EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6071
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Coefficient Results on Certain Subclasses of Sakaguchi
Type Bi-univalent Functions

Lee Chew Yee1,∗, Maslina Darus1,∗

1 Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, Bangi 43600 Selangor, Malaysia

Abstract. In this article, we study some subclasses of Sakaguchi type bi-univalent functions as-
sociated with Gegenbauer polynomials and Einstein function. We explore certain properties of
functions belonging to these subclasses, including coefficient bounds and the Fekete–Szegö func-
tionals. This research generalise and improves the related works of several earlier authors.

2020 Mathematics Subject Classifications: 30C10, 30C45, 30C50, 33C45

Key Words and Phrases: Analytic functions, bi-univalent functions, convolution, Gegenbauer
polynomial, Einstein function, Sakaguchi type function, coefficient estimate, Fekete-Szegö

1. Introduction and Preliminaries

Let A signify the class of analytic functions written in the form

F(ξ) = ξ +
∞∑
n=2

anξ
n, ξ ∈ U = {ξ ∈ C : |ξ| < 1}, (1)

with normalization F(0) = 0,F ′(0) = 1. Define S as the subclass of univalent functions
in A.

An function F ∈ A is said to be starlike, denoted as S∗, if and only if

Re

(
ξF ′(ξ)

F(ξ)

)
> 0.

Similarly, F is convex, denoted as K, if and only if

Re

(
1 +

ξF ′′(ξ)

F ′(ξ)

)
> 0.

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6071

Email addresses: p154492@siswa.ukm.edu.my (C.Y. Lee), maslina@ukm.edu.my (M. Darus)

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



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 2 of 12

If a function G ∈ A is given by

G(ξ) = ξ +
∞∑
n=2

bnξ
n, ξ ∈ U,

then the Hadamard product of two functions F and G is given by

(F ∗ G)(ξ) = ξ +

∞∑
n=2

anbnξ
n, ξ ∈ U.

If there exists an analytic Schwarz function Ω in U, such that for all ξ ∈ U, Ω satisfies
Ω(0) = 0, |Ω(ξ)| < 1 and F(ξ) = G(Ω(ξ)), then two analytic functions F and G can be
described as F is subordinate to G or G is superordinate to F . This relationship is written
as F ≺ G. If the function G is univalent in unit disk, then

F(ξ) ≺ G(ξ) ⇔ F(0) = G(0) and F(U) ⊂ G(U).

For more details on subordination principles, please refer to [1].
Koebe’s one-quarter theorem [2] stated that, if U is mapped by F biholomorphically

onto a domain ∆ in the complex plane, then each of its tangent disks of radius r will be
mapped onto a domain containing a disk of radius 1

4r. Hence, the following conditions
must be satisfied by function F ∈ A for its inverse map F−1 to exist:

F−1(F(ξ)) = ξ (ξ ∈ U) and F(F−1(w)) = w

(
|w| < r0(F); r0(F) ≥ 1

4

)
.

Inverse function F−1 of (1) can be conveyed as a power series of the form

G(ω) = F−1(ω) = ω − a2ω
2 + (2a22 − a3)ω

3 − (5a32 − 5a2a3 + a4)ω
4 + ... .

The class of bi-univalent functions, Σ, is a set of F ∈ A, in which both F and F−1 are
univalent in U. Examples of functions in class Σ include

−log(1− ξ),
ξ

1− ξ
,
1

2
log

(
1 + ξ

1− ξ

)
, ... .

It is noteworthy that the Koebe function is not bi-univalent, since it does not contain
U. To be precise, the image of Koebe function does not contain the slit of negative real
axis from −1

4 to −∞.
The coefficient bounds for functions of class Σ have been investigated since 1967, where

Lewin [3] showed that |a2| < 1.51. In addition to that, Brannan and Clunie [4] showed
that max

f∈Σ
|a2| =

√
2. Subsequently, Netanyahu [5] improved this bound to |a2| ≤ 4

3 . The

best non-sharp estimate |a2| < 1.485 was obtained in 1984 by Tan [6]. On the other hand,
the estimate for coefficients other than |a1| and |a2| for each F ∈ Σ is yet to be explored.



