Initial Coefficient Estimates of Bi-Univalent Functions Linked with
Balancing Coefficients
ARZU AKGÜL
Department of Mathematics,
Kocaeli University,
Faculty of Arts and Science,
Umuttepe Campus,Kocaeli,
TURKEY
Abstract: - In the next study we introduce a new class
󰇛󰇛󰇜󰇜 of bi-univalent functions connected with
Balancing numbers. For functions in this class we have derived the estimates of the Taylor–Maclaurin
coefficients
2
a
and
3
a
and Fekete-Szegö functional problems for functions belonging to this new subclass.
The main corollaries are followed by some special manners, and the innovation of the definitions and the proofs
could involve other studies for such types of similarly investigated subclasses of the bi-univalent functions.
Key-Words: - Balancing polynomial, bi-univalent, analytic function, subordination, coefficient estimates,
Fekete-Szegö functional
1 Introduction
The notion of Balancing numbers 󰇛󰇜
introduced by, [1].These numbers have been studied
extensively in the last twenty years. Most new
studies on the topic include the articles, [2], [3], [4],
[5], [6], [7], [8]. Generalizations of Balancing
numbers can be obtained in various ways, [9], [10],
[11], [12], [13].
Definition 1. [11]. Assume that
Cx
and 
Balancing polynomials are defined with the
following recurrence relation
󰇛󰇜󰇛󰇜󰇛󰇜 (1)
where
󰇛󰇜 (2)
󰇛󰇜.
By using the recurrence relation given by (1) it is
easily obtained that 2(x) = 6x, (3)
󰇛󰇜
Lemma 2. [12]. The ordinary generating function of
the Balancing polynomials is defined by
󰇛󰇜
∞

2
61 zxz
z
(4)
Let us denote by
A
the class of functions of the
form:
2
,)(
n
n
nzazzf
(5)
which are analytic in the open unit disc
1z and: CzzD
normalized by
0)0( f
and
. Further, denote by
S
the class of
analytic normalized and univalent functions in
D
.
The Koebe-one quarter theorem, [14], ensures that
the image of
D
under every univalent function
Af
contains a disc of radius 1/4. Thus every
univalent function
f
has an inverse
1
f
satisfying
Dzzzff
,))((
1
and
).41)(),((,))(( 00
1
frfrwwwff
The inverse function
1
f
is given by
5HFHLYHG-XQH5HYLVHG6HSWHPEHU$FFHSWHG2FWREHU3XEOLVKHG2FWREHU
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.87
Arzu Akgül
E-ISSN: 2224-2880
792
Volume 22, 2023
.)55(
)2()(
3
432
3
2
3
3
2
2
2
2
1
waaaa
waawawwf
(6)
A function
Af
is said to be bi-univalent in
U
if
both
f
and
1
f
are univalent in
D
. Let
denote the class of bi-univalent functions defined in
D
. Some examples of functions in the class
are

󰇛󰇜
Koebe function is a member of but not in the class
.
The class was first studied by ,[15], and showed
that  Later, [16], conjectured that
 After that, [17], showed that 
For two analytic functions,
1
f
and
2
f
, such that
)0()0( 21 ff
, we say that
1
f
is subordinate to
2
f
in
U
and write
Uzzfzf ),()( 21
, if there
exists a Schwarz function
)(zv
with
0)0( v
and
Uzzzv ,)(
such that
Uzzvfzf )),(()( 21
. Furthermore, if the
function
2
f
is univalent in
U
, then we have the
following equivalence;
).()( and)0()0()()( 212121 UfUfffzfzf
In a recent study, using Balancing polynomials,
[18], the authors defined the class 󰇡󰇛󰇜󰇢
and
examined the initial coefficients of the functions
belonging to the class 󰇡󰇛󰇜󰇢
as follows:
Definition 3. [18]. The function is named to be in
the class 󰇡󰇛󰇜󰇢
if the following conditions are
satisfied:
′′󰇛󰇜
′󰇛󰇜󰇛󰇜

′′󰇛󰇜
′󰇛󰇜󰇛󰇜

and
)()( 1zfwg
is defined by (2).
Theorem 4. [18]. If
f
󰇛󰇜
and ℂ
󰇥
󰇦, then

,
󰇛󰇜
The following theorem gives the FeketeSzegö
type inequality for the functions in 󰇛󰇜
:
Theorem 4. [18]. If
f
󰇛󰇜
and 
󰇥
󰇦, then





