∈ Θ),

such that κ(0) = τ(0) = 0 and |κ(℘)| < 1, |τ(ϖ)| < 1 for all ℘,ϖ ∈ Θ.
From the equalities (13) and (14), we get

(1− τ)
EL(℘)

℘
+ τEL′(℘)+λ℘EL′′(℘) = 1+h2(y)j1℘+

(
h2(y)j2 + h3(y)j

2
1

)
℘2+ · · · (15)

and

(1− τ)
EV (ϖ)

ϖ
+ τEV ′(ϖ)+λϖEV ′′(ϖ) = 1+h2(y)d1ϖ+

(
h2(y)d2 + h3(y)d

2
1

)
ϖ2+ · · · .

(16)
It is common knowledge that if

|κ(℘)| =
∣∣j1℘+ j2℘

2 + j3℘
3 + · · ·

∣∣ < 1, (℘ ∈ Θ)

and
|τ(ϖ)| =

∣∣d1ϖ + d2ϖ
2 + d3ϖ

3 + · · ·
∣∣ < 1, ϖ ∈ Θ,

then
|ji| ≤ 1 and |di| ≤ 1 for all i ∈ N. (17)



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 7 of 12

Equating the coefficients of both sides in (15) and (16), we get

1

3
(2λ+ τ + 1) c2 = h2(y)j1, (18)

1

10
(6λ+ 2τ + 1) c3 = h2(y)j2 + h3(y)j

2
1 , (19)

−1

3
(2λ+ τ + 1) c2 = h2(y)d1, (20)

and
1

10
(6λ+ 2τ + 1)

[
2c22 − c3

]
= h2(y)d2 + h3(y)d

2
1. (21)

It follows from (18) and (20) that

j1 = −d1 (22)

and
2

9
(2λ+ τ + 1)2 c22 = [h2(y)]

2 (j21 + d21
)
. (23)

If we add (19) and (21), we get

1

5
(6λ+ 2τ + 1) c22 = h2(y) (j2 + d2) + h3(y)

(
j21 + d21

)
. (24)

Replacing the value of
(
c21 + d21

)
from (23) in the right hand side of (24), we have[

1

5
(6λ+ 2τ + 1)− 2

9
(2λ+ τ + 1)2

h3(y)

[h2(y)]
2

]
c22 = h2(y) (j2 + d2) . (25)

Using (3) and (17) in (25), we find that

|c2| ≤
ty
√
2ty√∣∣∣15 (6λ+ 2τ + 1) t2y2 − 2

9 (2λ+ τ + 1)2 (rty2 + su)
∣∣∣ .

Also, if we subtract (21) from (19), we obtain

1

5
(6λ+ 2τ + 1)

(
c3 − c22

)
= h2(y) (j2 − d2) + h3(y)

(
j21 − d21

)
. (26)

Then, from (22) and (23), equation (26) becomes

c3 =
9 [h2(y)]

2

2 (2λ+ τ + 1)2
(
j21 + d21

)
+

5h2(y)

6λ+ 2τ + 1
(j2 − d2) .

By applying (3), we conclude that

|c3| ≤
9t2y2

(2λ+ τ + 1)2
+

10ty

6λ+ 2τ + 1
.

Using the values of c2 and c3, we prove the functional
∣∣c3 − φc22

∣∣ for family functions
FΩ(s, r, u, y, t, λ, τ).



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 8 of 12

Theorem 2. Let L ∈ Ω given by (5) belongs to the family FΩ(s, r, u, y, t, λ, τ). Then

∣∣c3 − φc22
∣∣ ≤


10|ty|

6λ+2τ+1

2t3y3|1−φ|
| 15 (6λ+2τ+1)t2y2− 2

9
(2λ+τ+1)2(rty2+su)|

|1− φ| ≤ Π1,

|1− φ| ≥ Π1.

where

Π1 = 1− 10 (2λ+ τ + 1)2 (rty2 + su)

(6λ+ 2τ + 1) t2y2
.

Proof. From (25) and (26)

c3 − φc22 =
5h2(y)

6λ+ 2τ + 1
(j2 − d2)

+ (1− φ)
[h2(y)]

3 (j2 + d2)
1
5 (6λ+ 2τ + 1) [h2(y)]

2 − 2
9 (2λ+ τ + 1)2 h3(y)

= h2(y)

[
𭟋(φ) +

5

6λ+ 2τ + 1

]
j2 + h2(y)

[
𭟋(φ)− 5

6λ+ 2τ + 1

]
d2,

where

𭟋(φ) =
[h2(y)]

2 (1− φ)
1
5 (6λ+ 2τ + 1) [h2(y)]

2 − 2
9 (2λ+ τ + 1)2 h3(y)

,

Then, from (3), we deduce that

∣∣c3 − φc22
∣∣ ≤


10|h2(y)|
6λ+2τ+1

2 |h2(y)| |𭟋(φ)|

|𭟋(φ)| ≤ 5
6λ+2τ+1 ,

|𭟋(φ)| ≥ 5
6λ+2τ+1 .

