EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2538-2549
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Inclusive Subclasses of Bi-univalent Functions Specified
by Euler Polynomials

Basem Aref Frasin1, Tariq Al-Hawary2,4,∗, Ala Amourah3,7, Jamal Salah5,
Oqlah Al-Refai6

1 Faculty of Science, Department of Mathematics, Al al-Bayt University, P.O. Box: 130095
Mafraq, Jordan
2 Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun
26816. Jordan
3 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar
3111, Oman
4 Jadara Research Center, Jadara University, Irbid 21110, Jordan
5 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42, Post
Code No. 400 Ibra, Sultanate of Oman
6 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13132, Jordan
7 Applied Science Research Center. Applied Science Private University, Amman, Jordan

Abstract. Our research delineates novel two subclasses FΠ(κ, ε, ℓ) and LΠ(φ, ℓ) of analytical
functions using Euler polynomials. Afterwards, we estimate the Fekete–Szegö functional problem
and the Maclaurin coefficients for this subclasses, namely |c2| and |c3|. Additionally, several new
results are shown to follow after specializing the parameters employed in our main results.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, Univalent and Bi-univalent functions, Euler poly-
nomials, Fekete-Szegö

1. preliminaries

Euler polynomials, which date back to Leonhard Euler’s research in the eighteenth cen-
tury, are fundamental components for articulating complex functions and comprehending
their geometric characteristics. They play a significant role in the characterization of con-
formal mappings in geometric function theory that preserve angles locally. They are also
useful in the study of univalent and analytic functions.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5334

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

https://www.ejpam.com 2538 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2539

Because Euler polynomials are used so widely in pure mathematics, many academics
have begun to work in a number of domains. The geometric properties of special func-
tions and several other related functions are the main focus of current study in geometric
function theory. For a few of these functions geometric characteristics, we refer to [4, 17]
and any pertinent references.

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

p(ℶ) = ℶ+

∞∑
i=2

ciℶi, (ℶ ∈ ∆). (1)

We also let Φ the class of univalent functions in ∆.
Every function p ∈ Φ has an inverse p−1, defined by

p−1(p(ℶ)) = ℶ and ϖ = p(p−1(ϖ))(ℶ ∈ ∆, |ϖ| < s10(p) ≥
1

4
)

where

p−1(ϖ) = h(ϖ) = ϖ − c2ϖ
2 + (2c22 − c3)ϖ

3

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

Let Γ the class of bi-univalent functions in ∆ given by (1) (a function p is bi-univalent in
∆ if p and p−1 are univalent in ∆).

Example in the class Γ is h(ℶ) = ℶ
1−ℶ but h(ℶ) = ℶ

1−ℶ2 not members of Γ (see [3]).
The first differential subordination problem introduced Miller and Mocanu [10], see

[11] and [12]. The function p is subordinate to h, written as p ≺ h, if p and h are analytic
in ∆ and exists function ϖ ∈ 𭟋 in ∆ with

ϖ(0) = 0 and |ϖ(ℶ)| < 1, (ℶ ∈ Ω)

such that
p(ℶ) = h(ϖ(ℶ)).

Also, if h is univalent in ∆, then

p(ℶ) ≺ h(ℶ) if and only if p(0) = h(0) and p(∆) ⊂ h(∆).

Many authors have deduced multiple subordination between certain classes of analytic
functions by applying a subordination theorem for analytic functions., for example, see [7]
and [16].

Geometric function theory offers fascinating uses for Euler polynomials, a basic tool
in mathematical analysis, especially when studying conformal mappings.

In this paper, we take a specific special function, the Euler polynomial, and we build
two new and comprehensive subclasses of bi-univalent functions.



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2540

the Eulers polynomials Φi(υ) are defined using the generating function (see, e.g., [9,
15]):

K(υ, h) =
2ehυ

eh + 1

=
∞∑
i=0

Φi(υ)
hi

i!
,

(
1

2
< υ ≤ 1, |h| < π

)
.