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 3 of 12

Introduced by Sakaguchi [7], a function F ∈ S is described as starlike about symmetric
points, provided it satisfies the inequality

Re

(
ξF ′(ξ)

F(ξ)−F(−ξ)

)
> 0 (ξ ∈ U).

This class of functions is denoted by S∗
S .

On the other hand, Das and Singh [8] established another class of univalent functions
F which are convex about symmetric points. This family of functions, KS satisfy the
inequality

Re

(
(ξF ′(ξ))′

(F(ξ)−F(−ξ))′

)
> 0 (ξ ∈ U).

Two polynomials Pn and Pm of order n and m are said to be orthogonal, provided∫ b

a
w(x)Pn(x)Pm(x)dx = 0 for n ̸= m,

where w(x) is non-negative function in the interval (a, b). As described by [9], Gegenbauer
polynomials are special orthogonal polynomials which are typically associated with typi-
cally real functions. For α ∈ R − {0}, Gegenbauer polynomial is defined by a generating
function

Hα(x, ξ) =
1

(1− 2xξ + ξ2)α
,

where ξ ∈ U and x ∈ [−1, 1]. When x is fixed, the function Hα is analytic in U, therefore
it is possible to express Hα in the form of Taylor series expansion

Hα(x, ξ) =
∞∑
n=0

Cα
n (x)ξ

n,

where Cα
n (x) is a Gegenbauer polynomial of degree n.

Since H0 generates nothing, it is presumed to be

H0(x, ξ) = 1− log(1− 2xξ + ξ2) =
∞∑
n=0

C0
n(x)ξ

n.

By using the recurrence relations, Gegenbauer polynomial can also be represented as [10–
12]

Cα
n (x) =

1

n
[2x(n+ α− 1)Cα

n−1(x)− (n+ 2α− 2)Cα
n−2(x)].

As indicated in [9], the Gegenbauer polynomial possesses initial values as follows

Cα
0 (x) = 1, Cα

1 (x) = 2αx and Cα
2 (x) = 2α(1 + α)x2 − α.

Meanwhile, the name Einstein function is sometimes applied for one of the functions

[13, 14]: E1(ξ) =
ξ

eξ−1
, E2(ξ) =

ξ2eξ

(eξ−1)2
, E3(ξ) = log(1−e−ξ) or E4(ξ) =

ξ
eξ−1

−log(1−e−ξ).



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 4 of 12

Among these functions, E1(ξ) = ξ
eξ−1

, has some nice properties, such as E1 is a convex
function, with its real part, Re(E1(ξ)) > 0, ∀ξ ∈ U. Its image domain is starlike about
E1(0) = 1 and is symmetric along the real axis. However, since E′

1(0) ̸> 0, a new function
E(ξ) = E1(ξ) + ξ is defined to make E ∈ P (see [15]). This function has the series
representation

E(ξ) = 1 + ξ +
∞∑
n=1

Bn

n!
ξn,

whereBn is known as the nth Bernoulli number. By traversing the contour which possesses
a radius which is less than 2πi and encloses the origin in positive (counterclockwise)
direction, the values of Bn can be determined by contour integral [16]

Bn =
n!

2πi

∮
ξ

eξ − 1

dξ

ξn+1
.

It is known that the first few terms of Bn are

B0 = 1,B1 = −1

2
,B2 =

1

6
,B4 = − 1

30
,B6 =

1

42
and B2n+1 = 0,∀n ∈ N.

Some similar work has been done with regard to coefficient bounds for bi-univalent
functions, including [17–19]. Several authors specifically studied this on Gegenbauer poly-
nomial, such as [9, 20]. So far we have not seen any work with classes associated with
Gegenbauer and Einstein functions of Sakaguchi type. Therefore, in this article, we are
solving coefficient bounds and the Fekete-Szegö functional for the aforementioned classes.

Definition 1.1. A bi-univalent function F is presumed to be in class HE∗(α) if for all
ξ, ω ∈ U, the following subordinations hold,

2ξF ′(ξ)

F(ξ)−F(−ξ)
≺ (H ∗ E)(ξ),

and
2ωG′(ω)

G(ω)− G(−ω)
≺ (H ∗ E)(ω).

