EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3899-3914
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On the Study of Bi-Univalent Functions Defined by the
Generalized Sălăgean Differential Operator

Waleed Al-Rawashdeh

Department of Mathematics, Zarqa University, 2000 Zarqa, 13110 Jordan

Abstract. In this paper, we make use of the generalized Sălăgean differential operator to define a
novel class of bi-univalent functions that is associated with the generalized hyperbolic sine function
in the open unit disk D. The prime goal of this paper to derive sharp coefficient bounds in open
unit disk D, especially the first two coefficient bounds for the functions belong to this class . The
investigation also focuses on studying the classical Fekete-Szegö functional problem for functions
belong to this class. Furthermore, some known corollaries are highlighted based on the unique
choices of the parameters involved in this class.

2020 Mathematics Subject Classifications: 30C45, 30C50, 33C45, 33C05, 11B39

Key Words and Phrases: Bi-Univalent Functions, Generalized Sălăgean Differential Opera-
tor, Sălăgean Differential Operator, Generalized Hyperbolic Sine Function, Coefficient estimates,
Fekete-Szegö functional problem

1. Introduction

The research conducted in geometric function theory sheds a light on the intricate
relationships between coefficients and the geometric properties of functions. By examin-
ing the bounds placed on the modulus of a function’s coefficients, researchers can gain a
deeper understanding of how these functions behave and interact within the mathematical
framework. This analytical approach not only enhances our comprehension of the under-
lying principles governing geometric function theory but also paves the way for further
exploration and discovery in this dynamic field of study.

Many operators have been used ever since the beginning of the study of analytic func-
tions. The differential and integral operators are the most fascinating of them, using these
operators has made it simpler to add new kinds of univalent and bi-univalent functions.
Sălăgean introduced the differential and integral operators, that bear his name, in his
1983 publication. These operators were immensely motivating, and many mathematicians

DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5548

Email address: walrawashdeh@zu.edu.jo (W. Al-Rawashdeh)

https://www.ejpam.com 3899 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3900

have used them to get novel and intriguing results in the field of geometric function theory.

In this paper, as an application of the Sălăgean operator, we introduce a new class
of bi-univalent functions and discuss certain characteristic properties of this generalized
function class. Consider the set H, which consists of all functions f(ζ) that are analytic
within the open unit disk denoted as D = {ζ ∈ C : |ζ| < 1} and normalized by the
conditions f(0) = 0 = 1 − f ′(0). Moreover, any function f belongs to the set H can be
written as

f(ζ) = ζ +
∞∑
n=2

anζ
n, where ζ ∈ D. (1)

Let the functions f and g be analytic in the open unit disk D. We say that f is
subordinated by g in D, denoted as f(z) ≺ g(ζ) for all ζ ∈ D, if there exists a Schwarz
function w satisfying w(0) = 0 and |w(ζ)| < 1 for all ζ ∈ D, such that f(ζ) = g(w(ζ))
for all ζ ∈ D. This relationship between f and g is a fundamental concept in complex
analysis, providing a way to compare the behavior of two analytic functions within the
unit disk. Notably, when the function g is univalent over D, the condition f(ζ) ≺ g(ζ) is
equivalent to f(0) = g(0) and f(D) ⊂ g(D). This equivalence highlights the significance of
the subordination principle in understanding the relationship between analytic functions.
For further insights and detailed discussions on the Subordination Principle, interested
readers are encouraged to explore the monographs [10], [11], [23], and [25]. These sources
provide comprehensive explanations and applications of this principle in the context of
complex analysis and geometric function theory.

In this paper, S represents the set of functions that are univalent in the open unit disk
D and belong to the set H. As known univalent functions are injective functions. Hence,
they are invertible and the inverse functions may not be defined on the entire unit disk
D. In fact, according to Koebe one-quarter Theorem, the image of D under any function
f ∈ S contains the disk D(0, 1/4) of center 0 and radius 1/4. Accordingly, every function
f ∈ S has an inverse f−1 = g which is defined as

g(f(z)) = z, z ∈ D

f(g(w)) = w, |w| < r(f); r(f) ≥ 1/4.

