EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6294
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Representation of Bi-Univalent Functions to Lucas
Balancing Polynomials with Geometric Properties and

Coefficient Bounds
Stalin Thangamani1, Pshtiwan Othman Mohammed2,3,∗, Jeno Francis Devadoss4,
Majeed A. Yousif5, Meraa Arab6,∗, Dumitru Baleanu7

1 Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R & D Institute of
Science and Technology, Avadi, Chennai 600062, India
2 Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq
3 Research Center, University of Halabja, Halabja 46018, Iraq
4 Department of Mathematics, Prathyusha Engineering College, Anna University, Chennai
602025, India
5 Department of Mathematics, College of Education, University of Zakho, Zakho 42002, Iraq
6 Department of Mathematics and Statistics, College of Science, King Faisal University,
Hofuf 31982, Al Ahsa, Saudi Arabia
7 Department of Computer Science and Mathematics, Lebanese American University, Beirut
11022801, Lebanon

Abstract. In this paper, we introduce and analyze a new subclass of bi-univalent functions defined
in the open unit disk, associated with Lucas and Lucas balancing polynomials. By employing the
Taylor-Maclaurin series expansion, precise bounds for the second and third coefficients, |a2| and
|a3|, are obtained. These estimates lead to important geometric interpretations related to the
distortion, growth, and structural behavior of the functions near the origin. Furthermore, the
mapping characteristics of these functions are examined in connection with existing bi-univalent
subclasses. Special emphasis is placed on deriving a Fekete-Szegö type inequality for the proposed
class, thereby extending earlier contributions in the field. The findings presented here are valuable
for applications in geometric function theory and areas like fluid mechanics, conformal mappings,
and engineering models where the analytic structure of bi-univalent functions plays a significant
role.
2020 Mathematics Subject Classifications: 30C45, 30C50
Key Words and Phrases: Bi-univalent functions, Lucas polynomial, Fekete-Szegö inequality

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6294

Email addresses: drstalint@veltech.edu.in (S. Thangamani),
pshtiwansangawi@gmail.com (P.O. Mohammed), jenofrancis29@gmail.com (J. F. Devadoss),
majeed.yousif@uoz.edu.krd (M.A. Yousif), marab@kfu.edu.sa (M. Arab),
dumitru.baleanu@lau.edu.lb (D. Baleanu)

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



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 2 of 19

1. Introduction

In the open unit disk D = {Θ ∈ C : |Θ | < 1}, A often denotes the class of normalized
and analytic functions f (Θ) of the form

f (Θ) = Θ +
∞∑

k=2

akΘ
k (1)

and S act as a subclass of all functions in A. As stated by Koebe one-quarter theorem [1],
if f ∈ S is a univalent function, then the image of unit disk under f will contain a disk of
radius 1

4 . As a results, for all univalent functions f ∈ S, there exists an inverse function
f −1 that satisfies f −1[f (Θ)] = Θ and f [f −1(Φ)] = g(Φ) for |Φ| < r0 (f ) : r0 (f ) ≥ 1

4 . A
power series in f (Θ) given by Equation (1) can be used to define the inverse function,
where a2, a3, . . . are complex coefficients. If both a function f and a function f −1 are
univalent in D, the function is said to be bi-univalent in the unit disc. Let Σ signify the
class of bi-univalent functions in D written by (1). Some familiar function for the class Σ

are specified below: Θ
1−Θ , 1

2 log
(
1+Θ
1−Θ

)
, −log(1 −Θ).

Lewin [2] initially derived the bi-univalent functions class Σ, and found that |a2| < 1.51.
For the generalized subclass of class Σ Brannan and Clunie [3] later enhanced this outcome
as |a2| ≤

√
2. Brannan and Taha [4] focused on a specific subclass of class Σ, also estimate

the initial coefficients |a2| and |a3|. For convex and starlike functions, Ma Minda [5]
employed a number of subclasses that meet the requirement that either 1 + Θf ′′(Θ)

f ′(Θ) or
Θf ′(Θ)
f(Θ) is subordinate to a more comprehensive subordinate function. In order to do this,

Ma Minda looked at an analytic function φ, it uses the unit disk to map D onto an
area that is starlike with regard to 1 and also meets the requirements given that it is
symmetrical relating to the real axis, also full files the conditions φ(0) = 1 and φ′(0) > 0.
The function f in the Ma Minda starlike class satisfies subordination Θf ′(Θ)

f(Θ) ≺ φ(Θ).
The Ma Minda convex class , in a similar vein, consists of functions f that fulfill the
subordination 1 + Θf ′′(Θ)

f ′(Θ) ≺ φ(Θ).

