EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5731
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

An Application of Legendre Polynomials to Bi-Bazilevic
Functions associated with q-Ruscheweyh Operator

Waleed Al-Rawashdeh

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

Abstract. In this paper, we make use of the concept of fractional q-calculus to introduce a novel
class of bi-Bazilevic functions involving q-Ruscheweyh differential operator that are subordinate
to Legendre Polynomials. This study explores the characteristics and behaviors of these functions,
providing estimates for the modulus of the initial Taylor series coefficients a2 and a3 within this
specific class and its various subclasses. Additionally, the research delves into the traditional
Fekete-Szegö functional problem of functions f belong to the newly defined class and several of its
subclasses.

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

Key Words and Phrases: Bi-Univalent Functions; Bi-Bazilevic; q-Ruscheweyh Differential Op-
erator; Jackson q-Derivative Operator; q-Gamma Function; Fractional q-Calculus Operator; Legen-
dre Polynomials; Coefficient estimates; Fekete-Szegö functional problem; Convolution; Hadamard
Product

1. Introduction

The q-calculus offers essential tools that are widely utilized to examine different cat-
egories of analytic functions. Various geometric properties, such as coefficient estimates,
convexity, near-convexity, distortion bounds, and radii of starlikeness, have been investi-
gated within these classes of functions. Moreover, q-analysis has garnered considerable
attention in operator theory, as highlighted by the extensive research documented in [10].
The progress made in operator theory within this domain has inspired many researchers,
leading to the publication of a variety of scholarly articles.

Recently, Srivastava [46] has released a comprehensive survey and expository review
paper, which serves as a significant resource for researchers interested in the field of geomet-
ric function theory. This survey meticulously investigates the mathematical frameworks
and applications of fractional q-derivative operators and fractional q-calculus, particularly
in relation to geometric function theory. It addresses the complexities involved in utilizing

DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5731

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

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



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 2 of 20

these fractional operators and calculus concepts to characterize mathematical functions
and their geometric attributes. Furthermore, the review highlights the practical applica-
tions and ramifications of fractional q-derivative operators within the expansive scope of
geometric function theory, thereby offering an in-depth analysis of both the theoretical
underpinnings and practical implementations of these mathematical instruments in the
relevant domain.

Many researchers have employed the concept of q-calculus to establish novel subclasses
of analytic and univalent functions. This investigation seeks to enhance the comprehen-
sion of the properties and attributes of analytic and univalent functions, particularly in
relation to the newly introduced q-derivative, thereby elucidating the criteria that deter-
mine membership within the specified subclasses, see, for example, the articles [9], [22],
[28], [38], [40], [47], [51] and the related references included therein.

In this research paper, the central focus lies in the application of the concept of the
q-derivative to derive specific differential operator. This operator is introduced with the
aim of generalizing the class of qanalogue of the Ruscheweyh operator within the set of
univalent functions. By utilizing the newly defined operator, we define a novel class of
bi-Bazilvic functions associated with the Legendre polynomials.

Now, consider the setH, which consists of all functions f(z) that are analytic within the
open unit disk denoted as D = {z ∈ C : |z| < 1} and normalized by the conditions f(0) =
0 = 1 − f ′(0). The exploration of such functions contributes to a deeper comprehension
of complex analysis and its applications. Moreover, any function f belongs to the set H

can be written as

f(z) = z +

∞∑
n=2

anz
n, where z ∈ D. (1)

The Hadamard product (or convolution) of two analytic functions f(z) given by Equa-

tion (1) and h(z) = z +
∞∑
n=2

bnz
n is defined as:

(f ∗ h)(z) = z +
∞∑
n=2

anbnz
n.

The convolution facilitates deeper mathematical exploration and enhances better un-
derstanding of the geometric and symmetric properties of f ∈ H. The significance of
convolution, within operator theory and geometric function theory, is well-documented in
the literature. For more information about convolution in the geometric function theory,
we invite the interested reader to see the monograph [10], the articles [23], [39] [47], and
the related references provided therein.

Let us consider two functions, f and g, which are analytic within the open unit disk D.
We say that f is subordinated to g in D, denoted as f(z) ≺ g(z) for every z in D, if there



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 3 of 20

exists a Schwarz function w that meets the criteria of w(0) = 0 and |w(z)| < 1 for all z
in D. This means that for every z in D, the relationship f(z) = g(w(z)) holds true. This
concept of subordination is essential in complex analysis, as it allows us to analyze and
compare the behaviors of two analytic functions within the unit disk. Importantly, when
g is a univalent function in D, the condition f(z) ≺ g(z) translates to the equivalence of
f(0) = g(0) and the inclusion f(D) ⊂ g(D). This equivalence underscores the importance
of the subordination principle in elucidating the connections between analytic functions.
For those seeking a deeper understanding and more detailed discussions on the Subordi-
nation Principle, it is recommended to consult the monographs [18], [17], [34], and [36],
which offer thorough explanations and applications of this principle within the realms of
complex analysis and geometric function theory.