An explicit formula for Φi(υ) is given by

Φj(υ) =

j∑
i=0

1

2i

i∑
u=0

(−1)u
(
i

u

)
(υ + u)j . (3)

Now Φi(υ) in terms of Φu, obtained from (3) as:

Φi(υ) =

i∑
u=0

Φu
2u

(
i

u

)
(υ − 1

2
)i−u.

Initial Euler polynomial values are:

Φ0(υ) = 1;

Φ1(υ) =
2υ − 1

2
;

Φ2(υ) = υ2 − υ; (4)

Φ3(υ) =
4υ3 − 6υ2 + 1

4
;

Φ4(υ) = υ4 − 2υ3 + υ.

Several subclasses of the class Γ were introduced and non-sharp estimates on the
coefficients |c2| and |c3| in the Taylor series expansion (1). For example, Al-Hawary et al.
[18] defined the novel subclass Kγ

Σ(σ, δ, µ, x) using Gegenbauer polynomials. Amourah et
al. [1] defined the class K(ϑ, δ) by means of (p, h)−Lucas polynomials. Amourah et al. [2]
defined the class s1(α, β, t) by means of Chebyshev polynomials. Peng et al. [19] defined
the class Sa,p,cΣ (γ, λ, ϕ) using Hohlov operator. Yousef, et al. [5] defined some subclasses
by Frasin differentia operator. Bulut et al. [14] introduced a subclass Kµ

Σ(λ, t) using the
Chebyshev polynomials. Srivastava et al. [8] investigated two interesting subclasses Hα

Σ

and HΣ(β).
In this paper, we define new two subclasses of Γ utilizing Euler polynomials which are

denote by FΓ(κ, ϵ, υ) and LΓ(ψ, υ), and derive bounds for the coefficients |c2| and |c3| and
Fekete–Szegö problems. Additionally, several new results are shown.



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2541

2. Bounds of the classes FΓ(κ, ϵ, υ) and LΓ(ψ, υ)

A definitions of the new subclasses FΓ(κ, ϵ, υ) and LΓ(ψ, υ) connected to Euler poly-
nomials is provided at the beginning of this section.

Definition 1. If the next subordinations are satisfied for a function p ∈ ∆ given by (1),
then p ∈ FΓ(κ, ϵ, υ):

(1− κ)
p(ℶ)
ℶ

+ κp′(ℶ) + ϵℶp′′(ℶ) ≺ K(υ,ℶ) =
∞∑
i=0

Φi(υ)
ℶi

i!
(5)

and

(1− κ)
h(ϖ)

ϖ
+ κh′(ϖ) + ϵϖh′′(ϖ) ≺ K(υ,ϖ) =

∞∑
i=0

Φi(υ)
ϖi

i!
, (6)

where κ ≥ 1, ϵ ≥ 0, 1
2 < υ ≤ 1 ℶ, ϖ ∈ ∆ and h = p−1.

Definition 2. If the nex subordinations are satisfied for a function p ∈ ∆ given by (1),
then p ∈ LΓ(ψ, υ):

p′(ℶ) + ℶ
eiψ + 1

2
p′′(ℶ) ≺ K(υ,ℶ) =

∞∑
i=0

Φi(υ)
ℶi

i!
(7)

and

h′(ϖ) +ϖ
eiψ + 1

2
h′′(ϖ) ≺ K(υ,ϖ) =

∞∑
i=0

Φi(υ)
ϖi

i!
, (8)

where −π < ψ ≤ π, 1
2 < υ ≤ 1 ℶ, ϖ ∈ ∆ and h = p−1.

Remark 1. Many subclasses can be found by taking special values for the parameters κ, λ
and υ in Definition 1, and for the parameters ψ and υ in Definition 2.