A large number of integer number sequences have surfaced due to the elegance of their
recurrence relations, including Fibonacci, Jacobsthal, Lucas, Fermat, Pell, Chebyshev,
Telephone and others. The Lucas numbers are a brand-new integer series that Behera
and Panda [6] revealed.In recent years,[7–11] several authors have explored subclasses of
biunivalent (or univalent) functions Over the past few decades, the features of this unique
number sequence have been intensively investigated, and certain generalizations have been
developed. A definition and a number of interesting properties of the polynomials are given
by Lee et al. [12] and Ray [13]. The polynomials are a innate consequence of the Lucas
numbers and Balancing Lucas numbers. Due to their applications in fluid dynamics, ge-
ometric function theory, and conformal mapping, bi-univalent functions have attracted a
lot of attention in complex analysis. These functions have intriguing geometric proper-
ties that can provide important information on growth behavior, distortion, and mapping



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 3 of 19

properties. A greater comprehension of the structural characteristics of bi-univalent func-
tions and how they interact with classical inequalities can be gained by studying their
subclasses. Lucas polynomials are one such class of functions, and their special charac-
teristics can be used to study how these functions behave. In recent years, a multitude of
writers have made great strides in implementing sharp bounds for a variety of bi-univalent
function subclasses that commonly interact with specific polynomial families, including
Lucas and balancing Lucas polynomials [14–24] and [25] .

The primary objective of this work is to determine the Taylor-Maclaurin coefficients |a2|
and |a3| and their influence on the geometric and analytic features of the bi-univalent
subclass of Lucas and Lucas balancing polynomial functions. By presenting new results
and building on previous work, this study contributes to our understanding of bi-univalent
functions.

The main objective of this study is to provide upper bounds and initial coefficients for
the Taylor-Maclaurin and Fekete-Szegö functionals of the function of the stated subclass.
This research is divided into three parts. In the first section, some basic concepts of
bi-univalent function theory are discussed, including information on Lucas and balancing
Lucas polynomials. Parts two and three introduce new subclasses of bi-univalent functions
using the Lucas and balancing Lucas polynomials, respectively.

The motivation for this study stems from the need to better understand the intricate geo-
metric and analytic properties of bi-univalent functions. These functions play a crucial role
in complex analysis, particularly in areas such as conformal mappings and fluid dynamics,
where the behavior of functions near boundaries and their growth rates are of significant
importance. By focusing on the Taylor-Maclaurin coefficients and their implications, this
work aims to provide deeper insights into how these functions behave in the unit disk and
their connections to other classes of functions.

In addition to their theoretical significance, the initial coefficients have practical implica-
tions. These include calculating the degree to which bi-univalent functions can resemble
other kinds of functions, creating numerical algorithms to find bi-univalent functions using
the initial coefficients, and using these functions in complex analysis, particularly when
examining boundary behavior of mappings from the unit disk and conformal mappings.
Furthermore, a function’s growth rate is frequently estimated using the initial coefficients.
In particular, they help define limits on the function’s maximum modulus and behav-
ior near the unit disk’s edge. By setting upper constraints for these coefficients, we can
regulate how much the function ”spreads” or ”compresses” inside the unit disk.
The structure of the paper is organized systematically to facilitate a coherent flow of ideas
and results. Section 2 presents the initial coefficient estimates for the subclass GΣ,G

u,v (τ),
laying the foundation for the subsequent analysis. Building on this, Section 3 derives the
Fekete-Szegö inequality for the same subclass, highlighting important functional bounds.
In Section 4, attention shifts to a related subclass GΣ,G(τ), where initial coefficient es-
timates are again established. Following this, Section 5 investigates the Fekete-Szegö



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 4 of 19

inequality for GΣ,G(τ), offering a comparative view of the functional behavior. Section 6
is dedicated to a discussion on the influence of the parameter s, supported by surface plot
interpretations that provide visual insights into the parameter’s effect. Finally, Section 7
concludes the paper by summarizing the key findings and highlighting potential directions
for future research.

2. The initial Coefficient estimates for the subclass GΣ,G
u,v (τ)

The Lucas polynomials Ln(u(s), v(s), s) are defined by the recurrence relation that
follows:

Ln(s) = u(s)Ln−1(s) + v(s)Ln−2(s)

with initial conditions:
L0(s) = 2, L1(s) = u(s),

The generating function for Ln(s) is given by:

G(s,Θ) =
2− u(s)Θ

1− u(s)Θ− v(s)Θ2
.

The generating function G(s,Θ) is related to the sequence Ln(u(s), v(s), s) by the expan-
sion:

G(s,Θ) =
∞∑
n=0

Ln(u(s), v(s), s)Θ
n

Thus, we can express the generating function as:

G(s,Θ) =
2− u(s)Θ

1− u(s)Θ− v(s)Θ2
= 2+u(s)Θ+(2v(s)+u2 (s))Θ2+(3u(s)v(s)+u3 (s))Θ3+...,

(2)

∴ G(s,Θ) =

∞∑
n=0

Ln(u(s), v(s), s)Θ
n.

where

L0(s) = 2, L1(s) = u(s), L2(s) = 2v(s) + u2 (s) L3(s) = u3 (s) + 3u(s)v(s), ...