In the last decades, to they can applicable to
number theory, numerical analysis, combinatorics,
and other fields, theory and applications of
Fibonacci, Lucas, Chebyshev, LucasLehmer,
LucasBalancing polynomials, Gregory numbers,
telephone numbers in modern science has gained
very importance. Nowadays, these kind of
polynomials have been investigated by many
authors in, [18], [19], [20], [21], [22], [23], [24],
[25], [26], [27], [28].
2 Coefficient Bounds of the Class
󰇛󰇛󰇜󰇜 and the Fekete-Szegö
İnequality
Consider in the next section analytic bi-univalent
function class 󰇛󰇛󰇜󰇜 deal with the
Balancing polynomials to obtain the estimates of the
coefficients and and Fekete Szegö
functional problems, [29].
Definition 6. A function is named to be in the
class 󰇛󰇛󰇜󰇜 if the following
subordinations
′′󰇛󰇜
′󰇛󰇜 󰇛󰇜
󰇛󰇜
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.87
Arzu Akgül
E-ISSN: 2224-2880
793
Volume 22, 2023
′′󰇛󰇜
′󰇛󰇜󰇛󰇜
󰇛󰇜
,  and
)()( 1zfwg
is defined by (2).
Remark 7. If , we say that  is in
󰇛󰇜󰇛󰇜
󰇛󰇜
This class was introduced by, [18].
Remark 8. If , we sat that  is in
󰇛󰇛󰇜󰇜 İf the following conditions hold
true ′′󰇛󰇜󰇛󰇜
′′󰇛󰇜󰇛󰇜
The next lemma will be used in our study. This
lemma is a generalization of Lemma 6 in, [30],
which could be obtained for 
Lemma 9. [30]. Let  and . If
 and  then
󰇛󰇜󰇛󰇜


.
The next result explains the upper bounds for the
first two coefficients of the functions in
󰇛󰇛󰇜󰇜.
Theorem 10. If
f
󰇛󰇛󰇜󰇜 and 
, then

󰇛󰇜, (10)
󰇛󰇜 (11)
Proof. Let
f
󰇛󰇛󰇜󰇜Then from the
Definition 3, the subordinations (8) and (9) satisfy.
Thus, there exists an analytic function in with
󰇛󰇜, 󰇛󰇜
(12)
such that
′′󰇛󰇜
󰇡′󰇛󰇜󰇢 󰇛󰇜 (13)
Also, there exists an analytic function in with
󰇛󰇜 , 󰇛󰇜 < 1,
(12)
such that
′′󰇛󰇜
′󰇛󰇜󰇛󰇜
where  and the analytical functions and
have the form
󰇛󰇜
・
・
・

󰇛󰇜
・
・
・

Hence, the functions 󰇛󰇜 and 󰇛󰇜 are
of the form
󰇛󰇜 󰇛󰇜󰇛󰇜
󰇟󰇛󰇜󰇛󰇜
󰇠
󰇟󰇛󰇜󰇛󰇜
󰇛󰇜󰇠
・
・
・
 (16)
and
󰇛󰇜 󰇛󰇜󰇛󰇜
󰇟󰇛󰇜󰇛󰇜󰇠
󰇟󰇛󰇜󰇛󰇜
󰇛󰇜󰇠
・
・
・
 (17)
So, comparing the corresponding coefficients in
(13) by (16) and (15) by (17), we obtain that
󰇛󰇜 (18)
󰇛󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
(19)
󰇛󰇜 (20)
󰇛󰇜
=󰇛󰇜󰇛󰇜
󰇛󰇜
(21)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.87
Arzu Akgül
E-ISSN: 2224-2880
794
Volume 22, 2023
From (18) and (20)
 (22)
and

󰇛󰇜󰇛
󰇜 (23)
Adding (18) and (20) we get
󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜󰇛
󰇜 (24)
By using (23) in (24) we have

󰇛󰇜󰇛󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜 (25)
Considering relations (2) and (3) and using them in
(25), we get
󰇛󰇜
󰇛󰇜 (26)
Using (12) and (14) together with the triangle’s
inequality in the equality (26) it follows

