EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1948-1958
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Euler Polynomials and Bi-univalent Functions

Ala Amourah1,2,∗, Dunia Alawi Jarwan3, Jamal Salah4, M. J. Mohammed3,
Saad A. Meqdad5, Nidal Anakira1

1 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar
3111, Oman.
2 Jadara Research Center, Jadara University, Irbid 21110, Jordan.
3 Department of Mathematics College of Science University Of Anbar Ramadi, Iraq.
4 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post
Code No. 400 Ibra, Sultanate of Oman.
5 Applied Science Private University, Amman, Jordan.

Abstract. Our research introduces new subclasses of analytical functions that are defined by Euler
polynomials. We then proceed to estimate the Fekete-Szegö functional problem and the Maclaurin
coefficients for this specific subfamily, denoted as |a2| and |a3|. Furthermore, we demonstrate
several new results that emerge when we specialize the parameters used in our main findings.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, Univalent functions, Bi-univalent functions, Euler
polynomials, Fekete-Szegö problem.

1. Preliminaries

Euler polynomials, which have their origins in Leonhard Euler’s eighteenth-century re-
search, are essential for understanding complex functions and their geometric properties.
They play a key role in characterizing conformal mappings that preserve angles locally
in geometric function theory. Additionally, they are widely utilized in various areas of
geometric function theory, including the study of univalent functions, Schwarz-Christoffel
mappings, and Riemann surface theory. These applications shed light on the intricate re-
lationship between geometric transformations and analytic functions facilitated by Euler
polynomials. This text explores the fundamental properties of Euler polynomials, provid-
ing an explanation of how they are employed to represent solutions to specific differential
equations and to generate functions for different types of analytic functions.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5314

Email addresses: AAmourah@su.edu.om (A. Amourah), dunia.alawi@uoanbar.edu.iq (D. A. Jarwan),
damous73@yahoo.com (J. Salah), mohadmath87@uoanbar.edu.iq (M. J. Mohammed),
smeqdad@su.edu.om (S. A. Meqdad), nanakira@su.edu.om (N. Anakira)

https://www.ejpam.com 1948 © 2024 EJPAM All rights reserved.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1949

Due to the extensive usage of Euler polynomials in pure mathematics, numerous aca-
demics have started to explore various domains. The current research in geometric func-
tion theory primarily revolves around the geometric properties of special functions and
their related counterparts. For further information on the geometric properties of these
functions, please refer to [16, 28], and other relevant sources.

Let 𭟋 be the class of analytic functions b in the unit disk Λ = {κ ∈ C : |κ| < 1} and
normalized by b(0) = b′(0)− 1 = 0 of the form:

b(κ) = κ+
∞∑
i=2

ciκ
i, (κ ∈ Λ). (1)

We also let Ψ consisting of functions univalent in Λ.
Every mathematical function b ∈ Ψ has an inverse b−1, defined by

b−1(b(κ)) = κ and w = b(b−1(w)) (κ ∈ Λ, |w| < r0(b); r0(b) ≥
1

4
)

where

b−1(w) = q(w) = w − c2w
2 + (2c22 − c3)w

3 − (c4 + 5, c32 − 5c3c2)w
4 + · · · . (2)

A function b is said to be bi-univalent in Λ if both b and b−1 are univalent in Λ. Let Π
denote the class of all bi-univalent functions in Λ given by (1).

Example in the class Π is h(κ) = κ
1−κ but h(κ) = κ

1−κ2 not members of Π. For
interesting function classes in class Π, (see [1]).

Miller and Mocanu [21] introduced the first differential subordination problem, see [22]
and [23]. We say that the function b is subordinate to q, written as b ≺ q, if b and q are
analytic in Λ and exists function w ∈ 𭟋 in Λ with

w(0) = 0 and |w(κ)| < 1, (κ ∈ Ω)

such that
b(κ) = q(w(κ)).

Also, if q is univalent in Λ, then

b(κ) ≺ q(κ) if and only if b(0) = q(0) and b(Λ) ⊂ q(Λ).

Geometric function theory makes effective use of Euler polynomials, which is a fun-
damental tool in mathematical analysis. They are particularly important in the study of
complex analysis and conformal mappings.

In this study, our focus is on the Euler polynomial, a specific special function. Our
aim is to construct a new and comprehensive subclass of bi-univalent functions.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1950

The generating function is commonly used to define the Eulers polynomials Θi(ℓ) (see,
[19, 27]):

B(ℓ, h) = 2ehℓ

eh + 1
=

∞∑
i=0

Θi(ℓ)
hi

i!
,

(
1

2
< ℓ ≤ 1, |h| < π

)
.