Remark 1.
It should be noted that the (u, v)-polynomial Ln(s) leads to different polynomials for certain
values of u and v. Here, we highlight a few examples from those:

(i) For u(s) = 2s and v(s) = 1, then obtain the polynomials of Pell-Lucas Qn(s).

(ii) For u(s) = 1 and v(s) = 2s, then acquire polynomials of Jacobsthal-Lucas jn(s).

(iii) For u(s) = 3s and v(s) = −2, then obtain the polynomials of Fermat-Lucas fn(s).



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 5 of 19

(iv) For u(s) = 2s and v(s) = −1, then acquire first kind Chebyshev polynomials Tn(s).

The following lemma is to be kept forefront in order to get at our primary conclusions.

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

Re (p(Θ)) > 0, p(Θ) = 1 + p1Θ + p2Θ
2 + ...; Θ ∈ D.

In the following, it is assumed that G(s,Θ) is form of Taylor series nature

G(s,Θ)−1 = 1+u(s)Θ+2v(s)+u2 (s))Θ2+(3u(s)v(s)+u3 (s))Θ3+(u4 (s)+4u2 (s)+2v(s))Θ4+...,
(3)

G(t,Φ)−1 = 1+u(t)Φ+2v(t)+u2 (t))Φ2+(3u(t)v(t)+u3 (t))Φ3+(u4 (t)+4u2 (t)+2v(t))Φ4+...,
(4)

Definition 1. If the function f ∈ Σ is considered to be in the class GΣ,G
u,v (τ), then the

resulting subordination holds,

(1− τ)

(
Θf ′(Θ)

f(Θ)

)
+ τ

(
1 +

Θf ′′(Θ)

f ′(Θ)

)
≺ G(s,Θ)− 1 (5)

and

(1− τ)

(
Φf ′(Φ)

f(Φ)

)
+ τ

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
≺ G(t,Φ)− 1 (6)

Example 1. When replacing τ = 0 in definition (1) then the particular subordination is
hold for the class GΣ,G

u,v (0). (
Θf ′(Θ)

f(Θ)

)
≺ G(s,Θ)− 1

(
Φf ′(Φ)

f(Φ)

)
≺ G(t,Φ)− 1

Example 2. When replacing τ = 1 in definition (1) then the particular subordination is
hold for the class GΣ,G

u,v (1). (
1 +

Θf ′′(Θ)

f ′(Θ)

)
≺ G(s,Θ)− 1

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
≺ G(t,Φ)− 1



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 6 of 19

Theorem 1. If f (Θ) defined by the equation (1) is belongs to the class GΣ,G
u,v (τ). Then

|a2| ≤

√
u2 (s) + u2 (t) + 2(v(s) + v(t))

(2 + 2τ)

and
|a3| ≤

u2 (s) + u2 (t)

2(1 + τ)2
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + 2τ)
(7)

Proof. Let f ∈ GΣ,G
u,v (τ). Then there are two analytic functions f, g : D → D given by

(5) and (6) such that

(1− τ)

(
Θf ′(Θ)

f(Θ)

)
+ τ

(
1 +

Θf ′′(Θ)

f ′(Θ)

)
= G(s,Θ)− 1

and
(1− τ)

(
Φf ′(Φ)

f(Φ)

)
+ τ

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
= G(t,Φ)− 1

Since

(1−τ)

(
Θf ′(Θ)

f(Θ)

)
+τ

(
1 +

Θf ′′(Θ)

f ′(Θ)

)
= 1+(1+τ)a2Θ+(2(1+2τ)a3− (1+3τ)a22)Θ

2+ ...

(8)
and

(1−τ)

(
Φf ′(Φ)

f(Φ)

)
+τ

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
= 1− (1+τ)a2Φ+((3+5τ)a22−2(1+2τ)a3)Φ

2+ ...

(9)
From the equations (3) and (8), By comparing coefficients of Θ and Θ2 respectively, we
get coefficient of Θ :

(1 + τ)a2 = u(s) (10)

coefficient of Θ2 :
2(1 + 2τ)a3 − (1 + 3τ)a22 = u2 (s) + 2v(s) (11)

From the equations (4) and (9), equating the coefficients of Φ and Φ2 respectively, we get
coefficient of Φ :

−(1 + τ)a2 = u(t) (12)

coefficient of Φ2 :
(3 + 5τ)a22 − 2(1 + 2τ)a3 = u2 (t) + 2v(t) (13)

Now, Adding the equations (10) and (12), we have

u(s) = −u(t) (14)

Squaring and adding the equations (10) and (12), we get

2(1 + τ)2a22 = u2 (s) + u2 (t)



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 7 of 19

a22 =
u2 (s) + u2 (t)

2(1 + τ)2
(15)

subtracting the equations (11) and (13), we get

2(1 + 2τ)a3 − 4(1 + 2τ)a22 + 2(1 + 2τ)a3 = u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + 2τ)a3 = 4(1 + 2τ)a22 − u(t)2 + u2 (s) + 2(v(s)− v(t))

a3 = a22 +
u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + 2τ)
(16)

substituting the equations (11) in equation (13), we get

(2 + 2τ)a22 = u2 (s) + 2v(s) + u2 (t) + 2v(t)

a22 =
u2 (s) + u2 (t) + 2(v(s) + v(t))