Moreover, the inverse function is given by

g(w) = w − a2w
2 + (2a22 − a3)w

3 − (5a32 − 5a2a3 + a4)w
4 + · · ·· (2)

For this reason, we define the class Σ as follows. A function f ∈ H is said to be
bi-univalent if both f and f−1 are univalent in D. Therefore, let Σ denote the class of all
bi-univalent functions in H which are given by Equation (1). For example, the following
functions belong to the class Σ:

z

1− z
, − log(1− z), log

√
1 + z

1− z
.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3901

However, Koebe function,
2z − z2

2
and

z

1− z2
do not belong to the class Σ. For more

information about univalent and bi-univalent functions we refer the readers to the articles
[18], [22], [26] the monograph [11], [13], [34] and the references provided therein.

For example, within the class S, it is established that the modulus of the coefficient
an is bounded by the value of n. These bounds on the modulus of coefficients provide
valuable insights into the geometric characteristics of these functions. Specifically, the
restriction on the second coefficients of functions belonging to the class S offers crucial
details regarding the growth and distortion bounds within this class.

The exploration of coefficient-related properties of functions within the bi-univalent
class Σ commenced in the 1970s. Notably, Lewin’s work, in 1967 [18], marked a significant
milestone as he examined the bi-univalent function class and established a bound for the
coefficient |a2|. Following this, Netanyahu’s research, in 1969 [26], determined that the
maximum value of |a2| is 4

3 for functions categorized under Σ. Furthermore, Brannan and

Clunie, in 1979 [5], demonstrated that for functions in this class, the inequality |a2| ≤
√
2

holds true. This foundational work has spurred numerous investigations into the coefficient
bounds for various subclasses of bi-univalent functions. Despite the extensive research con-
ducted on the coefficient bounds for bi-univalent functions, there remains a significant gap
in knowledge regarding the general coefficients |a2| for cases where n ≥ 4. The challenge
of estimating the coefficients, particularly the general coefficient |an|, continues to be an
unresolved issue in the field. This ongoing inquiry highlights the complexity and richness
of the bi-univalent function class, suggesting that further exploration is necessary to fully
understand the behavior of these coefficients in higher dimensions.

Fekete and Szegö, in 1933 [12], determined the maximum value of |a3 − λa22| for a uni-
valent function f , with the real parameter 0 ≤ λ ≤ 1. This result led to the establishment
of the Fekete-Szegö problem, which involves maximizing the modulus of the functional
Ψλ(f) = a3 − λa22 for f ∈ H with any complex number λ. Numerous researchers have
delved into the Fekete-Szegö functional and other coefficient estimates problems. For in-
stance, relevant articles include [2], [3], [4], [6], [8], [12], [16], [17], [21], [22], [33], and the
references provided therein. These studies have contributed to a deeper understanding of
the Fekete-Szegö problem and its implications in the field of geometric function theory.
Furthermore, the results presented in this paper are anticipated to yield a diverse array
of results for subclasses associated with orthogonal polynomials, including Legendre, La-
grange, Laguerre, Gegenbauer, and Horadam polynomials. For more information about
orthogonal polynomials, we encourage the interested readers to consult the papers [7], [9]
and the related references included therein.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3902

2. Preliminaries, Examples and Lemmas

The information presented in this section are essential for understanding the principal
outcomes of this paper. Let f be an analytic function represented in the form (1). The
Sălăgean differential operator, which was introduced in [35], is defined as

Dmf(z) = z +

∞∑
n=2

nmanz
n.

This operator has proven to be a significant source of inspiration, leading numerous
mathematicians to achieve novel and intriguing results through their applications. Many
researchers, using Sălăgean operators, have developed a variety of new operators. Then,
they investigated their characteristics and subsequently employed these newly defined op-
erators to establish classes of univalent functions that exhibit exceptional properties. For
examples, see the articles [15], [30], [31], [36] and the references provided therein.

In the year 2004, Al-Oboudi [1] has developed the generalized Sălăgean differential
operator which we defined as follows. Let f be an analytic function, m ∈ N0 = {0, 1, 2, ···},
and q ≥ 0, then we have

D0
qf(z) = f(z),

D1
qf(z) = (1− q)f(z) + qzf ′(z), and

Dm+1
q f(z) = (1− q)Dm

q f(z) + qz
(
Dm

q f(z)
)′
= Dq

(
Dm

q f(z)
)
.

Moreover, if f is in the form (1), then this operator can be written as

Dm
q f(z) = z +

∞∑
n=2

[1 + (n− 1)q]manz
n.

It is clear that when q = 1; we have the Sălăgean differential operator [35] that is men-
tioned above. For more information about the generalized Sălăgean differential operator,
we encourage the interested readers to consult the articles [14], [19], [20], [24], [28], [29]
and the references provided therein.