Lemma 1. ([13]) If g ∈ G, then |mn| ≤ 2 for each n ∈ N, where G is the family of
analytic functions in ∆ such that

Re (g(ℶ)) > 0, g(ℶ) = 1 +m1ℶ+m2
2ℶ+ · · · (ℶ ∈ ∆).

For a function p ∈ ∆, we solve Fekete-Szegö and provide the coefficient estimations
(see [6]) for the classes FΓ(κ, ϵ, υ) and LΓ(ψ, υ), respectively.

Theorem 1. Let p ∈ Γ given by (1) in the class FΓ(κ, ϵ, υ) where κ ≥ 1, ϵ ≥ 0, 1
2 < υ ≤ 1

ℶ, ϖ ∈ ∆ and h = p−1. Then

|c2| ≤
√
𭟋(ϵ,κ, υ),



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2542

|c3| ≤
(2υ − 1)2

4 (2ϵ+ κ + 1)2
+

2υ − 1

2 (6ϵ+ 2κ + 1)
.

and ∣∣c3 − ζc22
∣∣

≤


2υ−1

6ϵ+2κ+1 if 0 ≤ |1− ζ|𭟋(ϵ,κ, υ) < 2υ−1
2(6ϵ+2κ+1) ,

2 |1− ζ|𭟋(ϵ,κ, υ) if |1− ζ|𭟋(ϵ,κ, υ) ≥ 2υ−1
2(6ϵ+2κ+1) .

where

𭟋(ϵ,κ, υ)

=
(2υ − 1)3

2
∣∣∣(6ϵ+ 2κ + 1) (2υ − 1)2 − 2 (2ϵ+ κ + 1)2 (υ2 − 3υ + 1)

∣∣∣ .
Proof. Since p(ℶ) = ℶ+

∞∑
i=2

ciℶi ∈ FΓ(κ, λ, υ), So from Definition 1, we have

(1− κ)
p(ℶ)
ℶ

+ κp′(ℶ) + ϵℶp′′(ℶ) ≺ K(υ,ℶ) (9)

and

(1− κ)
h(ϖ)

ϖ
+ κh′(ϖ) + ϵϖh′′(ϖ) ≺ K(υ,ϖ). (10)

We may think of two functions r1, r2 : ∆ → ∆, with r1(0) = r2(0) = 0 and |r1(ℶ)| < 1,
|r2(ϖ)| < 1 for all ℶ, ϖ ∈ ∆. So we can define γ, λ ∈ D as:

γ(ℶ) =
s1(ℶ) + 1

1− s1(ℶ)
= 1 + γ1ℶ+ γ2ℶ2 + γ3ℶ3 + · · · , |γi| ≤ 2, i ∈ N.

⇒ s1(ℶ) =
γ(ℶ)− 1

γ(ℶ) + 1
=
γ1
2
ℶ+

(
γ2
2

− γ21
4

)
ℶ2

+
1

2

(
γ3 − γ1γ2 +

γ31
4

)
ℶ3 + · · · (11)

and

λ(ϖ) =
r2(ϖ) + 1

1− r2(ϖ)

= 1 + λ1ϖ + λ2ϖ
2 + λ3ϖ

3 + · · · , |λi| ≤ 2, i ∈ N.



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2543

⇒ r2(ϖ) =
λ(ϖ)− 1

λ(ϖ) + 1
=
λ1
2
ϖ +

(
λ2
2

− λ21
4

)
ϖ2

+
1

2

(
λ3 − λ1λ2 +

λ31
4

)
ϖ3 + · · · . (12)

Using (11) and (12), we get

K(υ, s1(ℶ)) = Φ0(υ) +
Φ1(υ)

2
γ1ℶ+

(
Φ1(υ)

2

(
γ2 −

γ21
2

)
+

Φ2(υ)

8
γ21

)
ℶ2

+

 Φ1(υ)
2

(
γ3 − γ1γ2 +

γ31
4

)
+Φ2(υ)

4

(
γ1γ2 −

γ31
2

)
+ Φ3(υ)