(2 + 2τ)
(17)

|a2| ≤

√
u2 (s) + u2 (t) + 2(v(s) + v(t))

(2 + 2τ)
(18)

substituting the equations (15) in equation (16), we get

|a3| ≤
u2 (s) + u2 (t)

2(1 + τ)2
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + 2τ)
(19)

substituting the equation (17) in (16), we attain

a3 =
u2 (s) + u2 (t) + 2(v(s) + v(t))

(2 + 2τ)
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + 2τ)

|a3| ≤
u2 (s) + 2v(s)

(2 + 2τ)
(20)

Hence,

|a2| ≤

√
u2 (s) + u2 (t) + 2(v(s) + v(t))

(2 + 2τ)

and
|a3| ≤

u2 (s) + u2 (t)

2(1 + τ)2
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + 2τ)

Specializing the parameter values of τ = 0 and τ = 1 in the above theorem (1), then
the subsequent corollaries are obtained, in that order.



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 8 of 19

Corollary 1. If f (Θ) is derived by (1) which is belongs to class GΣ,G
u,v (τ). Then

|a2| ≤
√

u2 (s) + u2 (t) + 2(v(s) + v(t))

2

and
|a3| ≤

u2 (s) + u2 (t)

2
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

4

Corollary 2. If the function f (Θ) ∈ GΣ,G
u,v (τ). Then

|a2| ≤
√

u2 (s) + u2 (t) + 2(v(s) + v(t))

4

and
|a3| ≤

u2 (s) + u2 (t)

8
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

12

3. Fekete-Szegö inequality for the subclass GΣ,G
u,v (τ)

Theorem 2. Assume f (Θ) is provided by (1) and is a member of the class GΣ,G
u,v (τ). Then

|a3 − ηa22| ≤

{
v(s)

(2+2τ) if 0 ≤ |h1(η)| ≤ 1
(2+2τ) ,

v(s)| 2(1−η)
(2+2τ)−4(1+τ)2

| if |h1(η)| ≥ 1
(2+2τ) ,

(21)

where
h1(η) =

(1− η)

(2 + 2τ)− 4(1 + τ)2
.

Proof. From the equation (16) and (17), we get

a3 − ηa22 = a22 +
u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + τ)2
− ηa22

a3 − ηa22 = (1− η)a22 +
u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + τ)2

a3 − ηa22 = (1− η)
2(v(s) + v(t)

(2 + 2τ))− 4(1 + τ)2
+

u2 (s)− u2 (t) + 2(v(s)− v(t))

4(1 + τ)2

a3 − ηa22 = (1− η)
2v(s)

(2 + 2τ)− 4(1 + τ)2
+

u2 (s)− u2 (t)

4(1 + τ)2
+

2(v(s)− v(t))

4(1 + τ)2

a3 − ηa22 = v(s)

(
2(1− η)

(2 + 2τ)− 4(1 + τ)2
+

1

2(1 + τ)2

)
+ v(t)

(
2(1− η)

(2 + 2τ)− 4(1 + τ)2
− 1

2(1 + τ)2

)



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 9 of 19

+
u2 (s)− u2 (t)

4(1 + τ)2

a3 − ηa22 = v(s)

(
h1(η) +

1

2(1 + τ)2

)
+ v(t)

(
h1(η)−

1

2(1 + τ)2

)
where

h1(η) =
2(1− η)

(2 + 2τ)− 4(1 + τ)2
(22)

Hence,

|a3 − ηa22| ≤

{
v(s)

(2+2τ) if 0 ≤ |h1(η)| ≤ 1
(2+2τ) ,

v(s)| 2(1−η)
(2+2τ)−4(1+τ)2

| if |h1(η)| ≥ 1
(2+2τ) .

Specializing the parameter values of τ = 0 and τ = 1 in the theorem (2), we obtain
the following corollaries respectively.

Corollary 3. Suppose that f (Θ) belongs to the class GΣ,G
u,v (τ) and is provided by (1). Then

|a3 − ηa22| ≤

{
v(s) if 0 ≤ |h1(η)| ≤ 1

2 ,

v(s)|(1− η)| if |h1(η)| ≥ 1
2 ,

where
|h1(η)| =

(1− η)

2
.

Corollary 4. Suppose that f (Θ) is belonging to the class GΣ,G
u,v (τ) and is provided by (1).

Then

|a3 − ηa22| ≤

{
v(s)
4 if 0 ≤ |h1(η)| ≤ 1

4 ,

v(s)| (1−η)
6 | if |h1(η)| ≥ 1

4 ,

where
|h1(η)| =

(1− η)

6
.