Now, we aim to establish a new class which consists of bi-univalent functions that
are defined using the generalized Sălăgean differential operator and that associated to the
generalized hyperbolic sine function, which we denote as Sq(λ,m, β, sinh), which we define
as follows.

Definition 1. A function f(z) belongs to the family Σ is considered to be part of the class
Sq(λ,m, β, sinh) if it obeys the following subordination conditions:

(1− λ)

(
Dm

q f(z)

z

)
+ λ

(
Dm

q f(z)
)′ ≺ 1 + sinh(βz)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3903

and

(1− λ)

(
Dm

q g(w)

w

)
+ λ

(
Dm

q g(w)
)′ ≺ 1 + sinh(βw),

where the function g(w) = f−1(w) is given by the Equation (2), the parameters q ≥ 0,
λ ≥ 0, β ≥ 0 and m ∈ N0 = {0, 1, 2, · · ·}.

Choosing λ = 0 and λ = 1, we get the following two subclasses of our presenting class,
respectively.

Example 1. A bi-univalent function f that is represented by Equation (1) belongs to the
subclass S

q
0(m,β, sinh) if the following subordinations hold:(

Dm
q f(z)

z

)
≺ 1 + sinh(βz) (3)

and (
Dm

q g(w)

w

)
≺ 1 + sinh(βw), (4)

where the function g(w) = f−1(w) is given by the Equation (2), the parameters q ≥ 0,
β ≥ 0 and m ∈ N0 = {0, 1, 2, · · ·}.

Example 2. A bi-univalent function f that is represented by Equation (1) belongs to the
subclass S

q
1(m,β, sinh) if the following subordinations hold:(

Dm
q f(z)

)′ ≺ 1 + sinh(βz) (5)

and (
Dm

q g(w)
)′ ≺ 1 + sinh(βw), (6)

where the function g(w) = f−1(w) is given by the Equation (2), the parameters q ≥ 0,
β ≥ 0 and m ∈ N0 = {0, 1, 2, · · ·}.

If we take q = 0, m = 0, or m = 1 and q = 0, then Dm
q f(z) = f(z) for any f ∈ H that

is given by Equation (1). Hence we obtain the following subclass.

Example 3. A bi-univalent function f that is represented by Equation (1) belongs to the
subclass S∗(λ, β, sinh) if the following subordinations hold:

(1− λ)

(
f(z)

z

)
+ λ (f(z))′ ≺ 1 + sinh(βz) (7)

and

(1− λ)

(
g(w)

w

)
+ λ (g(w))′ ≺ 1 + sinh(βw), (8)

where the function g(w) = f−1(w) is given by the Equation (2), the parameters λ ≥ 0 and
β ≥ 0.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3904

Taking q = 1 and m = 1, we get Dm
q f(z) = zf ′(z) for any f ∈ H that is given by

Equation (1). Hence we obtain the following subclass.

Example 4. A bi-univalent function f that is represented by Equation (1) belongs to the
subclass S(λ, β, sinh) if the following subordinations hold:

f ′(z) + λ
(
(zf ′′(z)

)
≺ 1 + sinh(βz), (9)

and
g′(w) + λ

(
wg′′(w)

)
≺ 1 + sinh(βw), (10)

where the function g(w) = f−1(w) is given by the Equation (2), the parameters λ ≥ 0
and β ≥ 0. This class of starlike functions has been studied by many reserchers, see, for
example [27], [32] and the references provided therein.

The lemma outlined below is well-documented in the literature (see, for instance,
[17]), is considered a fundamental principle that plays a crucial role in the research we are
undertaking.

Lemma 1. if p(z) belongs to the Caratheodory class P, then for z ∈ D the function p

can be written as p(z) = 1 +

∞∑
n=1

pnz
n. Moreover, |pn| ≤ 2 for each natural number n. In

addition for any complex number ζ, we have

|p2 − ζp21| ≤ 2max{1, |2ζ − 1|}.

The following lemma, which are thoroughly detailed in literature (see, for instance,
[17]), are widely recognized principles that are of considerable relevance to the research
we are presenting.

Lemma 2. Let K and L be real numbers. Let p and q be complex numbers. If |p| < r and
|q| < r,

|(K + L)p+ (K − L)q| ≤

{
2r|K|, if |K| ≥ |L|
2r|L|, if |K| ≤ |L|.