In this context, 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)

Now, we introduce the class Σ in the following manner. A function f ∈ H is considered
bi-univalent if both the function itself and its inverse, f−1, are univalent within the unit
disk D. Consequently, we define Σ as the collection of all bi-univalent functions in H that
are represented by equation (1). For more information about univalent and bi-univalent
functions we refer the readers to the articles [30], [32], [37] the monograph [18], [21] and
the references provided therein.

Research in geometric function theory illuminates the complex connections between
coefficients and the geometric properties of functions. By analyzing the constraints on the
modulus of a function’s coefficients, we can better understand the behavior and interac-
tions of these functions within the mathematical landscape. This analytical perspective
not only deepens our grasp of the fundamental principles of geometric function theory but
also opens avenues for further investigation and innovation in this vibrant area of study.
For instance, within the class S, it is shown that the modulus of the coefficient an is limited
by the value of n. These constraints on the modulus of coefficients yield important insights
into the geometric features of these functions. In particular, the bounds on the second
coefficients of functions in the class S provide essential information about the growth and



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 4 of 20

distortion bounds relevant to this class.

The study of coefficient-related characteristics of functions within the bi-univalent class
Σ began in the 1970s. A pivotal moment occurred in 1967 when Lewin [30] investigated
the bi-univalent function class and set a limit for the coefficient |a2|. In 1969, Netanyahu
[37] furthered this research by establishing that the maximum value of |a2| for functions in
Σ is 4

3 . Later, in 1979, Brannan and Clunie [11] proved that for functions belonging to this

class, the inequality |a2| ≤
√
2 is valid. This foundational research has led to a multitude

of studies focused on the coefficient bounds for various subclasses of bi-univalent func-
tions. However, despite the extensive investigations into coefficient bounds, there is still a
considerable lack of understanding regarding the general coefficients |a2| when n ≥ 4. The
difficulty in estimating these coefficients, especially the general coefficient |an|, remains
an open question in the field, underscoring the complexity and depth of the bi-univalent
function class and indicating that further research is essential to grasp the behavior of
these coefficients in higher dimensions.

In 1933, Fekete and Szegö [19] established the upper bound of the expression |a3−λa22|
for univalent functions f , where the 0 ≤ λ ≤ 1. This pivotal finding gave rise to
the Fekete-Szegö problem, which focuses on maximizing the modulus of the functional
Ψλ(f) = a3 − λa22 for functions f belonging to the class H, with λ being any complex
number. A significant body of research has since been dedicated to exploring the Fekete-
Szegö functional and related coefficient estimation issues. Noteworthy contributions to
this area can be found in various publications, including [3], [4], [6], [12], [15], [24], [26],
[31], [32], [48] and the references provided therein. These investigations have significantly
enhanced the comprehension of the Fekete-Szegö problem and its relevance within the
domain of geometric function theory.

2. Preliminaries and Lemmas

The information provided in this section is crucial for comprehending the key findings
of this study. In 1975, Ruscheweyh [43] introduced the operator R, which is defined
through the convolution (Hadamard product) of two power series. In particular, for a
function f ∈ H, a variable z ∈ D, and a real number α > −1, the Ruscheweyh operator is
articulated as follows:

Rαf(z) = f(z) ∗ z

(1− z)α+1
.

For α = n ∈ N0 = N ∪ {0}, we get the Ruscheweyh derivative Rα of the function f :

Rαf(z) = z

(
zα−1f(z)

)(α)
Γ(α+ 1)

.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 5 of 20

Moreover, the Taylor-Maclaurin series (see, for example [27]) of Rαf is given by

Rαf(z) = z +
∞∑
n=2

Γ(α+ n)

Γ(n)Γ(α+ 1)
anz

n.

In this framework, we revisit the concept of q-difference operators, which play a crucial
role in the fields of hypergeometric series, quantum mechanics, and the theory of geometric
functions. The introduction of q-calculus can be traced back to Jackson [23]. Following
this, Kanas and Răducanu [25] utilized fractional q-calculus operators to explore particu-
lar categories of analytic functions associated with conic domains.

The q-integer number, for 0 < q < 1 and non-negative integer n, is defined as follows

[n]q =
1− qn

1− q
=

n−1∑
k=0

qk, with [0]q = 0.

In general, for any non-negative real number x, we have [x]q =
1− qx

1− q
. Moreover, the

q-shifted factorial is defined by

[n]q! = [n]q[n− 1]q[n− 2]q · · · [2]q[1]q, with [0]q! = 1.

It is obvious that lim
q→1−

[n]q = n and lim
q→1−

[n]q! = n!.