4. The initial Coefficient estimates for the subclass GΣ,G(τ)

For u(s) = s and v(s) = 1 in (2), the following Lucas polynomials Ln(s) were produced.
In the following, it is assumed that G(s,Θ) is a Taylor series of the form

G(s,Θ)− 1 = 1 + sΘ+ (2 + s2)Θ2 + (3s+ s3)Θ3 + (s4 + 4s2 + 2)Θ4 + ..., (23)

G(t,Φ)− 1 = 1 + tΦ+ (2 + t2)Φ2 + (3t+ t3)Φ3 + (t4 + 4t2 + 2)Φ4 + ..., (24)



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 10 of 19

Definition 2. If a function f ∈ Σ is regarded as a member of the class GΣ,G(τ), then the
resulting subordination is valid.

(1− τ)

(
Θf ′(Θ)

f(Θ)

)
+ τ

(
1 +

Θf ′′(Θ)

f ′(Θ)

)
≺ G(s,Θ)− 1 (25)

and

(1− τ)

(
Φf ′(Φ)

f(Φ)

)
+ τ

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
≺ G(t,Φ)− 1 (26)

Example 3. For the function f to be classified as belonging to the class GΣ,G(τ) when
τ = 0, it must meet a subordination condition. This constrains how f behaves in relation
to other functions, frequently in terms of growth, distortion, or mapping properties in the
unit disc. (

Θf ′(Θ)

f(Θ)

)
) ≺ G(s,Θ)− 1

and (
Φf ′(Φ)

f(Φ)

)
≺ G(t,Φ)− 1

Example 4. A function f ∈ Σ is regarded as belonging to the class GΣ,G(τ) when τ = 1
in definition (2) if it meets the subsequent subordination condition:(

1 +
Θf ′′(Θ)

f ′(Θ)

)
≺ G(s,Θ)− 1

and (
1 +

Φf ′′(Φ))

f ′(Φ)

)
≺ G(t,Φ)− 1

Theorem 3. If a function f ∈ Σ is regarded as belonging to the class GΣ,G(τ), then the
resulting subordination condition is valid:

|a2| ≤

√
s2 + t2 + 4

4(1 + 2τ)
(27)

and
|a3| ≤

s2 + t2

8(1 + τ)2
+

s2 − t2

4(1 + 2τ)
(28)

Proof: Let f ∈ GΣ,G(τ). Then f, g : D → D given by (25) and (26) such that

(1− τ)

(
Θf ′(Θ)

f(Θ)

)
+ τ

(
1 +

Θf ′′(Θ)

f ′(Θ)

)
= G(s,Θ)− 1 (29)



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 11 of 19

and
(1− τ)

(
Φf ′(Φ)

f(Φ)

)
+ τ

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
= L(t,Φ)− 1 (30)

Since

(1−τ)

(
Θf ′(Θ)

f(Θ)

)
+τ

(
1 +

Θf ′′(Θ)

f ′(Θ)

)
= 1+(1+τ)a2Θ+(2(1+2τ)a3− (1+3τ)a22)Θ

2+ ...

(31)
and

(1−τ)

(
Φf ′(Φ)

f(Φ)

)
+τ

(
1 +

Φf ′′(Φ))

f ′(Φ)

)
= 1− (1+τ)a2Φ+((3+5τ)a22−2(1+2τ)a3)Φ

2+ ...

(32)
From the equations (23) and (31), By comparing coefficients of Θ and Θ2 respectively, we
get coefficient of Θ :

(1 + τ)a2 = s (33)

coefficient of Θ2 :
2(1 + 2τ)a3 − (1 + 3τ)a22 = 2 + s2 (34)

From the equations (24) and (32), equating the coefficients of Φ and Φ2 respectively, we
get coefficient of Φ :

−(1 + τ)a2 = t (35)

coefficient of Φ2 :
(3 + 5τ)a22 − 2(1 + 2τ)a3 = 2 + t2 (36)

Now, adding the equations (33) and (35), we have

s = −t (37)

Squaring and adding the equations (33) and (35), we get

2(1 + τ)2a22 = s2 + t2

a22 =
s2 + t2

2(1 + τ)2
(38)

subtracting the equations (34) and (36), we get

2(1 + 2τ)a3 − (1 + 3τ)a22 − (3 + 5τ)a22 + 2(1 + 2τ)a3 = s2 − t2

4(1 + 2τ)a3 = s2 − t2 + (4 + 8τ)a22

a3 = a22 +
s2 − t2

4(1 + 2τ)
(39)

substituting the equations (38) in equation (39), we get

a3 =
s2 − t2

4(1 + 2τ)
+

s2 + t2

2(1 + τ)2
(40)



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 12 of 19

substituting the equations (34) in equation (36), we get

2(1 + 2τ)a3 − (1 + 3τ)a22 + (3 + 5τ)a22 + (2 + 2τ)a3 = 2 + t2

(2 + 2τ)a22 = 2 + t2 + 2 + s2

a22 =
s2 + t2 + 4

2(1 + τ)
(41)

substituting the equation (41) in (39), we get

a3 =
s2 + t2 + 4

(2 + 2τ)
+

s2 − t2

4(1 + 2τ)

|a3| ≤
s2 + 2

2(1 + τ)
(42)

Hence,

|a2| ≤

√
s2 + t2 + 4

(2 + 2τ)

|a3| ≤
s2 + t2

2(1 + τ)2
+

s2 − t2

4(1 + 2τ)

The next couple of consequences are obtained by specializing the parameter values of
τ = 0 and τ = 1 in the theorem (3), respectively.