Definition 1.2. A bi-univalent function F is presumed to be in class HEK(α) if for all
ξ, ω ∈ U, the following subordinations hold,

2(ξF ′(ξ))′

(F(ξ)−F(−ξ))′
≺ (H ∗ E)(ξ),

and
2(ωG′(ω))′

(G(ω)− G(−ω))′
≺ (H ∗ E)(ω).

Remark 1. If F(−ξ) = −F(ξ), then HE∗(α) and HEK(α) is a class of starlike and convex
bi-univalent functions associated with H ∗ E, respectively.



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 5 of 12

2. Coefficient Bounds for the Class HE∗(α)

Within this section and Section 3, coefficient bounds will be investigated for bi-
univalent functions in the families HE∗(α) and HEK(α), respectively.

Theorem 2.1. Suppose that F ∈ Σ belongs to the class HE∗(α), then

|a2| ≤
|α|x

√
6x√

|8αx2 − 4x2 + 2|
,

and

|a3| ≤
α2x2

4
+

αx

2
.

Proof. Let u, v be Schwarz functions such that u(ξ) =
∑∞

k=1 dkξ
k, v(ω) =

∑∞
k=1 dkω

k,
then

(H ∗ E)(u(ξ)) = 1 +
Cα
1 (x)

2
d1ξ +

(
Cα
1 (x)

2
d2 +

Cα
2 (x)

12
d21

)
ξ2 + ... , (2)

and

(H ∗ E)(v(ω)) = 1 +
Cα
1 (x)

2
d1ω +

(
Cα
1 (x)

2
d2 +

Cα
2 (x)

12
d21

)
ω2 + ... . (3)

A Sakaguchi type function of the class A can be expanded as follows

2ξF ′(ξ)

F(ξ)−F(−ξ)
= 1 + 2a2ξ + 2a3ξ

2 + ... , (4)

and
2ωG′(ω)

G(ω)− G(−ω)
= 1− 2a2ω + (4a22 − 2a3)ω

2 + ... . (5)

Comparing coefficients of (2) with (4), and (3) with (5), the followings are obtained

2a2 =
Cα
1 (x)

2
d1, (6)

2a3 =
Cα
1 (x)

2
d2 +

Cα
2 (x)

12
d21, (7)

−2a2 =
Cα
1 (x)

2
d1, (8)

and

4a22 − 2a3 =
Cα
1 (x)

2
d2 +

Cα
2 (x)

12
d21. (9)

From (6) and (8), we have
d1 = −d1. (10)

By adding the squares of (6) and (8),

8a22 =
[Cα

1 (x)]
2

4
(d21 + d21), (11)



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 6 of 12

and

d21 + d21 =
32a22

[Cα
1 (x)]

2
. (12)

Sum up (7) and (9), we have

4a22 =
Cα
1 (x)

2
(d2 + d2) +

Cα
2 (x)

12
(d21 + d21). (13)

Substituting (12) into (13), we obtain

4a22 =
Cα
1 (x)

2
(d2 + d2) + Cα

2 (x)

(
8a22

3[Cα
1 (x)]

2

)
,

and so (
8− 16Cα

2 (x)

3[Cα
1 (x)]

2

)
a22 = Cα

1 (x)(d2 + d2). (14)

Substitute the values of Cα
n , (14) becomes(

8− 32α(1 + α)x2 − 16α

12α2x2

)
a22 = 2αx(d2 + d2)

and

a22 =
3α2x3

8αx2 − 4x2 + 2
(d2 + d2). (15)

Since |u(ξ)| < 1 and |v(ω)| < 1, we have

|dk| ≤ 1 and |dk| ≤ 1, ∀k ∈ N. (16)

Therefore,

|a2| ≤
|α|x

√
6x√

|8αx2 − 4x2 + 2|
.

