EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 105-115
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fekete-Szegö Functional of a Subclass of Bi-Univalent
Functions Associated with Gegenbauer Polynomials

Waleed Al-Rawashdeh

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

Abstract. In this paper, we introduce and investigate a class of bi-univalent functions, denoted
by F(n, α, β), that depends on the Ruscheweyh operator and defined by the use of Gegenbauer
Polynomials. For functions in this class, we derive the estimations for the initial Taylor-Maclaurin
coefficients |a2| and |a3|. Moreover, we obtain the classical Fekete-Szegö inequality of functions
belonging to this class.

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

Key Words and Phrases: Analytic Functions, Taylor-Maclaurin Series, Univalent and Bi-
Univalent Functions, Principle of Subordination, Hadamard Product, Ruscheweyh Operator, Ruscheweyh
Derivative, Gegenbauer Polynomials, Chebyshev polynomials, Coefficient estimates, Fekete-Szegö
Inequality

1. Introduction

Let A be the family of all analytic functions f that are defined on the open unit disk
D = {z ∈ C : |z| < 1} and normalized by the conditions f(0) = 0 and f ′(0) = 1. Any
function f ∈ A has the following Taylor-Maclarin series expansion:

f(z) = z +

∞∑
n=2

anz
n, where z ∈ D. (1)

Let S denote the class of all functions f ∈ A that are univalent in D. Let the functions
f and g be analytic in D, we say the function f is subordinate by the function g in D,
denoted by f(z) ≺ g(z) for all z ∈ D, if there exists a Schwartz function w, with w(0) = 0
and |w(z)| < 1 for all z ∈ D, such that f(z) = g(w(z)) for all z ∈ D. In particular, if the
function g is univalent over D then f(z) ≺ g(z) equivalent to f(0) = g(0) and f(D) ⊂ g(D.
For more information about the Subordination Principle we refer the readers to to the
monographs [9], [23] and [24].

DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5004

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

https://www.ejpam.com 105 © 2024 EJPAM All rights reserved.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 106

As known univalent functions are injective (one-to-one) functions. Hence, they are
invertible and the inverse functions may not be defined on the entire unit disk D. In fact,
the Koebe one-quarter Theorem tells us that 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 ∈ A 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 A 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
.

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
[19], [22], [25], the monograph [10], [12] and the references therein.

In the year 1784, Legendre [18] introduced and studied the orthogonal polynomials.
Traditionally, orthogonal polynomials are crucial in approximation theory where are used
in polynomial interpolation. Moreover, under specific restrictions, orthogonal polynomials
are frequently used in the study of differential equations. In particular in some special
cases of Sturm-Liouville differential equation. An example of orthogonal polynomials is a
Gegenbauer polynomial. Special cases of Gegenbauer polynomials are Legendre polyno-
mials and the Chebyshev polynomials of the first and second kind. For more information
about orthogonal polynomials we refer the readers to the monograph [8]. We define Gegen-
bauer polynomials in the next section.

The subject of the geometric function theory in complex analysis has been investi-
gated by many researchers in recent years, the typical problem in this field is studying
a functional made up of combinations of the initial coefficients of the functions f ∈ A.
For a function in the class S, it is well-known that |an| is bounded by n. Moreover, the
coefficient bounds give information about the geometric properties of those functions. For
instance, the bound for the second coefficients of the class S gives the growth and distor-
tion bounds for the class. In addition, the Fekete-Szegö functional arises naturally in the
investigation of univalency of analytic functions. In the year 1933, Fekete and Szegö [11]



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 107

found the maximum value of |a3−λa22|, as a function of the real parameter 0 ≤ λ ≤ 1 for a
univalent function f . Since then, the problem of dealing with the Fekete-Szegö functional
for f ∈ A with any complex λ is known as the classical Fekete-Szegö problem. There
are many researchers investigated the Fekete-Szegö functional and the other coefficient
estimates problems, for example see the articles [20], [15], [21], [17], [11], [22], [14] and the
references therein.

2. Preliminaries

In this section we present some information that are curial for the main results of this
paper. we start by defining our subclass. In the year 1994, Szynal [28] introduced and
studied a subclass F(α) of the class A consisting of functions of the form

f(z) =

∫ 1

−1
K(z, x) dσ(x), (3)

where K(z, x) =
z

(z2 − 2xz + 1)α
, α ≥ 0, −1 ≤ x ≤ 1, and σ is the probability measure on

[−1, 1]. Moreover, the functionK(z, x) has the following Taylor-Maclaurin series expansion

K(z, x) = z + Cα
1 (x)z

2 + Cα
2 (x)z

3 + Cα
3 (x)z

4 + · · ·,

where Cα
n (x) denotes the Gegenbauer polynomials of order α and degree n in x. Fur-

thermore, for any real numbers α, x ∈ R ,with α ≥ 0 and −1 ≤ x ≤ 1, and z ∈ D the
generating function of Gegenbauer polynomials is given by

Hα(z, x) = (z2 − 2xz + 1)−α.