Corollary 5. The function f (Θ) satisfies a couple of conditions to be included in this
subclass of bi-univalent functions if it is provided by (1) and is a member of the class
GΣ,G(τ). Then

|a2| ≤
√

s2 + t2 + 4

2

and
|a3| ≤

s2 + t2

2
+

s2 − t2

4

Corollary 6. The function f (Θ) satisfies a couple of conditions to be included in this
subclass of bi-univalent functions if it is provided by (1) and is a member of the class
GΣ,G(τ). Then

|a2| ≤
√

s2 + t2 + 4

4

and
|a3| ≤

s2 + t2

8
+

s2 − t2

12



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 13 of 19

5. Fekete-Szegö inequality for the subclass GΣ,G(τ)

Theorem 4. If f (Θ) is given by (1) be in the class GΣ,G(τ). Then

|a3 − ηa22| ≤

{
s2

(1+2τ) if 0 ≤ |h1(η)| ≤ 1
4(1+2τ) ,

2s2| (1−η)
2(1+τ)2

| if |h1(η)| ≥ 1
4(1+2τ) ,

(43)

where
h1(η) =

(1− η)

2(1 + τ)2
.

Proof. From the equations (38) and (39), we get

a3 − ηa22 = a22 +
s2 − t2

4(1 + 2τ)
− ηa22

a3 − ηa22 = (1− η)a22 +
s2 − t2

4(1 + 2τ)

a3 − ηa22 = (1− η)
s2 + t2

2(1 + τ2)
+

s2 − t2

4(1 + 2τ)

a3 − ηa22 = s2
(

(1− η)

2(1 + τ)2
+

1

4(1 + 2τ)

)
+ t2

(
(1− η)

2(1 + τ)2
− 1

4(1 + 2τ)

)

a3 − ηa22 = s2
(
h1(η) +

1

4(1 + 2τ)

)
+ t2

(
h1(η)−

1

4(1 + 2τ)

)
where

h1(η) =
(1− η)

2(1 + τ)2
(44)

Hence,

|a3 − ηa22| ≤

{
s2

(1+2τ) if 0 ≤ |h1(η)| ≤ 1
4(1+2τ) ,

2s2| (1−η)
2(1+τ)2

| if |h1(η)| ≥ 1
4(1+2τ) .

The two subsequent consequences are obtained by changing the parameter values in
the theorem (4) to τ = 0 and τ = 1, respectively.



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 14 of 19

Corollary 7. If the function f (Θ) is defined by (1) and belongs to the class GΣ,G(τ), then
it exhibits specific geometric and analytic property |a3−ηa22| that characterize this subclass
of functions. i.e.,

|a3 − ηa22| ≤

{
s2 if 0 ≤ |h1(η)| ≤ 1

4 ,

2s2| (1−η)
2 | if |h1(η)| ≥ 1

4 ,

This implies

|a3 − ηa22| ≤

{
s2 if 0 ≤ |h1(η)| ≤ 3

4 ,

s2|(1− η)| if |h1(η)| ≥ 3
4 ,

where
h1(η) =

(1− η)

2
.

Corollary 8. The function f (Θ) possesses particular geometric feature |a3 − ηa22| of
function f (Θ), it is defined by (1) and it is a member of the class GΣ,G(τ). Then

|a3 − ηa22| ≤

{
s2

3 if 0 ≤ |h1(η)| ≤ 1
12 ,

2s2| (1−η)
8 | if |h1(η)| ≥ 1

12 ,

This implies

|a3 − ηa22| ≤

{
s2

3 if 0 ≤ |h1(η)| ≤ 7
8 ,

s2| (1−η)
4 | if |h1(η)| ≥ 7

8 ,

where
h1(η) =

(1− η)

8
.

6. Discussion: Influence of Parameter s and Surface Plot Interpretation

For special values of s, the expansion may reduce to well-known families of polynomials:

• When s = 0: only even powers of Θ survive with relatively simple coefficients —
suggesting a resemblance to Chebyshev-type behavior.

• When s = 1: the sequence does not exactly match Fibonacci or Lucas, but the
structure mimics a nonlinear recurrence.

• When s = 2: the pattern aligns closely with growth seen in Bell or Motzkin-like
structures, potentially capturing partition-related or moment-generating behavior.

From a generating function perspective, the parameter s may encode a deformation,
weight, or even a flow parameter in a functional or combinatorial system. However, because
the expansion is polynomial in s, the Fibonacci sequence is not recovered directly for
any constant value of s. Moreover, the appearance of higher powers of s than allowed in
classical recursions indicates that the structure diverges from a standard Lucas recurrence.