Let the function f belong to the set H and represented by Equation (1). The q-Jackson
derivative operator (or q-difference operator) is defined by

Dqf(z) =


f(qz)−f(z)

(q−1)z , if z ̸= 0

f ′(0), if z = 0

f ′(z), as q → 1−.

Therefore, for a function f ∈ H that is given by Equation (1), it is easy to see that

Dqf(z) = 1 +
∞∑
n=2

[n]qanz
n−1.

For example, if n ∈ N = {1, 2, · · ·} and z ∈ D, then

Dq (z
n) =

(qn − 1)zn−1

(q − 1)
= [n]qz

n−1.

Also, lim
q→1−

Dq (z
n) = lim

q→1−
[n]qz

n−1 = nzn−1, which is the ordinary derivative of the func-

tion zn.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 6 of 20

Moreover, for m ∈ N, we have the following

D0
qf(z) = f(z), and Dm

q f(z) = Dq

(
Dm−1
q f(z)

)
.

It is known that, for f, g ∈ H, we have the following rules for the q-difference operator

(i) Dq(mf(z)± ng(z)) = mDqf(z)± nDqg(z), for m,n ∈ C.

(ii) Dq(fg)(z) = f(z)Dqg(z) + g(z)Dqf(z).

(iii) Dq

(
f(z)
g(z)

)
=

g(z)Dqf(z)−f(z)Dqg(z)
g(z)g(qz) , where g(z)g(qz) ̸= 0.

For any real number x and natural number n, the q-generalized Pochhammer symbol
is defined as follows

[x;n]q = [x]q[x+ 1]q[x+ 2]q · · · [x+ n− 1]q.

Moreover, for x > 0, the q-Gamman function is defined as follows

Γq(x+ 1) = [x]qΓq(x), with Γq(1) = 1.

Now, we present a q-analogue of the Ruscheweyh differential operator by employing
the convolution alongside the q-difference operator Rαq : H → H. Thus, for any f ∈ H

and α > −1, this linear operator is defined as Rαq f(z) = Fq,α+1(z) ∗ f(z), where

Fq,α+1(z) = z +
∞∑
n=2

Γq(n+ α)

[n− 1]q!Γq(α+ 1)
zn.

More precisely, the q-Rucheweyh differential operator can be written as follows

Rαq f(z) = z +
∞∑
n=2

ψn(q, α)anz
n,

where

ψn = ψn(q, α) =
Γq(α+ n)

[n− 1]q!Γq(α+ 1)
.

It is clear that,
R0
qf(z) = f(z), R1

qf(z) = zDqf(z), and

Rnq f(z) =
zDn

q

(
zn−1f(z)

)
[n]q!

, n ∈ N

It is worth mention that,

lim
q→1−

Fq,α+1(z) =
z

(1− z)α+1
,



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 7 of 20

and
lim
q→1−

Rαq f(z) = f(z) ∗ z

(1− z)α+1
= Rαf(z).

For more information about q-Rucheweyh differential operator and q-derivative opera-
tor, we refer the interested readers to consult the articles [8], [9], [13], [16], [23], [25], [27],
[44], [46], [50], [51], [52] and the references provided therein.

Legendre polynomials belong to a well-established family of classical orthogonal poly-
nomials. They are defined by their compliance with a second-order linear differential
equation, which emerges naturally in the context of solving initial value problems in three-
dimensional spaces exhibiting spherical symmetry. The equation associated with Legendre
polynomials is classified as a Legendre second-order differential equation:

(1− x2)y′′ − 2xy′ + λy = 0, −1 < x < 1. (3)

The process of identifying the parameters λ ∈ R that allows Equation (3) to possess
a bounded solution is referred to as a singular Sturm-Liouville problem. The significance
of these eigenvalues λ lies in their role in determining the nature of the solutions to the
differential equation. In this context, the necessity for boundary conditions is eliminated,
as the boundedness of the solution itself serves as a substitute for these conditions. It has
been established that the only permissible values of λ that yield bounded solutions are of
the form λ = n(n + 1), where n is a natural number. These values of λ are called the
eigenvalues of the Sturm-Liouville problem.

The polynomial solutions of Legendre’s differential equation can be explicitly expressed
as follows. These solutions play a significant role in various applications, particularly in
mathematical physics and engineering, where they are utilized to solve problems involving
spherical symmetry.

Pn(x) =
1

2n

⌊n/2⌋∑
k=0

(−1)k
(
n

k

)(
2n− 2k

n

)
xn−2k, (4)

where ⌊z⌋ is the floor of z, i.e. the greatest integerm ≤ z. It is worth to mention that when
n is even, the polynomial Pn(x) exclusively comprises even powers of x, while for odd n it
contains only odd powers. Consequently, Pn(x) is classified as an even function for even
n and as an odd function for odd n. Their unique properties, such as orthogonality and
recurrence relations, further enhance their utility in both theoretical and applied mathe-

matics. The first few of them are: P0(x) = 1, P1(x) = x, P2(x) =
1

2
(3x2 − 1), P3(x) =

1

2
(5x3 − 3x), P4(x) =

1

8
(35x4 − 30x2 + 3), P5(x) =

1

8
(63x5 − 70x3 + 15x).