An explicit formula for Θi(ℓ) is given by

Θj(ℓ) =

j∑
i=0

1

2i

i∑
u=0

(−1)u
(
i

u

)
(ℓ+ u)j .

Now Θi(ℓ) in terms of Θu can be obtained from the above equation as:

Θi(ℓ) =
i∑

u=0

Θu

2u

(
i

u

)
(ℓ− 1

2
)i−u.

The initial values of Euler polynomials are:

Θ0(ℓ) = 1;

Θ1(ℓ) =
2ℓ− 1

2
;

Θ2(ℓ) = ℓ2 − ℓ; (3)

Θ3(ℓ) =
4ℓ3 − 6ℓ2 + 1

4
;

Θ4(ℓ) = ℓ4 − 2ℓ3 + ℓ.

A lot of studies have looked at the geometric function theory in recent years, including
coefficient estimates. Several subclasses of the class Π were introduced and non-sharp
estimates on the coefficients |a2| and |a3| in the Taylor-Maclaurin series expansion (1)
were obtented in ([2–15, 18, 20, 24, 25, 29–31]).

In this study, we define new subclass of Π involving the Euler polynomials which are
denote by FΠ(ζ, ℓ), and derive bounds for the |a2| and |a3| Taylor-Maclaurin coefficients
and Fekete–Szegö functional problems. Furthermore, Several novel findings are shown to
ensue.

2. Definition and Examples

At the beginning of this section, we present a definition of the new subclasses FΠ(ζ, ℓ)
that is associated with Euler polynomials.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1951

Definition 1. If the following subordinations are met for a function b ∈ Λ given by (1),
then b ∈ FΠ(ζ, ℓ):

κb′(κ) + ζκ2b′′(κ)

(1− ζ)b(κ) + ζκb′(κ)
≺ B(ℓ, κ) =

∞∑
i=0

Θi(ℓ)
κi

i!
(4)

and
wg′(w) + ζw2g′′(w)

(1− ζ)g(w) + ζwq′(w)
≺ B(ℓ, w) =

∞∑
i=0

Θi(ℓ)
wi

i!
, (5)

where 0 ≤ ζ ≤ 1, 1
2 < ℓ ≤ 1 κ,w ∈ Λ and q = b−1.

Example 1. If the following subordinations are met for a function b ∈ Λ given by (1),
then b ∈ FΠ(0, ℓ):

κb′(κ)

b(κ)
≺ B(ℓ, κ) =

∞∑
i=0

Θi(ℓ)
κi

i!

and
wg′(w)

g(w)
≺ B(ℓ, w) =

∞∑
i=0

Θi(ℓ)
wi

i!
,

where 1
2 < ℓ ≤ 1 κ,w ∈ Λ and q = b−1.

Example 2. If the following subordinations are met for a function b ∈ Λ given by (1),
then b ∈ FΠ(1, ℓ):

1 +
κb′′(κ)

b′(κ)
≺ B(ℓ, κ) =

∞∑
i=0

Θi(ℓ)
κi

i!

and

1 +
wg′′(w)

q′(w)
≺ B(ℓ, w) =

∞∑
i=0

Θi(ℓ)
wi

i!
,

where 1
2 < ℓ ≤ 1 κ,w ∈ Λ and q = b−1.

Lemma 1. ([26]) If d ∈ D, then |mn| ≤ 2 for each n, where D is the family of all analytic
functions in Λ for which

Re (d(κ)) > 0, d(κ) = 1 +m1κ+m2
2κ+ · · · (κ ∈ Λ).



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1952

3. Bounds of the class FΠ(ζ, ℓ)

For a function b ∈ Λ, we give the coefficient estimates and solve Fekete-Szegö prob-
lem(see [17]) for the class FΠ(ζ, ℓ), respectively.

Theorem 1. Let b ∈ Π given by (1) belongs to the class FΠ(ζ, ℓ) where 0 ≤ ζ ≤ 1,
1
2 < ℓ ≤ 1 κ,w ∈ Λ and q = b−1. Then

|c2| ≤
√
Υ(ζ, ℓ),

|c3| ≤
(2ℓ− 1)2

(1 + ζ)2
+

2ℓ− 1

4(1 + 2ζ)
.

and ∣∣c3 − κc22
∣∣ ≤


2ℓ−1

2(1+2ζ)

2 |1− κ|Υ(ζ, ℓ)

0 ≤ |1− κ|Υ(ζ, ℓ) < 2ℓ−1
4(1+2ζ) ,

|1− κ|Υ(ζ, ℓ) ≥ 2ℓ−1
4(1+2ζ) .

where

Υ(ζ, ℓ) =
2 (2ℓ− 1)3∣∣∣[(1 + 2ζ − ζ2) (2ℓ− 1)2 − 2(1 + ζ)2 (ℓ2 − 3ℓ+ 1)