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 15 of 19

The parameter s plays a crucial role in shaping the behavior of the Fekete–Szegö
inequality, particularly in its effect on the sharpness and structure of the bound for the
third coefficient. From the theorem (4) bound is given by:

|a3 − ηa22| ≤


s2

1 + 2τ
, if |h1(η)| ≤

1

4(1 + 2τ)
,

2s2
∣∣∣∣ 1− η

2(1 + τ)2

∣∣∣∣ , if |h1(η)| ≥
1

4(1 + 2τ)
,

where h1(η) =
1− η

2(1 + τ)2
.

The corresponding surface plot of the bound |a3−ηa22| over the complex η-plane reveals
a piecewise smooth, radially structured surface centered at η = 1. The height of the surface
at each point corresponds to the maximum allowable deviation of the coefficient a3 in terms
of a2, modulated by the geometry of η and the analytic parameter τ . The region defined
by:

|1− η| ≤ 1

2

marks a flat plateau where the bound remains constant. Beyond this threshold, the
bound increases linearly with |1− η|, creating a rising surface outward from η = 1.

For a fixed value of τ = 1, the shape and scale of the surface are strongly influenced
by the parameter s. This dependency is clearly shown in Figures 1-3, where the behavior
of the surface is illustrated for different values of s. Variations in s significantly alter the
surface characteristics, showcasing how the bound’s structure evolves as the parameter
changes.

The surface plot of the bound |a3 − ηa22| provides a clear visual interpretation of this
analytic bound. When s = 0, the bound becomes zero, reflecting a highly constrained
structure similar to Chebyshev-like behavior in the generating function. As s increases
to 1, the surface shows a mild elevation and nonlinear coefficient growth, but it does not
align with Fibonacci or Lucas forms. For s = 2, the surface rises more sharply, exhibiting
coefficient growth that resembles Bell or Motzkin-like structures.

The interaction between s and η demonstrates how geometric deviations and deforma-
tions control the flexibility and complexity of the function class GΣ,G(τ). This interplay
is crucial in determining the structure of the function, providing a deeper understanding
of the role of second-order coefficient bounds in geometric function theory.



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 16 of 19

-1

0

1

2

3

-2

-1

0

1

2

-1

-0.5

0

0.5

1

Figure 1: For s = 0, the bound is zero, indicating constrained geometry with a Chebyshev-like generating
function.

0

2

0.2

1 3

0.4

2

0.6

0

0.8

1
-1

0

-2 -1

Figure 2: For s = 1, the surface shows mild elevation and nonlinear coefficient growth, without matching
Fibonacci or Lucas forms.



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 17 of 19

0.5

2

1

1.5

1 3

2

2

2.5

0

3

1
-1

0

-2 -1

Figure 3: For s = 2, the surface rises steeply, resembling coefficient behavior in Bell or Motzkin-like structures.

7. Conclusion

In conclusion, the Taylor-Maclaurin coefficients |a2| and |a3| for two new subclasses of
bi-univalent functions related to Lucas polynomial functions are precisely determined in
this study. The obtained coefficients and Fekete-Szegö estimates reveal that our subclass
of bi-univalent functions demonstrates improved control over its first few coefficients com-
pared to the previously studied subclasses. This has implications for understanding the
geometric properties of these functions, particularly in their growth and distortion behav-
ior. The classical theory is interestingly improved by the novel subclass of bi-univalent
functions, especially in relation to initial coefficients and the Fekete-Szegö inequality. Fu-
ture studies might concentrate on applying these discoveries to particular geometric issues
in univalent function theory or expanding them to higher-order coefficients. Analyzing
these initial coefficients helps us better understand these functions’ geometric, growth,
and distortion characteristics, as well as how they behave close to the origin and relate to
other bi-univalent subclasses. This work builds upon and improves upon earlier studies
by using the Fekete-Szegö inequality, providing a better understanding of the functional
and mapping properties. These results have wide-ranging implications in fields where the
exact behavior of bi-univalent functions is crucial, such as engineering, fluid dynamics,
and complex analysis.

Acknowledgements

This work was supported by the Deanship of Scientific Research, Vice Presidency for
Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 18 of 19

No. KFU252106].

Credit authorship contribution statement

Stalin Thangamani: Investigation, Writing – review & editing. Dumitru Baleanu:
Conceptualization, Project administration, Software. Jeno Francis: Conceptualization,
Project administration. Majeed Ahmad Yousif: Software, Writing – review & editing.
Meraa Arab: Funding acquisition, Methodology, Writing – review & editing. Pshtiwan
Othman Mohammed: Conceptualization, Writing – original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.

Availability of data and material

No data was used for the research described in the article.

References

[1] P. Koebe. Ueber die uniformisierung beliebiger analytischer kurven ii. Nachr. k. Ges.
Wissenschaft. Gottinger Math. Phys., pages 177–198, 1907.

[2] M. Lewin. On a coefficient problem for bi-univalent functions. Proc. Amer. Math.
Soc., 18:63–68, 1967.