Moreover, for any fixed x the function Hα is analytic on the unit disk D and its
Taylor-Maclaurin series is given by

Hα(z, x) =

∞∑
n=0

Cα
n (x)z

n.

In addition, if f ∈ F(α) that is given by (3), the nth coefficient can be written as

an =

∫ 1

−1
Cα
n−1(x) dσ(x).

In addition, Gegenbauer polynomials can be defined in terms of the following recurrence
relation:

Cα
n (x) =

2x(n+ α− 1)Cα
n−1(x)− (n+ 2α− 2)Cα

n−1(x)

n
, (4)

with initial values

Cα
0 (x) = 1, Cα

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



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 108

It is well-known that the Gegenbauer polynomials and their special cases such as
Legendre polynomials Ln(x) and the Chebyshev polynomials of the second kind Tn(x),
are orthogonal polynomials, where the values of α are α = 1/2 and α = 1 respectively,
more precisely

Ln(x) = C1/2
n (x), and Tn(x) = C1

n(x).

For more information about the Gegenbauer polynomials and their special cases, we refer
the readers to the articles [4], [3], [7], [6], [5], [21], [16], [22], [13], [14], the monograph [10],
[12], [27], and the references therein.

In the year 1975, Ruscheweyh [26] introduced the operator R which defined, using the
Hadamard product, as follows

Rλf(z) = f(z) ∗ z

(1− z)1−λ
,

where f ∈ A, z ∈ D and real number λ ≥ −1. For λ = n ∈ N0 = N ∪ {0}, we get the
Rscheweyh derivative Rn of order n of the function f :

Rnf(z) = z

(
zn−1f(z)

)(n)
n!

.

Moreover, the Taylor-Maclaurin series of Rnf is given by

Rnf(z) = z +
∞∑
k=2

σ(n, k)akz
k,

σ(n, k) =
Γ(n+ k)

(k − 1)!Γ(n+ 1)
. (5)

We say that a function f ∈ Σ in the subclass F(n, α, β) if it satisfies the following
subordination conditions, associated with the Gegenbauer Polynomials, for all z, w ∈ D:

(Rnf(z))′ + βz(Rnf(z))′′ ≺ Hα(z, x) (6)

and
(Rng(w))′ + βw(Rng(w))′′ ≺ Hα(w, x), (7)

where α > 0, β > 0, n ∈ N0, x ∈ (12 , 1] and g(w) is defined by equation (2).

The following lemma (see[17]) is a well-known fact, so we omit its proof.

Lemma 1. Let K, L ∈ R and p, q ∈ C. If |p| < R and |q| < R,

|(K + L)p+ (K − L)q| ≤

{
2|K|R, if |K| ≥ |L|
2|L|R, if |K| ≤ |L|



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 109

Our investigation in this paper is motivated by the work of the researchers presented
in the papers [1], [2], and [14]. In this presenting paper, we investigate a subclass of bi-
univalent functions Σ in the open unit disk D, which we denote by F(n, α, β) with α > 0,
β > 0 and n ∈ N0. For functions in this subclass, we obtain the estimates for the initial
Taylor-Maclarin coefficients |a2| and |a3|. Furthermore, we examine the corresponding
Fekete-Szegö functional problem for functions in this subclass.

3. Initial Coefficient estimates for the function class F(n, α, β)

In this section, we provide bounds for the initial Taylor-Maclaurin coefficients for the
functions belong to the class F(n, α, β) which are given by equation (1).

Theorem 1. Let the function f given by (1) be in the class F(n, α, β). Then

|a2| ≤
2αx

√
x(n!)√

|(3α(n+ 2)!(1 + 2β)x2 − 4(1 + β)2(n+ 1)(n+ 1)!{(2 + 2α)x2 − 1}|
(8)

and

|a3| ≤
4αx(n!)