The purpose of this article is to explore a new class of bi-univalent functions defined
using the generalized Sălăgean differential operator that is related to the generalized hy-
perbolic sine function. The central objective is to establish estimates for the moduli of
the initial coefficients of the Taylor series representation of functions within this cate-
gory. Additionally, the article delves into the Fekete-Szegö functional problem pertinent
to this particular class of functions, thereby enhancing the comprehension of their inherent
properties.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3905

3. Coefficient bounds of the function class Sq(λ,m, β, sinh)

This section of the paper is devoted to explore the bounds for the modulus of the
initial coefficients of functions that are part of the class Sq(λ,m, β, sinh), as denoted by
Equation (1).

Theorem 1. Let a function f be in the family Σ. If the function f belongs to the class
Sq(λ,m, β, sinh) and is represented by the equation (1), then the following inequalities
hold:

|a2| ≤
β√

β(1 + 2λ)(1 + 2q)m + (1 + λ)2(1 + q)2m
, (11)

and

|a3| ≤
β

(1 + 2λ)(1 + 2q)m
+

β2

(1 + λ)2(1 + q)2m
. (12)

Proof. Suppose a function f belongs to the class Sq(λ,m, β, sinh). According to the
Definition 1 and the Subordination Principle, we can find two Schwarz functions u(z) and
v(w) defined on the open unit disk D such that

(1− λ)

(
Dm

q f(z)

z

)
+ λ

(
Dm

q f(z)
)′
= 1 + sinh(βu(z)), (13)

and

(1− λ)

(
Dm

q g(w)

w

)
+ λ

(
Dm

q g(w)
)′
= 1 + sinh(βv(w)). (14)

Now, using those Schwarz functions, we define two new analytic functions h(z) and
k(w) as follow:

h(z) =
1 + u(z)

1− u(z)
and k(w) =

1 + v(w)

1− v(w)
.

It is clear that, these functions h(z) and k(w) are analytic in the open unit disk D and
belong to the Caratheodory class. Thus, we can write them as follows

h(z) =
1 + u(z)

1− u(z)
= 1 + h1z + h2z

2 + · · ·

and

k(w) =
1 + v(w)

1− v(w)
= 1 + k1w + k2w

2 + · · ·

Moreover, h(0) = 1 = k(0), they have positive real parts, |hj | ≤ 2 and |kj | ≤ 2 for all j ∈ N.

Equivalently, we get the following representations of u(z) and v(w)

u(z) =
h(z)− 1

h(z) + 1
=

1

2

[
h1z +

(
h2 −

h21
2

)
z2 + · · ·

]
, (15)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3906

and

v(w) =
k(w)− 1

k(w) + 1
=

1

2

[
k1w +

(
k2 −

k21
2

)
w2 + · · ·

]
. (16)

On one hand, by consulting Equation (15), the right-hand sides of Equations (13) can
be written as:

1 + sinh(βu(z)) = 1 +
βh1
2

z + β

(
h2
2

− h21
4

)
z2

+

(
βh31
8

− βh1h2
2

+
βh3
2

+
β3h31
48

)
z3

+

(
3βh21h2

8
− βh1h3

2
− βh41

16
− βh22

4
− β3h41

32
+

βh4
2

+
β3h21h2

16

)
z4 + · · ·

(17)

Thus, considering Equation (17) then comparing coefficients on both sides of Equation
(13), we get the following two equations

(1 + λ)(1 + q)ma2 =
β

2
h1, (18)

and

(1 + 2λ)(1 + 2q)ma3 = β

(
h2
2

− h21
4

)
. (19)

On the other hand, by consulting Equation (16), the right-hand side of Equation (14)
can be written as:

1 + sinh(βv(w)) = 1 +
βk1
2

w + β

(
k2
2

− k21
4

)
w2

+

(
βk31
8

− βk1k2
2

+
βk3
2

+
β3k31
48

)
w3

+

(
3βk21k2

8
− βk1k3

2
− βk41

16
− βk22

4
− β3k41

32
+

βk4
2

+
β3k21k2
16

)
w4 + · · ·

(20)

More over, considering Equation (20), then comparing coefficients on both sides of
Equation (14) we get the following two equations

−(1 + λ)(1 + q)ma2 =
β

2
k1, (21)

and

(1 + 2λ)(1 + 2q)m(2a22 − a3) = β

(
k2
2

− k21
4

)
. (22)

Now, using Equation (18) and Equation (21), we get the following equation



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3907

a2 =
βh1

2(1 + λ)(1 + q)m
=

−βk1
2(1 + λ)(1 + q)m

. (23)

Hence, the last equation gives the following equation

β2(h21 + k21) = 8(1 + λ)2(1 + q)2ma22. (24)

Moreover, adding Equation (19) to Equation (22), we obtain the following equation

β(h21 + k21) + 8(1 + 2λ)(1 + 2q)ma22 = 2β(h2 + k2).