It is important to highlight that the Legendre polynomials can be represented in a
more concise manner. Specifically, the nth Legendre polynomial, denoted as Pn, can be



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 8 of 20

formulated using Rodrigues’ formula (5), which serves as a foundational tool in the study
of these polynomials.

Pn(x) =
1

2nn!

dn

dxn
(x2 − 1)n. (5)

As a result of Rodrigue’s formula, one can observe a specific connection that exists
between three consecutive Legendre polynomials. This relationship plays a crucial role in
understanding the properties and behaviors of these polynomials and their applications in
mathematical physics.

(2n+ 1)xPn(x) = (n+ 1)Pn+1(x) + nPn−1(x).

It can be demonstrated that the Legendre polynomials are produced by the generating
function

g(x, t) =
1√

t2 − 2xt+ 1
.

This relationship highlights the significance of this function in generating these important
polynomials, which have numerous applications in physics and engineering. Additionally,
when the function g(x, t) is expanded as a Taylor series in terms of t, the coefficient
corresponding to tn is the Legendre polynomial Pn(x):

g(x, t) =
∞∑
n=0

Pn(x)t
n, where |x| < 1 and t ∈ D. (6)

In this paper, the symbol P denotes the Caratheodory class, which is formally defined
as

P = {Ω ∈ H : Ω(0) = 1, R(Ω(z)) > 0, z ∈ D}.

It is established in the literature (for example, see [21], page 102) that the function
ϕ(z) is a member of the class P for any real number θ, with ϕ expressed as

ϕ(z) =
1− z√

1− (2 cos θ)z + z2
.

Notably, the function ϕ(z) transforms the open unit disk D onto the right half-plane
R(w) > 0, with the exception of the slit along the positive real axis extending from
|cos(α/2)|−1 to infinity. Consequently, ϕ exhibits starlikeness with respect to the point 1.
By consulting Equation (6), it is straightforward to verify the following equation, for any
z within the open unit disk D.

L(z) =1 +

∞∑
n=1

[Pn(cos θ)− Pn−1(cos θ)] z
n (7)

=1 +

∞∑
n=1

βn(θ)z
n. (8)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 9 of 20

Using the Rodregue’s formula (5), we easily obtain the following initial values of
βn(θ) = Pn(cos θ)− Pn−1(cos θ) which are listed below:

β1(θ) = cos θ − 1, β2(θ) =
1

2
(cos θ − 1)(1 + 3 cos θ).

Additional information regarding the Legengre polynomials readers are encouraged to
consult the articles referenced as [1], [2], [5], [7], [13], [35] and [41], as well as the mono-
graphs [18], [21], [42], [49], and the related sources.

Expanding on these foundational concepts, our objective is to introduce a novel class.
This class is comprised of bi-Bazilevic functions characterized by the q-Ruscheweyh differ-
ential operator associated with Legendre polynomials. We denote this class asBλ(δ,Rαq , ϕ),
and we next provide a formal definition for this class.

Definition 1. A function f(z) belongs to the family Σ is considered to be part of the class
Bλ(δ,Rαq , ϕ) if it obeys the following subordination conditions:

eiδz1−λ
(
Rαq f(z)

)′(
Rαq f(z)

)1−λ ≺ ϕ(z) cos δ + i sin δ,

and
eiδw1−λ (Rαq g(w))′(

Rαq g(w)
)1−λ ≺ ϕ(w) cos δ + i sin δ,

where the function g(w) = f−1(w) is given by the Equation (2), the parameters λ ≥ 0,
0 < q < 1, α > −1, and δ ∈

(−π
2 ,

π
2

)
.

The following lemma, extensively elaborated upon in existing literature, represents
well-established principles that hold significant importance for the research we are cur-
rently presenting.

Lemma 1. [26] if Ω belongs to the Caratheodory class, then for z ∈ D the function Ω can
be written as

Ω(z) = 1 + c1z + c2z
2 + c3z

3 + · · ·

Moreover, |cn| ≤ 2 for each natural number n.

The lemma presented in the following discussion is extensively referenced in exist-
ing literature and is regarded as a foundational principle that significantly influences the
research we are conducting.