Now we proceed to evaluate the bounds of |a3|. Taking (7)−(9),

4a3 − 4a22 =
Cα
1 (x)

2
(d2 − d2) +

Cα
2 (x)

12
(d21 − d21). (17)

Substituting (10) into (17) and rearranging the terms, the following result is obtained

a3 = a22 +
Cα
1 (x)

8
(d2 − d2). (18)

Substitute (11) and the values of Cα
1 (x),

a3 =
4α2x2

32
(d21 + d21) +

2αx

8
(d2 − d2).

Taking (16) into consideration, we have

|a3| ≤
α2x2

4
+

αx

2
.



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 7 of 12

3. Coefficient Bounds for the Class HEK(α)

Theorem 3.1. Suppose that F ∈ Σ belongs to the class HEK(α), then

|a2| ≤
|α|x

√
3x√

|10αx2 − 8x2 + 4|
,

and

|a3| ≤
α2x2

16
+

αx

6
.

Proof. Let u, v be Schwarz functions such that u(ξ) =
∑∞

k=1 dkξ
k, v(ω) =

∑∞
k=1 dkω

k,
then functions F and G that are convex about symmetric points can be expanded as
follows

2(ξF ′(ξ))′

(F(ξ)−F(−ξ))′
= 1 + 4a2ξ + 6a3ξ

2 + ... , (19)

and
2(ωG′(ω)′)′

(G(ω)− G(−ω))′
= 1− 4a2ω + (12a22 − 6a3)ω

2 + ... . (20)

Comparing coefficients of (2) with (19), and (3) with (20), the followings are obtained

4a2 =
Cα
1 (x)

2
d1, (21)

6a3 =
Cα
1 (x)

2
d2 +

Cα
2 (x)

12
d21, (22)

−4a2 =
Cα
1 (x)

2
d1, (23)

and

12a22 − 6a3 =
Cα
1 (x)

2
d2 +

Cα
2 (x)

12
d21. (24)

From (21) and (23), we obtain
d1 = −d1. (25)

By adding the squares of (21) and (23),

32a22 =
[Cα

1 (x)]
2

4
(d21 + d21), (26)

and

d21 + d21 =
128a22

[Cα
1 (x)]

2
. (27)

Sum up (22) and (24), we obtain

12a22 =
Cα
1 (x)

2
(d2 + d2) +

Cα
2 (x)

12
(d21 + d21). (28)



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 8 of 12

Substituting (27) into (28), we have

12a22 =
Cα
1 (x)

2
(d2 + d2) + Cα

2 (x)

(
32a22

3[Cα
1 (x)]

2

)
,

and (
24− 64Cα

2 (x)

3[Cα
1 (x)]

2

)
a22 = Cα

1 (x)(d2 + d2). (29)

Substitute the values of Cα
n , (29) become(

24− 128α(1 + α)x2 − 64α

12α2x2

)
a22 = 2αx(d2 + d2)

and

a22 =
3α2x3

20αx2 − 16x2 + 8
(d2 + d2). (30)

By (16),

|a2| ≤
|α|x

√
3x√

|10αx2 − 8x2 + 4|
.

To evaluate the bounds of |a3|, take (22)−(24),

12a3 − 12a22 =
Cα
1 (x)

2
(d2 − d2) +

Cα
2 (x)

12
(d21 − d21). (31)

Substituting (25) into (31) and rearranging the terms,

a3 = a22 +
Cα
1 (x)

24
(d2 − d2). (32)

Substitute (26) and the values of Cα
1 (x),

a3 =
α2x2

32
(d21 + d21) +

αx

12
(d2 − d2).

Taking (16) into consideration, we obtain

|a3| ≤
α2x2

16
+

αx

6
.



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 9 of 12

4. Fekete-Szegö Inequality for the Class HE∗(α)

One of the most prominent problem affiliated to coefficient estimates of univalent
functions is Fekete-Szegö inequality. First investigated in [21], it states that for univalent
functions F , the inequality

|a3 − ηa22| ≤ 1 + 2e−2η/(1−µ)

is sharp when η ∈ R.
Within this section and Section 5, the sharp bounds of Fekete-Szegö functional for the

class HE∗(α) and HEK(α) are to be evaluated.