48 γ31

ℶ3 + · · · (13)

and

K(υ, s(ϖ)) = Φ0(υ) +
Φ1(υ)

2
λ1ϖ +

(
Φ1(υ)

2

(
λ2 −

λ21
2

)
+

Φ2(υ)

8
λ21

)
ϖ2

+

 Φ1(υ)
2

(
λ3 − λ1λ2 +

λ31
4

)
+Φ2(υ)

4

(
λ1λ2 −

λ31
2

)
+ Φ3(υ)

48 λ31

ϖ3 + · · · (14)

From (9), (10) and the previous two equations, we have

(2ϵ+ κ + 1) c2 =
Φ1(υ)

2
γ1, (15)

(6ϵ+ 2κ + 1) c3 =
Φ1(υ)

2

(
γ2 −

γ21
2

)
+

Φ2(υ)

8
γ21 , (16)

− (2ϵ+ κ + 1) c2 =
Φ1(υ)

2
λ1, (17)

and

(6ϵ+ 2κ + 1)
(
2c22 − c3

)
=

Φ1(υ)

2

(
λ2 −

λ21
2

)
+

Φ2(υ)

8
λ21. (18)

Adding two equations (15) and (17) and some simplifying, we obtain

γ1 = −λ1 and γ21 = λ21 (19)

and
8 (2ϵ+ κ + 1)2 c22 = Φ2

1(υ)(γ
2
1 + λ21). (20)

⇒ c22 =
Φ2
1(υ)(γ

2
1 + λ21)

8 (2ϵ+ κ + 1)2
(21)

Adding (16) to (18) gives

8 (6ϵ+ 2κ + 1) c22



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2544

= 2Φ1(υ)(γ2 + λ2) + (γ21 + λ21)

(
1

2
Φ2(υ)− Φ1(υ)

)
.

By (19), we have

8 (6ϵ+ 2κ + 1) c22

= 2Φ1(υ)(γ2 + λ2) + γ21 (Φ2(υ)− 2Φ1(υ)) (22)

Also, appling (19) in (20)

γ21 =
4 (2ϵ+ κ + 1)2 c22

Φ2
1(υ)

(23)

Replacing γ21 in (22)

c22 =
Φ3
1(υ) (γ2 + λ2)

2

[
2 (6ϵ+ 2κ + 1)Φ2

1(υ)

− (2ϵ+ κ + 1)2 (Φ2(υ)− 2Φ1(υ))

] (24)

⇒ |c2|2 =
Φ3
1(υ) (|γ2|+ |λ2|)

2

∣∣∣∣ 2 (6ϵ+ 2κ + 1)Φ2
1(υ)

− (2ϵ+ κ + 1)2 (Φ2(υ)− 2Φ1(υ))

∣∣∣∣
Applying (4) and Lemma 1, we obtain

|c2| ≤

√√√√√√ (2υ − 1)3

2

∣∣∣∣ (6ϵ+ 2κ + 1) (2υ − 1)2

−2 (2ϵ+ κ + 1)2
(
υ2 − 3υ + 1

) ∣∣∣∣
=

√
𭟋(ϵ,κ, υ).

Subtracting (18) from (16), then view (19) and after doing some calculations, we arrive
at

c3 = c22 +
Φ1(υ) (γ2 − λ2)

4 (6ϵ+ 2κ + 1)
(25)

By (21) and (19)

c3 =
Φ2
1(υ)γ

2
1

4 (2ϵ+ κ + 1)2
+

Φ1(υ) (γ2 − λ2)

4 (6ϵ+ 2κ + 1)
. (26)

Applying (4) and Lemma 1, we have:

|c3| ≤
(2υ − 1)2

4 (2ϵ+ κ + 1)2
+

2υ − 1

2 (6ϵ+ 2κ + 1)
.