󰇛󰇜
Also, if we subtract (21) from (19), considering
(22), we have
󰇛
󰇜󰇛󰇜󰇛󰇜,
then

󰇛󰇜󰇛󰇜
 (27)
This equation combined with (23) leads to
󰇛󰇜

󰇛󰇜󰇛󰇜
 (28)
Utilizing the triangle’s inequality, (12) , (14) and
(26) from (28) it follows
󰇛󰇜
For the special choices of the parameter α, we
obtain the following :
Corollary 11. If 󰇛󰇛󰇜󰇜, then our result
coincides with the result Thorem1 in, [8].
Corollary 12. If 󰇛󰇛󰇜󰇜 and ℂ
󰇥
󰇦 , then we obtain

,
and
󰇛󰇜
The next theorem gives us the Fekete-Szegö
İnequality :
Theorem 13. If
f
󰇛󰇛󰇜󰇜 and 
, then
󰇛󰇜


󰇛󰇜󰇛󰇜

(29)
Proof. If
f
󰇛󰇛󰇜󰇜has the form (5),
from the equations (25) and (27), we get


󰇛󰇜
 
󰇛󰇜
󰇛󰇜

󰇛󰇜󰇛󰇜

󰇛󰇜
󰇛󰇜
where 󰇛󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜 Using the
equalities (12) and (14) and applying Lemma 9, we
obtain the desired result.
Corollary 14. . If
f
󰇛󰇜then our result
coincides with the result Thorem1 in, [18].
Corollary 15. . If
f
󰇛󰇛󰇜󰇜 and
󰇥
󰇦 , then we obtain
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.87
Arzu Akgül
E-ISSN: 2224-2880
795
Volume 22, 2023