Theorem 4.1. Suppose that F ∈ Σ belongs to the class HE∗(α), then

|a3 − ηa22| ≤

{ |α|x
2 , |1− η| ≤ |4αx2−2x2+1

6αx2 |
3α2x3(1−η)
4αx2−2x2+1

, |1− η| ≥ |4αx2−2x2+1
6αx2 |

.

Proof. Let F ∈ HE∗(α). Using (15) and (18), for some η ∈ R,

a3 − ηa22 = a22 +
Cα
1 (x)

8
(d2 − d2)− ηa22

=
2αx

8
(d2 − d2) + (1− η)

(
3α2x3

8αx2 − 4x2 + 2

)
(d2 + d2)

= αx

{[
1

4
+

3αx2(1− η)

8αx2 − 4x2 + 2

]
d2 +

[
3αx2(1− η)

8αx2 − 4x2 + 2
− 1

4

]
d2

}
= αx

{[
h1(η) +

1

4

]
d2 +

[
h1(η)−

1

4

]
d2

}
where h1(η) =

3αx2(1−η)
8αx2−4x2+2

.
Using triangle inequality and considering (16), we are able to conclude that

|a3 − ηa22| ≤

{
|α|x
2 , |h1(η)| ≤ 1

4

2|α| x |h1(η)|, |h1η)| ≥ 1
4

.

Hence,

|a3 − ηa22| ≤

{ |α|x
2 , |1− η| ≤ |4αx2−2x2+1

6αx2 |
3α2x3(1−η)
4αx2−2x2+1

, |1− η| ≥ |4αx2−2x2+1
6αx2 |

.

Corollary 4.1. Suppose that F ∈ Σ belongs to the class HE∗(α), then

|a3 − a22| ≤
|α|x
2

.

Proof. Take η = 1 in Theorem 4.1.



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 10 of 12

5. Fekete-Szegö Inequality for the Class HEK(α)

Theorem 5.1. Suppose that F ∈ Σ belongs to the class HEK(α), then

|a3 − ηa22| ≤

{ |α|x
6 , |1− η| ≤ |5αx2−4x2+2

9αx2 |
3α2x3(1−η)

10αx2−8x2+4
, |1− η| ≥ |5αx2−4x2+2

9αx2 |
.

Proof. Let F ∈ HEK(α). Using (30) and (32), for some η ∈ R,

a3 − ηa22 = a22 +
Cα
1 (x)

24
(d2 − d2)− ηa22

=
αx

12
(d2 − d2) + (1− η)

(
3α2x3

20αx2 − 16x2 + 8

)
(d2 + d2)

= αx

{[
1

12
+

3αx2(1− η)

20αx2 − 16x2 + 8

]
d2 +

[
3αx2(1− η)

20αx2 − 16x2 + 8
− 1

12

]
d2

}
= αx

{[
h2(η) +

1

12

]
d2 +

[
h2(η)−

1

12

]
d2

}
where h2(η) =

3αx2(1−η)
20αx2−16x2+8

.
Using triangle inequality and considering (16), we are able to conclude that

|a3 − ηa22| ≤

{
|α|x
6 , |h2(η)| ≤ 1

12

2|α| x |h2(η)|, |h2(η)| ≥ 1
12

.

Hence,

|a3 − ηa22| ≤

{ |α|x
6 , |1− η| ≤ |5αx2−4x2+2

9αx2 |
3α2x3(1−η)

10αx2−8x2+4
, |1− η| ≥ |5αx2−4x2+2

9αx2 |
.

Corollary 5.1. Suppose that F ∈ Σ belongs to the class HEK(α), then

|a3 − a22| ≤
|α|x
6

.

Proof. Take η = 1 in Theorem 5.1.

6. Conclusion

By convoluting Gegenbauer polynomials and Einstein functions, some new subclasses
of Sakaguchi type bi-univalent functions are introduced in this paper. Some coefficient
bounds are evaluated and the Fekete-Szegö inequalities are assessed for these subclasses.



C.Y. Lee, M. Darus / Eur. J. Pure Appl. Math, 18 (2) (2025), 6071 11 of 12

Acknowledgements

The authors would like to thank UKM for providing support for this research. We
would also like to thank the referees for the constructive comments given to improve the
manuscript.

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