Lemma 2. [26] 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|.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 10 of 20

This paper seeks to explore two novel categories of bi-Bazilevic functions that are
defined through the q-Ruscheweyh operator within the open unit disk D, with a particular
connection to Legendre polynomials. The central objective is to establish estimates for the
magnitudes of the initial coefficients |a2| and |a3| that are linked to the Taylor-Maclaurin
series representation of functions belonging to this class. Additionally, the research delves
into the Fekete-Szegö functional problem pertinent to these functions, thereby enhancing
the comprehension of their inherent characteristics. Moreover, some known corollaries are
presented based on the choices of the parameters involved in defining our specific class.

3. Coefficient bounds of the function class

This section of the paper focuses on investigating the bounds pertaining to the modulus
of the initial coefficients of functions belonging to the class Bλ(δ,Rαq , ϕ), along with several
of its distinct subclasses, as delineated in Equation (1).

Theorem 1. Let a function f be in the family Σ. If the function f belongs to the class
Bλ(δ,Rαq , ϕ) and is represented by the equation (1), then the following inequalities hold:

|a2| ≤
√
2|1− cos θ| cos δ√

|A cos δ(cos θ − 1) + (1− 3 cos θ)(λ+ 1)2ψ2
2e
iδ|
, (9)

and

|a3| ≤
|1− cos θ| cos δ

(λ+ 2)ψ3
+

(1− cos θ)2 cos2 δ

(λ+ 1)2ψ2
2

, (10)

where
A = 2(λ+ 2)ψ3 + (λ− 1)(λ+ 2)ψ2

2.

Proof. Suppose a function f belongs to the class B(λ, δ,Rαη , β(t)). Consulting the
Definition 1 and Subordination Principle, we can find two Schwarz functions k(z) and
h(w) defined on the open unit disk D such that

eiδz1−λ
(
Rαq f(z)

)′(
Rαq f(z)

)1−λ = ϕ(k(z)) cos δ + i sin δ, (11)

and
eiδw1−λ (Rαq g(w))′(

Rαq g(w)
)1−λ = ϕ(h(w)) cos δ + i sin δ. (12)

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

η(z) =
1 + k(z)

1− k(z)
and ζ(w) =

1 + h(w)

1− h(w)
.

It is clear that, these functions η(z) and ζ(w) are analytic in the open unit disk D and
belong to the Caratheodory class. Therefore, they can be written as follows

η(z) =
1 + k(z)

1− k(z)
= 1 + η1z + η2z

2 + · · ·



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 11 of 20

and

ζ(w) =
1 + h(w)

1− h(w)
= 1 + ζ1w + ζ2w

2 + · · ·

Moreover, η(0) = 1 = ζ(0), ℜ(η) > 0, ℜ(ζ) > 0, |ηj | ≤ 2 and |ζj | ≤ 2 for all natural
numbers j.

Equivalently, we get the following representations of k(z) and h(w)

k(z) =
η(z)− 1

η(z) + 1
=
η1
2
z +

(
η2
2

− η21
4

)
z2 + · · ·, (13)

and

h(w) =
ζ(w)− 1

ζ(w) + 1
=
ζ1
2
w +

(
ζ2
2

− ζ21
4

)
w2 + · · ·. (14)

Therefore, by consulting Equation (7) and Equation (13) the right-hand side of Equa-
tion (11) can be written as:

ϕ(k(z)) cos δ + i sin δ

=

(
1 +

β1η1
2

z +

[
β1

(
η2
2

− η21
4

)
+
β2η

2
1

4

]
z2 + · · ·

)
cos δ + i sin δ.

(15)

Moreover, the left-hand side of Equation (11) can be written as:

eiδz1−λ
(
Rαq f(z)

)′(
Rαq f(z)

)1−λ
= eiδ(λ+ 1)ψ2a2z + eiδ

[
(λ− 1)(λ+ 2)

2
ψ2
2a

2
2 + (λ+ 2)ψ3a3

]
z2 + · · ·

(16)

Now, consulting Equation (11), we get the right-hand sides of Equation (15) and
Equation (16) are equal. Therefore comparing these equations coefficients we get the
following two equations:

2eiδ(λ+ 1)ψ2a2 = β1η1 cos δ, (17)

and

eiδ
[
2(λ− 1)(λ+ 2)ψ2

2a
2
2 + 4(λ+ 2)ψ3a3

]
= [2β1η2 + (β2 − β1)η

2
1] cos δ.