3(1 + 2β)(n+ 2)!
+

α2x2

(1 + β)2(n+ 1)2
(9)

Proof. Let f belong to the class F(n, α, β). Then Using (6) and (7) we can find two
analytic functions p and q on the unit disk D such that

(Rnf(z))′ + βz(Rnf(z))′′ ≺ Hα(x, p(z)), (10)

and
(Rng(w))′ + βw(Rng(w))′′ ≺ Hα(x, q(w)). (11)

where the analytic functions p and q are given by

p(z) = 1 + p1z + p2z
2 + p3z

3 + · · · where z ∈ D,

and
q(w) = 1 + q1w + q2w

2 + q3w
3 + · · · where w ∈ D

Such that
p(0) = q(0) = 0,

and for all z, w ∈ D
|p(z)| < 1 and |q(z)| < 1.

Moreover, it is well-known that (see, for details [10]) for all j ∈ N

|pj | ≤ 1 and |qj | ≤ 1.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 110

Now, upon comparing the coefficients in both sides of (10) and (11) we obtain the
following

2(1 + β)σ(n, 2)a2 = Cα
1 (x)p1, (12)

3(1 + 2β)σ(n, 3)a3 = Cα
1 (x)p2 + Cα

2 (x)p
2
1, (13)

−2(1 + β)σ(n, 2)a2 = Cα
1 (x)q1, (14)

and
3(1 + 2β)σ(n, 3)(2a22 − a3) = Cα

1 (x)q2 + Cα
2 (x)q

2
1 (15)

Using equations (12) and (14) we get

p1 = −q1 (16)

Moreover, adding the square of equations (12) and (14) we get

8(1 + β)2[σ(n, 2)]2a22 = [Cα
1 (x)]

2(p21 + q21) (17)

By adding equations (13) and (15) we get

6(1 + 2β)σ(n, 3)a22 = [Cα
1 (x)](p2 + q2) + [Cα

2 (x)](p
2
1 + q21) (18)

In view of equation (17), equation (18) can be written as

(
6(1 + 2β)σ(n, 3)[Cα

1 (x)]
2 − 8(1 + β)2[Cα

2 (x)][σ(n, 2)]
2
)
a22 = [Cα

1 (x)]
3(p2 + q2) (19)

Using equation (5), equation (19) becomes

a22 =
4α2x3(p2 + q2)

3(1 + 2β)(n+ 1)(n+ 2)αx2 − 4(1 + β)2(n+ 1)2[2(α+ 1)x2 − 1]
.

Using the facts |p2| ≤ 1 and |q2| ≤ 1, we get the desired estimate of a2:

|a2| ≤
2αx

√
x(n+ 1)√

(n+ 1)2|(3α(1 + 2β)(n+ 2)x2 − 4(1 + β)2(n+ 1){(2 + 2α)x2 − 1}|
.

Next, we look for the bound of |a3|. Subtracting equation (15) from equation (13) and
using equation (16), we get

a3 =
Cα
1 (x)(p2 − q2)

6(1 + 2β)σ(n, 3)
+ a22. (20)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 111

In view of equation (17), we obtain

a3 =
Cα
1 (x)(p2 − q2)

6(1 + 2β)σ(n, 3)
+

[Cα
1 (x)]

2p21
4(1 + β)2[σ(n, 2)]2

.

Hence, Using Equation (5) and the facts |p2| ≤ 1 and |q2| ≤ 1, we get the desired
estimate of a3:

|a3| ≤
4αx

3(1 + 2β)(n+ 2)(n+ 1)
+

α2x2

(1 + β)2(n+ 1)2
.

This completes the proof of Theorem 1.

Taking α = 1, we get the following interesting corollary of Theorem 1. These initial
coefficient estimates are related to Chebyshev polynomials of the second kind. The prove
is similar to the proof of previous theorem, so we omit the proof’s details.

Corollary 1. Let the function f given by (1) be in the class F(n, 1, β). Then

|a2| ≤
2x

√
x(n!)√

|(3(n+ 2)!(1 + 2β)x2 − 4(1 + β)2(n+ 1)(n+ 1)!(4x2 − 1)|
,

and

|a3| ≤
4x(n!)

3(1 + 2β)(n+ 2)!
+

x2

(1 + β)2(n+ 1)2
.

On the other hand, taking β = 1, we get the following corollary.

Corollary 2. Let the function f given by (1) be in the class F(n, α, 0). Then

|a2| ≤
2αx

√
x(n!)√

|(9α(n+ 2)!x2 − 16(n+ 1)(n+ 1)!{(2 + 2α)x2 − 1}|
,

and

|a3| ≤
4αx(n!)

9(n+ 2)!
+

α2x2

4(n+ 1)2
.

4. Fekete-Szegö problem for the function class F(n, α, β)

In this section, we consider the classical Fekete-Szegö problem for our presenting class
F(n, α, β).