Therefore, consulting Equation (24), the last equation can be written as

a22 =
β2(h2 + k2)

4β(1 + 2λ)(1 + 2q)m + 4(1 + λ)2(1 + 2q)2m
. (25)

Therefore, considering Equation (25), then using constraints |h2| ≤ 2 and |k2| ≤ 2 , we
get

|a2|2 ≤
β2

β(1 + 2λ)(1 + 2q)m + (1 + λ)2(1 + 2q)2m
, (26)

which gives the desired inequality (11) that represents the coefficient estimate of |a2|.

In the next step, we are looking to determine the coefficient estimate for |a3|. Sub-
tracting Equation (22) from Equation (19), we easily get the following equation

4(1 + 2λ)(1 + 2q)m(a3 − a22) = β(h2 − k2)−
β(h21 − k21)

2
.

Hence, consulting Equation (24), we get β(h21 − k21) = 0. Therefore, the last equation
can be written as

a3 =
β(h2 − k2)

4(1 + 2λ)(1 + 2q)m
+ a22. (27)

Moreover, consulting Equation (24), Equation (27) can be written as

a3 =
β(h2 − k2)

4(1 + 2λ)(1 + 2q)m
+

β2(h21 + k21)

8(1 + λ)2(1 + q)2m
. (28)

Thus, using the constraints |hj | ≤ 2 and |kj | ≤ 2 for all j ∈ N, simple calculations of
Equation (28) gives the required estimation of |a3|. Consequently, the proof of Theorem
1 is now concluded.

The following corollaries come out directly from Theorem 1, they are corresponding
to the examples presented in the previous section, respectively. The methods used in
establishing these corollaries bear a strong resemblance to those used in the proof of the
previous Theorem 1, which is why we have opted to omit the comprehensive proofs’ details.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3908

Corollary 1. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (3) and (4), then the following hold

|a2| ≤
β√

β(1 + 2q)m + (1 + q)2m
,

and

|a3| ≤
β

(1 + 2q)m
+

β2

(1 + q)2m
.

Corollary 2. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (5) and (6), then it can be concluded that

|a2| ≤
β√

3β(1 + 2q)m + 4(1 + q)2m
,

and

|a3| ≤
β

3(1 + 2q)m
+

β2

4(1 + q)2m
.

Corollary 3. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (7) and (8), then the following hold

|a2| ≤
β√

β(1 + 2λ) + (1 + λ)2
,

and

|a3| ≤
β

(1 + 2λ)
+

β2

(1 + λ)2
.

Corollary 4. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (9) and (10), then it can be concluded that

|a2| ≤
β√

3β(1 + 2λ) + 4(1 + λ)2
,

and

|a3| ≤
β

3(1 + 2λ)
+

β2

4(1 + λ)2
.

4. Fekete-Szegö problem of the function class Sq(λ,m, β, sinh)

In this section, we will establish the Fekete-Szegö inequalities for functions that are
members of our class Sq(λ,m, β, sinh) and some of its subclasses.

Theorem 2. If a function f is a member of the class Sq(λ,m, β, sinh) and is represented
by equation (1), then for β ̸= 0 and for a real number ζ the following inequality holds



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3909

|a3 − ζa22| ≤

{
β

(1+2λ)(1+2q)m , if ζ ∈ [ζ1, ζ2]
β2|1−ζ|
βA+B2 , if ζ /∈ [ζ1, ζ2],

(29)

where
A = (1 + 2λ)(1 + 2q)m, B = (1 + λ)(1 + q)m,

ζ1 =
−B2

βA
, and ζ2 = 2− ζ1.

Proof. For any real number ζ, using Equation (27), we get the following equation

a3 − ζa22 =
β(h2 − k2)

4(1 + 2λ)(1 + 2q)m
+ (1− ζ)a22. (30)

Therefore, by consulting Equation (25), the last equation can be written as follows

a3 − ζa22 =
β(h2 − k2)

4(1 + 2λ)(1 + 2q)m

+
β2(1− ζ)(h2 + k2)

4β(1 + 2λ)(1 + 2q)m + 4(1 + λ)2(1 + 2q)2m
.