(18)

On the other hand, by consulting Equation (7) and Equation (14) the right-hand side
of Equation (12) can be written as:

ϕ(h(w)) = 1 +
β1ζ1
2
w +

[
β1

(
ζ2
2

− ζ21
4

)
+
β2ζ

2
1

4

]
w2 + · · · (19)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 12 of 20

Moreover, the left-hand side of Equation (12) can be written as:

eiδw1−λ (Rαq g(w))′(
Rαq g(w)

)1−λ
= −eiδ(λ+ 1)ψ2a2w + eiδ

2(λ+ 2)ψ3

+
(λ− 1)(λ+ 2)

2
ψ2
2

 a22 − (λ+ 2)ψ3a3

w2 + · · ·

(20)

Now, considering Equation (12) and comparing coefficients on bothsides of Equation
(19) and Equation(20) we get the following two equations:

−2eiδ(λ+ 1)ψ2a2 = β1ζ1 cos δ, (21)

and

eiδ
([
8(λ+ 2)ψ3 + 2(λ− 1)(λ+ 2)ψ2

2

]
a22 − 4(λ+ 2)ψ3a3

)
= [2β1ζ2 + (β2 − β1)ζ

2
1 ] cos δ.

(22)

Therefore, using Equation (17) and Equation (21), we easily derive the following equa-
tion

eiδa2 =
β1η1 cos δ

2(λ+ 1)ψ2
=

−β1ζ1 cos δ
2(λ+ 1)ψ2

. (23)

On one hand, adding Equation (18) to Equation (22), we obtain the following equation

eiδ
[
4(λ− 1)(λ+ 2)ψ2

2 + 8(λ+ 2)ψ3

]
a22

= [2β1(η2 + ζ2) + (β2 − β1)(η
2
1 + ζ21 )] cos δ.

(24)

on the other hand, consulting Equation (23), we obtain the following equation:

η21 + ζ21 =
8(λ+ 1)2ψ2

2e
i(2δ)

β21 cos
2 δ

a22. (25)

Now, using Equation (24) and Equation (25), we easily derive the following equations

β21 cos δe
iδ
[
4(λ− 1)(λ+ 2)ψ2

2 + 8(λ+ 2)ψ3

]
a22

= 2β31 cos
2 δ(η2 + ζ2) + 8(β2 − β1)(λ+ 1)2ψ2

2e
i(2δ)a22.

Therefore, considering the initial values β1 = cos θ − 1 and 2(δ2 − δ1) = (3 cos θ −
1)(cos θ − 1), we easily get the following equation

a22 =
β21 cos

2 δ(η2 + ζ2)e
−iδ

β1 cos δ[2(λ− 1)(λ+ 2)ψ2
2 + 4(λ+ 2)ψ3] + 2(1− 3 cos θ)(λ+ 1)2ψ2

2e
iδ
. (26)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 13 of 20

Thus, using the constraints |η2| ≤ 2 and |ζ2| ≤ 2, then simple calculations give the
desired estimation of |a2| presented in Equation (9).

In the next step, we seek to determine the coefficient estimate for |a3|. By substituting
Equation (22) from Equation (18), we can derive the following equation:

8eiδ(λ+ 2)ψ3(a3 − a22) = [2β1(η2 − ζ2) + (β2 − β1)(η
2
1 − ζ21 )] cos δ.

Now, consulting Equation (23), we get η1 = −ζ1. Hence, the last equation can be
written as

a3 =
β1(η2 − ζ2) cos δ

4eiδ(λ+ 2)ψ3
+ a22. (27)

Moreover, using Equation (25), the last equation can be written as:

a3 =
β1(η2 − ζ2) cos δ

4eiδ(λ+ 2)ψ3
+
β21(η

2
1 + ζ21 ) cos

2 δ

8δei(2δ)(λ+ 1)2ψ2
2

. (28)

Finally, considering the value β1 = cos θ − 1, then using the constraints |ηj | ≤ 2
and |ζj | ≤ 2 for all j ∈ N, the last equation gives the required estimation of |a3| that is
represented by the Inequality (10). Consequently, the proof of Theorem 1 is now concluded.

By selecting particular values of λ in Definition 1, it is possible to obtain the subsequent
subclasses.

Example 1. A bi-univalent function f that represented as (1) belongs to the subclass
B0(δ,Rαq , ϕ) if the following subordinations hold:

eiδz
(
Rαq f(z)

)′
Rαq f(z)

≺ ϕ(z) cos δ + i sin δ, (29)

and
eiδw

(
Rαq g(w)

)′
Rαq g(w)

≺ ϕ(w) cos δ + i sin δ, (30)

where the function g(w) = f−1(w) is given by the Equation (2), the parameters 0 < q < 1,
α > −1, and δ ∈

(−π
2 ,

π
2

)
.

Example 2. A bi-univalent function f that represented as (1) belongs to the subclass
B1(δ,Rαq , ϕ) if the following subordinations hold:

eiδ
(
Rαq f(z)

)′ ≺ ϕ(z) cos δ + i sin δ, (31)

and
eiδ

(
Rαq g(w)

)′ ≺ ϕ(w) cos δ + i sin δ, (32)

where the function g(w) = f−1(w) is given by the Equation (2), the parameters 0 < q < 1,
α > −1, and δ ∈

(−π
2 ,

π
2

)
.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 14 of 20

Moreover, as q → 1− and taking α = 0, we get R0
qf(z) = f(z). Therefore, we get the

following close-to-starlike subclasses.