From (25), we obtain

c3 − ζc22 =
Φ1(υ) (γ2 − λ2)

4 (6ϵ+ 2κ + 1)
+ (1− ζ)c22



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2545

By using assist (4) in conjunction with the triangular inequality, we arrive at:∣∣c3 − ζc22
∣∣ ≤ 2υ − 1

2 (6ϵ+ 2κ + 1)
+ |1− ζ|𭟋(ϵ,κ, υ)

If

|1− ζ|𭟋(ϵ,κ, υ) ≤ 2υ − 1

2 (6ϵ+ 2κ + 1)

we obtain ∣∣c3 − ζc22
∣∣ ≤ 2υ − 1

6ϵ+ 2κ + 1

and if:

|1− ζ|𭟋(ϵ,κ, υ) ≥ 2υ − 1

2 (6ϵ+ 2κ + 1)

we obtain ∣∣c3 − ζc22
∣∣ ≤ 2 |1− ζ|𭟋(ϵ,κ, υ)

Which the Theorem 1 asserts.

Theorem 2. Let p ∈ Γ of the form (1) in the class LΓ(ψ, υ) where −π < ψ ≤ π, 1
2 < υ ≤ 1

ℶ, ϖ ∈ ∆ and h = p−1. Then
|c2| ≤

√
Φ(ψ, υ),

|c3| ≤
(2υ − 1)2

4 (eiψ + 3)
2 +

2υ − 1

6 (eiψ + 2)
.

and ∣∣c3 − ζc22
∣∣

≤


2υ−1

3(eiψ+2)
if 0 ≤ |1− ζ|Φ(ψ, υ) < 2υ−1

6(eiψ+2)
,

2 |1− ζ|Φ(ψ, υ) if |1− ζ|Φ(ψ, υ) ≥ 2υ−1
6(eiψ+2)

.

where

Φ(ψ, υ)

=
(2υ − 1)3

2
∣∣∣3 (eiψ + 2) (2υ − 1)2 − 2 (eiψ + 3)

2
(υ2 − 3υ + 1)

∣∣∣ .
Proof. Since p(ℶ) = ℶ+

∞∑
i=2

ciℶi ∈ LΓ(ψ, υ), So from equations (13), (14) and Definition

2, we are able to write

p′(ℶ) + ℶ
eiψ + 1

2
p′′(ℶ) ≺ K(υ,ℶ) (27)



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2546

and

h′(ϖ) +ϖ
eiψ + 1

2
h′′(ϖ) ≺ K(υ,ϖ). (28)

By compare the coefficients in (27) and (28), where K(υ,ℶ) and K(υ,ϖ) respectively
given by (13) and (14), we have (

eiψ + 3
)
c2 =

Φ1(υ)

2
γ1, (29)

3
(
eiψ + 2

)
c3 =

Φ1(υ)

2

(
γ2 −

γ21
2

)
+

Φ2(υ)

8
γ21 , (30)

−
(
eiψ + 3

)
c2 =

Φ1(υ)

2
λ1, (31)

and

3
(
eiψ + 2

) (
2c22 − c3

)
=

Φ1(υ)

2

(
λ2 −

λ21
2

)
+

Φ2(υ)

8
λ21. (32)

By the same technique proving of Theorem 1, we get the results given by Theorem 2.

3. Some Corollaries

When we set κ = 1 in Theorems 1, the next corollary is revealed.

Corollary 1. Let p ∈ Γ given by (1) in the class FΓ(1, ϵ, υ) where ϵ ≥ 0, 1
2 < υ ≤ 1

ℶ, ϖ ∈ ∆ and h = p−1. Then
|c2| ≤

√
𭟋(ϵ, 1, υ),

|c3| ≤
(2υ − 1)2

16 (ϵ+ 1)2
+

2υ − 1

6 (2ϵ+ 1)
.

and ∣∣c3 − ζc22
∣∣

≤


2υ−1

3(2ϵ+1) if 0 ≤ |1− ζ|𭟋(ϵ, 1, υ) < 2υ−1
6(2ϵ+1) ,

2 |1− ζ|𭟋(ϵ, 1, υ) if |1− ζ|𭟋(ϵ, 1, υ) ≥ 2υ−1
6(2ϵ+1) .

where

𭟋(ϵ, 1, υ)

=
(2υ − 1)3

2
∣∣∣3 (2ϵ+ 1) (2υ − 1)2 − 8 (ϵ+ 1)2 (υ2 − 3υ + 1)

∣∣∣ .