Theorem 2. Let the function f given by (1) be in the class F(n, α, β). Then for some
ζ ∈ R,

|a3 − ζa22| ≤

{
4αx
B , if |1− ζ| ≤ ∆(α,n,β)

4Bα2x2

16α3x3|1−ζ|
∆(α,n,β) , if |1− ζ| ≥ ∆(α,n,β)

4Bα2x2 ,
(21)

where

∆(α, n, β) = 4α[B − 4(1 + β)2(n+ 1)2(α+ 1)]x2 − 8α(1 + β)2(n+ 1)2,



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 105-115 112

and
B = 3(1 + 2β)(n+ 2)(n+ 1)

Proof. For some real number ζ, using equation (20) we have

a3 − ζa22 =
Cα
1 (x)(p2 − q2)

6(1 + 2β)σ(n, 3)
+ (1− ζ)a22.

In view of equation (19), we obtain

a3 − ζa22 =
Cα
1 (x)(p2 − q2)

6(1 + 2β)σ(n, 3)
+

(1− ζ)[Cα
1 (x)]

3(p2 + q2)

6(1 + 2β)σ(n, 3)[Cα
1 (x)]

2 − 8(1 + β)2[Cα
2 (x)][σ(n, 2)]

2
.

The last expression can be written as:

a3 − ζa22 = Cα
1 (x)[(K − L)p2 + (K + L)q2],

where

K =
1

6(1 + aβ)σ(n, 3)
,

and

L =
(1− ζ)[Cα

1 (x)]
2

△(α, n, β)
.

Using Lemma 1, we get the following

|a3 − ζa22| ≤

{
2| Cα

1 (x)
6(1+aβ)σ(n,3) |, if |K| ≥ |L|

2| (1−ζ)[Cα
1 (x)]3

△(α,n,β) |, if |K| ≤ |L|
.

Using the initial values (4) and equation (5), we get the desired inequality (21). This
completes the proof of Theorem 2.

The following corollaries are just consequences of Theorem 2. Taking α = 1, we get
the Fekete-Szegö inequality that is related to Chebyshev polynomials of the second kind.

Corollary 3. Let the function f given by (1) be in the class F(n, 1, β). Then for some
ζ ∈ R,

|a3 − ζa22| ≤

{
4x
B , if |1− ζ| ≤ G

16x3|1−ζ|
4B(n+2)(n+1)x2−8(1+β)(n+1)2(4x2−1)

, if |1− ζ| ≥ G,
(22)

where

G =
4B(n+ 2)(n+ 1)x2 − 8(1 + β)(n+ 1)2(4x2 − 1)

2Bx2
.

Taking β = 1, we get the following corollary.

Corollary 4. Let the function f given by (1) be in the class F(n, α, 0). Then for some
ζ ∈ R,

|a3 − ζa22| ≤

{
4(n!)αx
9(n+2)! , if |1− ζ| ≤ (n!)H(n,α)

36(n+2)!α2x2

16α3x3|1−ζ|
H(n,α) , if |1− ζ| ≥ (n!)H(n,α)

36(n+2)!α2x2 ,
(23)

where
H(n, α) = 4αx2(n+ 1) (9(n+ 2)− 16(n+ 1)(α+ 1))− 32α(n+ 1)2.



REFERENCES 113

5. Conclusion

This research paper has investigated a new subclass of bi-univalent functions, defined
in terms of the Ruscheweyh derivative Rn of order n, by the means of Gegenbauer polyno-
mials. For functions belong to this function class, the author has derived estimates for the
Taylor-Maclaurin initial coefficients and Fekete-Szegö functional problem. The work pre-
sented in this paper will lead to many different results for subclasses defined by the means

of Legendre polynomials Ln(x) = C
1/2
n (x) and the Chebyshev polynomials of the second

kind Tn(x) = C1
n(x). Moreover, the presented work in this paper will inspire researchers

to extend its concepts to harmonic functions and symmetric q-calculus.

Acknowledgements

The author would like to express his sincerest thanks to the referees for their valuable
comments and various useful suggestions.

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

References

[1] K. I. Abdullah and N. H. Mohammed. Bounds For the Coefficients of Two New
Subclasses of Bi-Univalent Functions. Science Journal of University of Zakho, 10:66–
69, 2022.

[2] W. Al-Rawashdeh. Horadam Polynomials and a Class of Bi-Univalent Functions
Defined by Ruscheweyh Operator. preprint.