Example 3. A bi-univalent function f that represented as (1) belongs to the subclass
S∗(δ, ϕ) if the following subordinations hold:

eiδ
(
zf ′(z)

f(z)

)
≺ ϕ(z) cos δ + i sin δ, (33)

and

eiδ
(
wg′(w)

g(w)

)
≺ ϕ(w) cos δ + i sin δ (34)

where g(w) = f−1(w) is given by the Equation (2) and δ ∈
(−π

2 ,
π
2

)
.

Example 4. A bi-univalent function f that represented as (1) belongs to the subclass
G∗(δ, ϕ) if the following subordinations hold:

eiδ
(
f ′(z)

)
≺ ϕ(z) cos δ + i sin δ, (35)

and
eiδ

(
g′(w)

)
≺ ϕ(w) cos δ + i sin δ, (36)

where g(w) = f−1(w) is given by the Equation (2) and δ ∈
(−π

2 ,
π
2

)
.

The subsequent corollaries are directly obtained from Theorem 1, contingent upon the
conditions specified in the earlier examples. The techniques employed in deriving these
corollaries closely mirror those applied in the proof of Theorem 1, which is the rationale
behind our decision to exclude the detailed proofs.

Corollary 1. If a function f ∈ Σ is represented by (1) and belong to the class B0(δ,Rαq , ϕ),
then it can be concluded that

|a2| ≤
√
2|1− cos θ| cos δ√

| cos δ(cos θ − 1)(4ψ3 − 2ψ2
2) + (1− 3 cos θ)ψ2

2e
iδ|
,

and

|a3| ≤
|1− cos θ| cos δ

2ψ3
+

(1− cos θ)2 cos2 δ

ψ2
2

.

Corollary 2. If a function f ∈ Σ is represented by (1) and belong to the class B1(δ,Rαq , ϕ),
then it can be concluded that

|a2| ≤
|1− cos θ| cos δ√

|2 cos δ(cos θ − 1)ψ3 + 2(1− 3 cos θ)ψ2
2e
iδ|
,

and

|a3| ≤
|1− cos θ| cos δ

3ψ3
+

(1− cos θ)2 cos2 δ

4ψ2
2

.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 15 of 20

Corollary 3. If a function f ∈ Σ is represented by (1) and belong to the class S∗(δ, ϕ),
then it can be concluded that

|a2| ≤
√
2|1− cos θ| cos δ√

|2 cos δ(cos θ − 1) + (1− 3 cos θ)eiδ|
,

and

|a3| ≤
|1− cos θ| cos δ

2
+ (1− cos θ)2 cos2 δ.

Corollary 4. If a function f ∈ Σ is represented by (1) and belong to the class G∗(δ, ϕ),
then it can be concluded that

|a2| ≤
|1− cos θ| cos δ√

|2 cos δ(cos θ − 1) + 2(1− 3 cos θ)eiδ|
,

and

|a3| ≤
|1− cos θ| cos δ

3
+

(1− cos θ)2 cos2 δ

4
.

4. Fekete-Szegö problem of the function class Bλ(α,Rα
q , ϕ)

In this section, we will derive the Fekete-Szegö inequalities for functions belonging to
the class Bλ(α,Rαq , ϕ), which encompasses bi-Bazilevic functions defined through the q-
Ruscheweyh differential operator and associated with Legendre Polynomials. Additionally,
we aim to establish Fekete-Szegö inequalities for several subclasses within our defined class.

Theorem 2. If a function f is a member of the class Bλ(α,Rαq , ϕ) and is represented by
equation (1), then for a real number γ the following inequality holds

|a3 − γa22| ≤

{ |1−cos θ| cos δ
2(λ+2)ψ3

, if |1− ζ| ≤ |∆|
|1−cos θ||1−γ| cos δ

|Aβ1 cos δ+B2(1−3 cos θ)eiδ| , if |1− ζ| ≥ |∆|,
(37)

where
A = 2(λ+ 2)ψ3 + (λ− 1)(λ+ 2)ψ2

2, B = (λ+ 1)ψ2, and

∆ =
|A(cos θ − 1) cos δ +B2(1− 3 cos θ)eiδ|

2|1− cos θ| cos δ(λ+ 2)ψ3
.

Proof. For any real number γ, using Equation (26) and Equation (27), we easily derive
the following equations

a3 − γa22 =
β1 cos δe

−iδ(η2 − ζ2)

4(λ+ 2)ψ3

+
β21 cos

2 δe−iδ(η2 + ζ2)(1− γ)

β1 cos δ[4(λ+ 2)ψ3 + 2(λ− 1)(λ+ 2)ψ2
2] + 2(1− cos θ)(λ+ 1)2ψ2

2e
iδ



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 16 of 20

=
(
β1 cos δe

−iδ
){(

µ+
1

4(λ+ 2)ψ3

)
η2 +

(
µ− 1

4(λ+ 2)ψ3

)
ζ2

}
,

where

µ =
(1− γ)β1 cos δe

−iδ

2Aβ1 cos δ + 2B2(1− cos θ)eiδ
.