(31)

Moreover, the last equation can be written as follows

a3 − ζa22 =

(
∆+

β

4A

)
h2 +

(
∆− β

4A

)
k2, (32)

where

∆ =
β2(1− ζ)

4β(1 + 2λ)(1 + 2q)m + 4(1 + λ)2(1 + 2q)2m
.

Now, by applying Lemma 2 on Equation (32), we easily arrive the following inequality

|a3 − ζa22| ≤

{
β

(1+2λ)(1+2q)m , if |∆| ≤ β
4A

β2|1−ζ|
βA+B2 , if |∆| ≥ β

4A .
(33)

Now, considering the following inequality∣∣∣∣ β2(1− ζ)

4β(1 + 2λ)(1 + 2q)m + 4(1 + λ)2(1 + 2q)2m

∣∣∣∣ ≤ β

4A
.

Then, for β ̸= 0, simple calculations gives us the following inequality

−B2

βA
≤ ζ ≤ 2βA+B2

βA
.

Therefore, taking ζ1 =
−B2

βA
and ζ2 =

2βA+B2

βA
, we easily get the desired inequality

that represented by (29). This completes the proof.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (4) (2024), 3899-3914 3910

The following corollaries are natural generated from the previously Theorem 2 under
the conditions presented in the Examples that are given in the second section, respectively.
The approach used to establish this corollary is quite similar to that of the earlier theorem;
hence, we have chosen to omit the comprehensive proof for this corollary.

Corollary 5. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (3) and (4), then for a real number ζ the following holds

|a3 − ζa22| ≤

{
β

(1+2q)m , if ζ ∈ [ζ1, ζ2]
β2|1−ζ|

β(1+2q)m+(1+q)2m
, if ζ /∈ [ζ1, ζ2],

where

ζ1 =
−(1 + q)2m

β(1 + 2q)m
, and ζ2 = 2− ζ1.

Corollary 6. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (5) and (6), then for a real number ζ the following holds

|a3 − ζa22| ≤

{
β

3(1+2q)m , if ζ ∈ [ζ1, ζ2]
β2|1−ζ|

3β(1+2q)m+4(1+q)2m
, if ζ /∈ [ζ1, ζ2],

where

ζ1 =
−4(1 + q)2m

3β(1 + 2q)m
, and ζ2 = 2− ζ1.

Corollary 7. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (7) and (8), then for a real number ζ the following holds

|a3 − ζa22| ≤

{
β

(1+2λ) , if ζ ∈ [ζ1, ζ2]
β2|1−ζ|

β(1+2λ)+(1+λ)2
, if ζ /∈ [ζ1, ζ2],

where

ζ1 =
−(1 + λ)2

β(1 + 2λ)
, and ζ2 = 2− ζ1.

Corollary 8. Let f be a bi-univalent function of the form (1). If the function f satisfies
the subordinations (9) and (10), then for a real number ζ the following holds

|a3 − ζa22| ≤

{
β

3(1+2λ) , if ζ ∈ [ζ1, ζ2]
β2|1−ζ|

3β(1+2λ)+4(1+λ)2
, if ζ /∈ [ζ1, ζ2],

where

ζ1 =
−4(1 + λ)2

3β(1 + 2λ)
, and ζ2 = 2− ζ1.



REFERENCES 3911

5. Conclusion

This research paper explored a novel class of bi-univalent functions characterized by
the generalized Sălăgean differential operator, which is linked to the generalized hyper-
bolic sine function. The author has established estimates for the initial coefficients of the
Taylor-Maclaurin series for functions within this class and has developed the Fekete-Szegö
inequalities relevant to these functions and their various subclasses. The findings of this
study are expected to yield numerous results for subclasses defined through orthogonal
polynomials, such as Legendre, Lagrange, Laguerre, Gegenbauer, and Horadam polyno-
mials. Furthermore, the presented work in this paper will inspire researchers to extend its
concepts to harmonic functions and symmetric q-calculus.

Acknowledgements

This research is partially funded by Zarqa University. The author would like to express
his sincerest thanks to Zarqa University for the financial support.

6. Conflicts of interest

The author confirms that there are no relevant conflicts of interest that are pertinent
to the content of this article.

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