]∣∣∣ .
Proof. Since b(κ) = κ+

∞∑
i=2

ciκ
i ∈ FΠ(ζ, ℓ), So from Definition 1, we can write

κb′(κ) + ζκ2b′′(κ)

(1− ζ)b(κ) + ζκb′(κ)
≺ B(ℓ, κ) (6)

and
wg′(w) + ζw2g′′(w)

(1− ζ)g(w) + ζwq′(w)
≺ B(ℓ, w). (7)

We can consider two functions r, s : Λ → Λ, with r(0) = s(0) = 0 and |r(κ)| < 1,
|s(w)| < 1 for all κ,w ∈ Λ. So we can define γ, λ ∈ D as following:

γ(κ) =
r(κ) + 1

1− r(κ)
= 1 + γ1κ+ γ2κ

2 + γ3κ
3 + · · · , |γi| ≤ 2 for all i ∈ N.

⇒ r(κ) =
γ(κ)− 1

γ(κ) + 1
=

γ1
2
κ+

(
γ2
2

− γ21
4

)
κ2 +

1

2

(
γ3 − γ1γ2 +

γ31
4

)
κ3 + · · · (8)

and

λ(w) =
s(w) + 1

1− s(w)
= 1 + λ1w + λ2w

2 + λ3w
3 + · · · , |λi| ≤ 2 for all i ∈ N.

⇒ s(w) =
λ(w)− 1

λ(w) + 1
=

λ1

2
w +

(
λ2

2
− λ2

1

4

)
w2 +

1

2

(
λ3 − λ1λ2 +

λ3
1

4

)
w3 + · · · . (9)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1953

Using (8) and (9), we get

B(ℓ, r(κ)) = Θ0(ℓ) +
Θ1(ℓ)

2
γ1κ+

(
Θ1(ℓ)

2

(
γ2 −

γ21
2

)
+

Θ2(ℓ)

8
γ21

)
κ2 (10)

+

(
Θ1(ℓ)

2

(
γ3 − γ1γ2 +

γ31
4

)
+

Θ2(ℓ)

4

(
γ1γ2 −

γ31
2

)
+

Θ3(ℓ)

48
γ31

)
κ3 + · · ·

and

B(ℓ, s(w)) = Θ0(ℓ) +
Θ1(ℓ)

2
λ1w +

(
Θ1(ℓ)

2

(
λ2 −

λ2
1

2

)
+

Θ2(ℓ)

8
λ2
1

)
w2 (11)

+

(
Θ1(ℓ)

2

(
λ3 − λ1λ2 +

λ3
1

4

)
+

Θ2(ℓ)

4

(
λ1λ2 −

λ3
1

2

)
+

Θ3(ℓ)

48
λ3
1

)
w3 + · · ·

From (6), (7) and the previous two equations, we have

(1 + ζ)c2 =
Θ1(ℓ)

2
γ1, (12)

2(1 + 2ζ)c3 − (1 + ζ)2c22 =
Θ1(ℓ)

2

(
γ2 −

γ21
2

)
+

Θ2(ℓ)

8
γ21 , (13)

−(1 + ζ)c2 =
Θ1(ℓ)

2
λ1, (14)

and

−2(1 + 2ζ)c3 − (ζ2 − 6ζ − 3)c22 =
Θ1(ℓ)

2

(
λ2 −

λ2
1

2

)
+

Θ2(ℓ)

8
λ2
1. (15)

Adding equations (12) and (14) and some simplification, we get

γ1 = −λ1 and γ21 = λ2
1 (16)

and
2(1 + ζ)2c22 = Θ2

1(ℓ)(γ
2
1 + λ2

1). (17)

⇒ c22 =
Θ2

1(ℓ)(γ
2
1 + λ2

1)

2(1 + ζ)2
(18)

Adding (13) to (15) gives

(
2 + 4ζ − 2ζ2

)
c22 = 2Θ1(ℓ)(γ2 + λ2) + (γ21 + λ2

1)

(
1

2
Θ2(ℓ)−Θ1(ℓ)

)
.