≡


10|ty|

6λ+2τ+1

2t3y3|1−φ|
| 15 (6λ+2τ+1)t2y2− 2

9
(2λ+τ+1)2(rty2+su)|

|1− φ| ≤ Π1,

|1− φ| ≥ Π1.

where

Π1 = 1− 10 (2λ+ τ + 1)2 (rty2 + su)

(6λ+ 2τ + 1) t2y2

3. Particular Cases

The following corollaries are obtained by specializing the parameters λ and τ in the
aforementioned theorems in section 2.



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 9 of 12

Corollary 1. Let L ∈ Ω given by (5) belongs to the family FΩ(s, r, u, y, t, τ). Then

|c2| ≤
ty
√
2ty√∣∣∣[1

5 (2τ + 1) t2y2 − 2
9 (τ + 1)2 (rty2 + su)

]∣∣∣ ,
|c3| ≤

9t2y2

(τ + 1)2
+

10ty

2τ + 1

and ∣∣c3 − φc22
∣∣ ≤


10|ty|
2τ+1

2t3y3|1−φ|
| 15 (2τ+1)t2y2− 2

9
(τ+1)2(rty2+su)|

|1− φ| ≤ Π2,

|1− φ| ≥ Π2.

where

Π2 = 1− 10 (τ + 1)2 (rty2 + su)

(2τ + 1) t2y2
.

Corollary 2. Let L ∈ Ω given by (5) belongs to the family FΩ(s, r, u, y, t). Then

|c2| ≤
ty
√
2ty√∣∣[3

5 t
2y2 − 8

9 (rty
2 + su)

]∣∣ ,
|c3| ≤

9t2y2

4
+

10ty

3
and ∣∣c3 − φc22

∣∣ ≤


10|ty|
3

2t3y3|1−φ|
| 35 t2y2− 8

9
(rty2+su)|

|1− φ| ≤ Π3,

|1− φ| ≥ Π3.

where

Π3 = 1− 40(rty2 + su)

3t2y2
.

Corollary 3. Let L ∈ Ω given by (5) belongs to the family FΩ(s, r, u, y, t, 0). Then

|c2| ≤
ty
√
2ty√∣∣[1

5 t
2y2 − 2

9 (rty
2 + su)

]∣∣ ,
|c3| ≤ 9t2y2 + 10ty

and ∣∣c3 − φc22
∣∣ ≤


10 |ty|

2t3y3|1−φ|
| 15 t2y2− 2

9
(rty2+su)|

|1− φ| ≤ 1− 10(rty2+su)
t2y2

,

|1− φ| ≥ 1− 10(rty2+su)
t2y2

.

The following corollary is obtained by specializing the parameters s, r, u and t in the
aforementioned theorems in section 2.



T. Al-Hawary et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5678 10 of 12

Corollary 4. Let L ∈ Ω given by (5) belongs to the family FΩ(1, 2, 1, y, 2, λ, τ). Then

|c2| ≤
4y

√
y√∣∣∣[4

5 (6λ+ 2τ + 1) y2 − 2
9 (2λ+ τ + 1)2 (4y2 + 1)

]∣∣∣
and

|c3| ≤
36y2

(2λ+ τ + 1)2
+

102y

6λ+ 2τ + 1
.

and ∣∣c3 − φc22
∣∣ ≤


20|y|

6λ+2τ+1

16y3|1−φ|
| 15 (6λ+2τ+1)t2y2− 2

9
(2λ+τ+1)2(4y2+1)|

|1− φ| ≤ Π4,

|1− φ| ≥ Π4.

where

Π4 = 1− 10 (2λ+ τ + 1)2 (4y2 + 1)

(6λ+ 2τ + 1) 4y2
.

4. Conclusions

In this paper, we defined a comprehensive family of analytic and bi-univalent functions
related to imaginary error function and subordinate to Horadam polynomials denoted by
FΩ(s, r, u, y, t, λ, τ). We estimated for the Maclaurin coefficients |c2|, |c3| and Fekete-Szegö
problems. Furthermore, by specializing the parameters s, r, u,t, λ and τ , one may deter-
mine the outcomes for the subfamilies FΩ(s, r, u, y, t, τ), FΩ(s, r, u, y, t), FΩ(s, r, u, y, t, 0)
and FΩ(1, 2, 1, y, 2, λ, τ) specified in Examples 1, 2, 3 and 4, respectively. Making use
of normalized error function could inspire researchers to find the estimates of the coef-
ficients |c2|, |c3| and Fekete-Szegö problems for functions belonging to new subfamily of
bi-univalent functions.

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