[3] D. A. Brannan and J. G. Clunie. Aspects of Contemporary Complex Analysis. Aca-
demic Press, New York and London, 1980.

[4] D. A. Brannan and T. S. Taha. On some classes of bi-valent functions. Studia Univ.
Babes-Bolyai Math., 31:70–77, 1986.

[5] W. C. Ma and D. Minda. A unified treatment of some special classes of univalent
functions. In Proceedings of the Conference on Complex Analysis, Tianjin, Conf.
Proc. Lecture Notes Anal., pages 157–169, Cambridge, MA, 1992.

[6] A. Behera and G. K. Panda. On the square roots of triangular numbers. Fibonacci
Quart., 37:98–105, 1999.

[7] B. A. Frasin and M. K. Aouf. New subclasses of bi-univalent functions. Appl. Math.
Lett., 24:1569–1573, 2011.

[8] D. Bansal and J. Sokol. Coefficient bound for a new class of analytic and bi-univalent
functions. J. Fract. Calc. Appl., 5(1):122–128, 2014.

[9] A. Y. Lashin. On certain subclasses of analytic and bi-univalent functions. J. Egyptian
Math. Soc., 24:220–225, 2016.

[10] A. O. Pall-Szabo and G. I. Oros. Coefficient related studies for new classes of bi-
univalent functions. Mathematics, 8:1110, 2020.



S. Thangamani et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6294 19 of 19

[11] K. A. Jassim, R. O. Rasheed, and R. H. Jassim. Generalized subclass of analytic bi-
univalent functions defined by differential operator. J. Interdiscip. Math., 24:961–970,
2021.

[12] G. Lee and M. Asci. Some properties of the (p, q)-fibonacci and (p, q)-lucas polyno-
mials. Journal of Applied Mathematics, 2012:264842, 2012.

[13] P. K. Ray. Certain matrices associated with balancing and lucas-balancing numbers.
Matematika, 28:15–22, 2012.

[14] S. Altinkaya and S. Yacin. On the (p, q)-lucas polynomial coefficient bounds of bi-
univalent function class. Boletín de la Sociedad Matemática Mexicana, 25:567–575,
2019.

[15] Y. Almalki, A. K. Wanas, T. G. Shaba, A. Alb Lupaş, and M. Abdalla. Coeffi-
cient bounds and fekete–szegö inequalities for a two families of bi-univalent functions
related to gegenbauer polynomials. Axioms, 12:1018, 2023.

[16] Arzu Akgül. (p, q)-lucas polynomial coefficient inequalities of the bi-univalent function
class. Turkish Journal of Mathematics, 43(5):2170–2176, 2019.

[17] S. R. Swamy, A. K. Wanas, and Y. Sailaja. Some special families of holomorphic
and salagean type bi-univalent functions associated with (m,n)-lucas polynomials.
Mathematics, 11(4):563–574, 2020.

[18] N. Magesh, C. Abirami, and Ş. Altınkaya. Initial bounds for certain classes of bi-
univalent functions defined by the (p, q)-lucas polynomials. TWMS Journal of Applied
and Engineering Mathematics, 11(1):282–288, 2021.

[19] A. K. Wanas and L. I. Cotîrlă. Applications of (m, n)-lucas polynomials on a certain
family of bi-univalent functions. Mathematics, 10(4):595, 2022.

[20] A. Hussen and M. Illafe. Coefficient bounds for a certain subclass of bi-univalent
functions associated with lucas-balancing polynomials. Mathematics, 11:4941, 2023.

[21] İ. Aktaş and İ. Karaman. On some new subclasses of bi-univalent functions defined
by balancing polynomials. Karamanoğlu Mehmetbey Univ. Journal of Engineering
and Natural Sciences, 5:25–32, 2023.

[22] A. K. Wanas, G. Ş. Sălăgean, and Á. P. S. Orsolya. Coefficient bounds and fekete-
szegő inequality for a certain family of holomorphic and bi-univalent functions defined
by (m, n)-lucas polynomials. Filomat, 37(4):1037–1044, 2023.

[23] A. K. Wanas, E. K. Wanas, A. Cătaş, and M. Abdalla. Applications of (m,n)-lucas
polynomials for a certain family of bi-univalent functions associating λ-pseudo-starlike
functions with sakaguchi type functions. Earthline Journal of Mathematical Sciences,
15(1):1–10, 2024.

[24] A. Hussen, M. S. A. Madi, and A. M. M. Abominjil. Bounding coefficients for certain
subclasses of bi-univalent functions related to lucas-balancing polynomials. AIMS
Mathematics, 9(7):18034–18047, 2024.

[25] M. Buyankara and M. Çağlar. Coefficient inequalities for two new subclasses of bi-
univalent functions involving lucas-balancing polynomials. Eastern Anatolian Journal
of Science, 10(2):5–11, 2024.

[26] P. L. Duren. Univalent Functions, volume 259 of Grundlehren der Mathematischen
Wissenschaften. Springer-Verlag, New York, 1983.