By (16), we have(
2 + 4ζ − 2ζ2

)
c22 = 2Θ1(ℓ)(γ2 + λ2) + γ21 (Θ2(ℓ)− 2Θ1(ℓ)) (19)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1954

Also, applying (16) in (17)

γ21 =
(1 + ζ)2c22
Θ2

1(ℓ)
(20)

Replacing γ21 in (19)

c22 =
2Θ3

1(ℓ)(γ2 + λ2)[
(2 + 4ζ − 2ζ2)Θ2

1(ℓ)− (1 + ζ)2 (Θ2(ℓ)− 2Θ1(ℓ))
] (21)

⇒ |c2|2 =
2Θ3

1(ℓ) (|γ2|+ |λ2|)∣∣[(2 + 4ζ − 2ζ2)Θ2
1(ℓ)− (1 + ζ)2 (Θ2(ℓ)− 2Θ1(ℓ))

]∣∣
Applying Lemma 1 and (3), we have:

|c2| ≤

√√√√ 2 (2ℓ− 1)3∣∣∣[(1 + 2ζ − ζ2) (2ℓ− 1)2 − 2(1 + ζ)2 (ℓ2 − 3ℓ+ 1)
]∣∣∣ =

√
Υ(ζ, ℓ).

Subtracting (15) from (13), then view (16) and with some computations, we obtain

c3 = c22 +
Θ1(ℓ) (γ2 − λ2)

8(1 + 2ζ)
(22)

By (18) and (16)

c3 =
Θ2

1(ℓ)γ
2
1

(1 + ζ)2
+

Θ1(ℓ) (γ2 − λ2)

8(1 + 2ζ)
. (23)

Applying Lemma 1 and (3), we have:

|c3| ≤
(2ℓ− 1)2

(1 + ζ)2
+

2ℓ− 1

4(1 + 2ζ)
.

From (22), we obtain

c3 − κc22 =
Θ1(ℓ) (γ2 − λ2)

8(1 + 2ζ)
+ (1− κ)c22

Applying the triangular inequality with assist (3), we obtain:

∣∣c3 − κc22
∣∣ ≤ 2ℓ− 1

4(1 + 2ζ)
+ |1− κ|Υ(ζ, ℓ)

If

|1− κ|Υ(ζ, ℓ) ≤ 2ℓ− 1

4(1 + 2ζ)

we obtain ∣∣c3 − κc22
∣∣ ≤ 2ℓ− 1

2(1 + 2ζ)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1948-1958 1955

and if:

|1− κ|Υ(ζ, ℓ) ≥ 2ℓ− 1

4(1 + 2ζ)

we obtain ∣∣c3 − κc22
∣∣ ≤ 2 |1− κ|Υ(ζ, ℓ)

Which are asserted by the Theorem 1.

4. Some Corollaries

If we set ζ = 1 in Theorems 1, we get the next corollary.

Corollary 1. Let b ∈ Π given by (1) belongs to the class FΠ(1, ℓ) where 1
2 < ℓ ≤ 1

κ,w ∈ Λ and q = b−1. Then
|c2| ≤

√
Υ(1, ℓ),

|c3| ≤
2 (2ℓ− 1) (3ℓ− 1)

12
.

and ∣∣c3 − κc22
∣∣ ≤


2ℓ−1
6

2 |1− κ|Υ(1, ℓ)

0 ≤ |1− κ|Υ(1, ℓ) < 2ℓ−1
12 ,

|1− κ|Υ(1, ℓ) ≥ 2ℓ−1
12 .

where

Υ(1, ℓ) =
(2ℓ− 1)3∣∣∣(2ℓ− 1)2 − 4 (ℓ2 − 3ℓ+ 1)

∣∣∣ .
If we set ζ = 0 in Theorems 1, we get the next corollary.

Corollary 2. Let b ∈ Π given by (1) belongs to the class FΠ(0, ℓ) where 1
2 < ℓ ≤ 1

κ,w ∈ Λ and q = b−1. Then
|c2| ≤

√
Υ(0, ℓ),

|c3| ≤ (2ℓ− 1)2 +
2ℓ− 1

4
.

and ∣∣c3 − κc22
∣∣ ≤


2ℓ−1
2

2 |1− κ|Υ(0, ℓ)

0 ≤ |1− κ|Υ(0, ℓ) < 2ℓ−1
4 ,

|1− κ|Υ(0, ℓ) ≥ 2ℓ−1
4 .

where

Υ(0, ℓ) =
2 (2ℓ− 1)3∣∣∣(2ℓ− 1)2 − 2 (ℓ2 − 3ℓ+ 1)

∣∣∣ .



REFERENCES 1956

5. Conclusions

Because polynomials and special functions are used in various mathematical and sci-
entific fields, many prominent mathematicians have recently focused on studying them.
This paper aims to define new subclasses of analytical and univalent functions using Euler
polynomials. For functions belonging to these classes FΠ(ζ, ℓ), FΠ(0, ℓ) and FΠ(1, ℓ), we
have established an upper bound estimate for the coefficients and successfully solved the
Fekete-Szegö problem. The sharp upper bounds for |c2|, |c3| and

∣∣c3 − κc22
∣∣ are still an

interesting challenge to discover, as well as the open problem regarding |ci|, i ≥ 3.

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