Tariq Al-Hawary et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2538-2549 2547

When we set ϵ = 0 in Theorems 1, the next corollary is revealed.

Corollary 2. Let p ∈ Γ given by (1) in the class FΓ(κ, 0, υ) where κ ≥ 1, 1
2 < υ ≤ 1

ℶ, ϖ ∈ ∆ and h = p−1. Then

|c2| ≤
√
𭟋(0,κ, υ),

|c3| ≤
(2υ − 1)2

4 (κ + 1)2
+

2υ − 1

2 (2κ + 1)
.

and ∣∣c3 − ζc22
∣∣

≤


2υ−1
2κ+1 if 0 ≤ |1− ζ|𭟋(0,κ, υ) < 2υ−1

2(2κ+1) ,

2 |1− ζ|𭟋(0,κ, υ) if |1− ζ|𭟋(0,κ, υ) ≥ 2υ−1
2(2κ+1) .

where

𭟋(0,κ, υ) =
(2υ − 1)3

2
∣∣∣(2κ + 1) (2υ − 1)2 − 2 (κ + 1)2 (υ2 − 3υ + 1)

∣∣∣ .
When ϵ = 0 in Corollary 1 or ψ = π in Theorems 2 simplifies to the following corollary.

Corollary 3. Let p ∈ Γ given by (1) in the class FΓ(1, 0, υ) ≡ LΓ(π, υ) where 1
2 < υ ≤ 1

ℶ, ϖ ∈ ∆ and h = p−1. Then
|c2| ≤

√
𭟋(0, 1, υ),

|c3| ≤
(2υ − 1)2

16
+

2υ − 1

6
.

and ∣∣c3 − ζc22
∣∣

≤


2υ−1
3 if 0 ≤ |1− ζ|𭟋(0, 1, υ) < 2υ−1

6 ,

2 |1− ζ|𭟋(0, 1, υ) if |1− ζ|𭟋(0, 1, υ) ≥ 2υ−1
6 .

where

𭟋(0, 1, υ) =
(2υ − 1)3

2 |4υ2 + 12υ − 5|
.

When we set ψ = 0 in Theorems 2, the next corollary is revealed.

Corollary 4. Let p ∈ Γ given by (1) in the class LΓ(0, υ) where 1
2 < υ ≤ 1 ℶ, ϖ ∈ ∆

and h = p−1. Then
|c2| ≤

√
Φ(0, υ),

|c3| ≤
(2υ − 1)2

64
+

2υ − 1

18
.



REFERENCES 2548

and ∣∣c3 − ζc22
∣∣

≤


2υ−1
9 if 0 ≤ |1− ζ|Φ(0, υ) < 2υ−1

18 ,

2 |1− ζ|Φ(0, υ) if |1− ζ|Φ(0, υ) ≥ 2υ−1
18 .

where

Φ(0, υ) =
(2υ − 1)3

2 |4υ2 + 60υ − 23|
.

4. Conclusions

polynomials and special functions are employed in so many different mathematical and
scientific domains, many eminent mathematicians have recently studied them. This paper
is concerned with defining new subclasses of analytical and univalent functions by the use
of Euler polynomials. For functions in these classes FΓ(κ, ϵ, υ) and LΓ(ψ, υ), We obtained
an upper bound for the coefficients and solved the Fekete-Szegö problem. It remains a
challenge to fined the upper bounds for |c2|, |c3| and

∣∣c3 − ζc22
∣∣, and still an interesting

open problem for |ci|, i ≥ 3.

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