[3] W. Al-Rawashdeh. Coefficient Bounds of a class of Bi-Univalent Functions Related
to Gegenbauer Polynomials. International Journal of Mathematics and Mathematical
Sciences, Article ID 2573044:7 pages, 2023.

[4] W. Al-Rawashdeh. Applications of Gegenbauer Polynomials on a Class of Non-
Bazilevic Functions. International Journal of Mathematics and Computer Science,
19:635–642, 2024.

[5] C. Cesarano. Identities and generating functions on Chebyshev polynomials. Georgian
Mathematical Journal, 19, 2012.

[6] C. Cesarano. Integral representations and new generating functions of Chebyshev
polynomials. Hacettpe Journal of Mathematics and Statistics, 44, 2015.

[7] C. Cesarano. Multi-dimensional Chebyshev polynomials: a non-conventional ap-
proach. Communications in Applied Industrial Mathematics, 10, 2019.

[8] B. Doman. The Classical Orthogonal Polynomials. World Scientific, Singapore, 2015.



REFERENCES 114

[9] P. Duren. Subordination in Complex Analysis, Lecture Notes in Mathematics.
Springer, Berlin, Germany, 599, 1977.

[10] P. Duren. Univalent functions. Grundlehren der Mathematischen Wissenschaften
259, Springer-Verlag, New York, 1983.

[11] M. Fekete and G. Szegö. Eine Bemerkung Über ungerade Schlichte Funktionen.
Journal of London Mathematical Society, s1-8, 1933.

[12] A. W. Goodman. Univalent functions. Mariner Publishing Co. Inc., Boston, 1983.

[13] N. Magesh H. Orhan and V. Balaji. Second Hankel determinant for certain class of
bi-univalent functions defined by Chebyshev polynomials. Asian-European Journal
of Mathematics, 12(2):1950017, 2019.

[14] M. Kamali H.M. Srivastava and A. Urdaletova. A study of the Fekete-Szegö func-
tional and coefficient estimates forvsubclasses of analytic functions satisfying a certain
subordination conditionv and associated with the Gegenbauer polynomials. AIMS
Mathematics, 7(2):2568–2584, 2021.

[15] Y.C. Kim J.H. Choi and T. Sugawa. A general approach to the Fekete-Szegö problem.
Journal of the Mathematical Society of Japan, 59, 2007.

[16] I. Naraniecka K. Kiepiela and J. Szynal. The Gegenbauer polynomials and typically
real functions. Journal of Computational and Applied Mathematics, 153(1-2):273–282,
2003.

[17] F.R. Keogh and E.P. Merkes. A Coefficient inequality for certain classes of analytic
functions. Proceedings of the American Mathematical Society, 20, 1969.

[18] A. Legendre. Recherches sur Laattraction des Sphéroides Homogénes; Mémoires
Présentes par Divers Savants a laAcadémie des Sciences de laInstitut de France.
Goethe Universitat, 10, 1785.

[19] M. Lewin. On a coefficient problem for bi-univalent functions. Proceedings of the
American Mathematical Society, 18(1):63–68, 1967.

[20] H. Orhan M. Cağlar and M. Kamali. Fekete-Szegö problem for a subclass of analytic
functions associated with Chebyshev polynomials. Bol. Soc. Paran. Mat., 40, 2022.

[21] E. Deniz M. Kamali, M. Cağlar and M. Turabaev. Fekete Szegö problem for a new
subclass of analytic functions satisfying subordinate condition associated with Cheby-
shev polynomials. Turkish J. Math., 45, 2012.

[22] N. Magesh and S. Bulut. Chebyshev polynomial coefficient estimates for a class of an-
alytic bi-univalent functions related to pseudo-starlike functions. Afrika Matematika,
29(1-2):203–209, 2018.



REFERENCES 115

[23] S. Miller and P. Mocabu. Differential Subordination: Theory and Applications. CRC
Press, New York, 2000.

[24] Z. Nehari. Conformal Mappings. McGraw-Hill, New York, 1952.

[25] E. Netanyahu. The minimal distance of the image boundary from the origin and the
second coefficient of a univalent function in |z| < 1. Archive for Rational Mechanics
and Analysis, 32(2):100–112, 1969.

[26] S. Ruscheweyh. New criteria for univalent functions. Proceedings of the American
Mathematical Society, 49, 1975.

[27] H.M. Srivastava and H.L. Manocha. A treatise on generating functions. Halsted
Press, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

[28] J. Szynal. An extension of typically real functions. Ann. Univ. Mariae Curie-
Sko lodowska Sect., 48, 1994.