Therefore, with the assistance of Lemma 2, we are able to achieve the following in-
equality

|a3 − γa22| ≤


2|β1 cos δe−iδ|

4(λ+2)ψ3
, if |µ| ≤ 1

4(λ+2)ψ3

2
∣∣β1 cos δe−iδ∣∣ |µ|, if |µ| ≥ 1

4(λ+2)ψ3
.

Finally, by streamlining the right-hand side of the final inequality, we arrive at the
expected result as presented in inequality (37). This signifies the completion of the proof.

The subsequent corollaries emerge as logical extensions of Theorem 2, given the con-
ditions outlined in the preceding examples. The methodology employed to derive this
corollary closely resembles that utilized in the earlier theorem; therefore, we have opted
to forgo a detailed proof for this corollary.

Corollary 5. If a function f ∈ Σ is represented by equation (1) and is obeying the
Subordination conditions (29) and (30), then for a real number γ the following holds

|a3 − γa22| ≤

{ |1−cos θ| cos δ
4ψ3

, if |1− ζ| ≤ |∆1|
|1−cos θ||1−γ| cos δ

|β1 cos δ(4ψ3−2ψ2
2)+(1−3 cos θ)ψ2

2e
iδ| , if |1− ζ| ≥ |∆1|,

where

∆1 =
|β1 cos δ(4ψ3 − 2ψ2

2) + (1− 3 cos θ)ψ2
2e
iδ|

4|1− cos θ| cos δψ3
.

Corollary 6. If a function f ∈ Σ is represented by equation (1) and is obeying the
Subordination conditions (31) and (32), then for a real number ζ the following holds

|a3 − γa22| ≤

{ |1−cos θ| cos δ
6ψ3

, if |1− ζ| ≤ |∆2|
|1−cos θ||1−γ| cos δ

|6β1 cos δψ3+4(1−3 cos θ)ψ2
2e

iδ| , if |1− ζ| ≥ |∆2|,

where

∆2 =
|3(cos θ − 1) cos δψ3 + 2(1− 3 cos θ)ψ2

2e
iδ|

3|1− cos θ| cos δψ3
.

Corollary 7. If a function f ∈ Σ is represented by equation (1) and is obeying the
Subordination conditions (33) and (34), then for a real number ζ the following holds

|a3 − γa22| ≤

{ |1−cos θ| cos δ
4 , if |1− ζ| ≤ |∆3|

|1−cos θ||1−γ| cos δ
|2β1 cos δ+(1−3 cos θ)2eiδ| , if |1− ζ| ≥ |∆3|,

where

∆3 =
|2β1 cos δ + (1− 3 cos θ)eiδ|

4|1− cos θ| cos δ
.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5731 17 of 20

Corollary 8. If a function f ∈ Σ is represented by equation (1) and is obeying the
Subordination conditions (35) and (36), then for a real number ζ the following holds

|a3 − γa22| ≤

{ |1−cos θ| cos δ
6 , if |1− ζ| ≤ |∆4|

|1−cos θ||1−γ| cos δ
|6β1 cos δ+4(1−3 cos θ)eiδ| , if |1− ζ| ≥ |∆4|,

where

∆4 =
|3(cos θ − 1) cos δ + 2(1− 3 cos θ)eiδ|

3|1− cos θ| cos δ
.

Remark 1. Assuming δ = 0, the results presented in this paper would give various new
and known results. Moreover, taking δ = 0 in Example 3, would lead to the known classes
of starlike bi-univalent functions that studied by many researchers see, for example, [7],
[14], [20], [29], [33], and [45]. More precisely, the results presented in those papers are
just special case of the class mentioned in Example 3.

5. Conclusion

This research paper investigates a new category of bi-Bazilevic functions that are
defined through the q-Ruscheweyh differential operator and are linked to Legendre poly-
nomials. The author has derived estimates for the initial coefficients and examined the
Fekete-Szegö functional problem concerning functions within these specific classes. In con-
clusion, potential avenues for future research are suggested, particularly the exploration
of substituting Legendre polynomials with other types of orthogonal polynomials, such as
Gegenbauer polynomials. Furthermore, the findings presented in this study are anticipated
to motivate researchers to expand the scope of this investigation to include meromorphic
bi-univalent functions.

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. The author expresses
sincere gratitude to the anonymous referees for their insightful comments and recommen-
dations, which significantly enhanced the quality of this paper’s presentation.

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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