4 Conclusion
In our new study, we have introduced and studied
the initial coefficient problems associated with the
new subclass 󰇛󰇛󰇜󰇜 of the well known
bi-univalent class . This bi-univalent function
subclass is given by Definition 6. For the functions
in our new class we have gained the estimates of the
Taylor–Maclaurin coefficients
2
a
,
3
a
, and we
obtained solutions for the Fekete-Szegö functional
problems. New results are shown to follow upon
specializing the parameters involved in our main
results as given in Remark 8 and Corollary 12 for
the class of 󰇛󰇜associated with
Balancing coefficients which are new and not yet
studied so far.
We hope that this work encourages the
researchers to obtain other characterization
properties and relevant connections in other classes
of univalent functions.
As an open problem, we can point out the following:
i) Other researchers may define the Hankel
determinant for this de_new class
ii) Radii of starlikeness can be investigated in this
class.
iii) A new class of fold symmetric analytic
functions introduced by using properties of this
class.
References:
[1] A. Behera, G.K. Panda, On the square roots of
triangular numbers, Fibonacci
Quarterly,Vol.37, 1999, pp.98–105.
[2] R. K. Davala, G. K. Panda, On sum and ratio
formulas for balancing numbers, Journal of
the Ind. Math. Soc., Vol.82, No.12, 2015,
pp.23–32.
[3] R. Frontczak, L. BadenWürttemberg, A note
on hybrid convolutions involving Balancing
and Lucas Balancing numbers, Appl. Math.
Sci.,Vol.12, No.25, 2018, pp.2001–2008
[4] R. Frontczak, L. BadenWürttemberg, Sums of
balancing and Lucas Balancing numbers with
binomial coefficients, Int. J. Math. Anal., Vol.
12, No.12, 2018, pp.585-594.
[5] R. Keskin, O. Karaatlı, Some new properties
of balancing numbers and square triangular
numbers, Journal of integer sequences,
Vol.15, No.1, 2012, pp.1–13.
[6] B.K. Patel, N. Irmak, P.K. Ray, Incomplete
balancing and Lucas-balancing numbers,
Math.Rep.,Vol.20, No.70, 2018, pp.59-72.
[7] T. Komatsu, G.K. Panda, On several kinds of
sums of balancing numbers,
arXiv:1608.05918, (2016).
[8] P.K Ray, Balancing and Lucas-balancing
sums by matrix methods, Math. Rep. (Bucur.),
Vol.17, No.2, 2015, pp.225-233.
[9] A. Berczes, K. Liptai and I. Pink, On
generalized balancing sequences, The
Fibonacci Quart., Vol.48, No.2, 2010, pp.121
128.
[10] K. Liptai, F. Luca, A. Pinter and L. Szalay,
Generalized balancing numbers, Ind. Math.
(N.S.), Vol.20, 2009, pp.87-100.
[11] P. K. Ray, Some Congruences for Balancing
and Lucas Balancing Numbers and Their
Applications, Integers,Vol.14, No.A8, 2014.
[12] R. Frontczak, On balancing polynomials,
Appl. Math. Sci., Vol.13, No.2,2019, pp.57-
66.
[13] R. K. Davala, G. K. Panda, On sum and ratio
formulas for balancing numbers, Journal of
the Ind. Math. Soc., Vol.82, No.12, 2015,
pp.23–32.
[14] P. L. Duren, Univalent Functions,
Grundlehren der Mathematischen
WissenschaftenSpringer, New York, USA 259
(1983).
[15] Lewin, M.: On a coefficient problem for
Biunivalent functions, Proc. Am.Math. Soc,
.Vol. 18, 1967, pp.63–68.
[16] D. A. Brannan,; J. G Clunie,. Aspects of
contemporary complex analysis, In
proceedings of the NATO Advanced Study
Institute held at the University of Durham,
Durham, July, 1979,Academic Press, New
York and London, 1980.
[17] Netanyahu, E. The minimal distance of the
image boundary from the origin and the
second coefficient of a univalent function in
|z| < 1. Arch. Ration. Mech. Anal., Vol.32,
1969,pp.100–112.
[18] İ. Aktaş, Inci Karaman, On Some New
Subclasses of BiUnivalent Functions Defined
by Balancing Polynomials, KMU Journal of
Engineering and Natural Sciences, Vol.5,
No.1, 2023, pp.2532.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.87
Arzu Akgül
E-ISSN: 2224-2880
796
Volume 22, 2023
[19] A. Akgül, F.M. Sakar, A new characterization
of 󰇛󰇜Lucas polynomial coefficients of
the bi-univalent function class associated with
-analogue of Noor integral operator. Afrika
Matematika, 2022 Sep; Vol.33, No.3,pp.87.
[20] A. Akgül, T.Shaba, 󰇛󰇜Lucas
polynomial coefficient relations of the
biunivalent function class, Communications
Faculty of Sciences University of Ankara
Series A1 Mathematics and Statistics, 2022
Dec 12; Vol.71 No.4, pp.112135.
[21] A. Akgül, On a family of bi-univalent
Functions related to the Fibonacci numbers.
Mathematica Moravica, 2022, Vol.26, No.1,
pp.10312.
[22] A. Akgül󰇛󰇜Lucas polynomial
coefficient inequalities of the bi-univalent
function class, Turkish Journal of
Mathematics, 2019, Vol. 43, No.5, pp.21706.
[23] A. Akgül, Coefficient estimates of a new bi-
univalent function class Introduced by Lucas-
Balancing polynomial, Int. J. Open Problems
Compt. Math., Vol.16, No.3, 2023, pp.3647.
[24] Ş. Altınkaya and S. Yalçın, On the
󰇛󰇜Lucas polynomial coefficient bounds
of the biunivalent function class, Boletín de la
Sociedad Matemática Mexicana, 2018, pp.19.
[25] Ş. Altinkaya, S. Yalçın, Some application of
the 󰇛󰇜−Lucas polynomials to the bi-
univalent function class 3a3, Mathematical
Sciences and Applications E Notes, Vol.8,
No.1, 2020, pp.134–141.
[26] F. M. Sakar, S. M.Aydoğan, Initial bounds for
certain subclass of certain subclass of
Generalized Salagean type biunivalent
functions associated with the Horadam
polynomials, Journal of Quality Measurement
and Analysis JQMA, Vol.15, No.1, 2019,
pp.89-100.
[27] R. Öztürk R, İ. Aktaş , Coefficient estimates
for two new subclasses of biunivalent
functions defined by LucasBalancing
polynomials, Turkish J. Ineq., Vol.7, No.1,
2023, pp.5564.
[28] G. Murugusundaramoorthy, K. Vijaya,
T.Bulboaca, Initial coefficient bounds for bi-
univalent functions related to Gregory
coefficients, Mathematics , Vol. 11, No. 13,
2023, pp. 2857.
[29] Fekete, M.; Szegö, G. Eine Bemerkung Über
Ungerade, Schlichte Funktionen, J.
Lond.Math. Soc., vol.18,No.2,1933,pp.85–89.
[30] P. Zaprawa, Estimates of initial coefficients
for bi-univalent functions, Abstr. Appl. Anal.,
2014, Article ID: 357480
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Arzu Akgül worked solely for Conceptualization,
methodology, software, validation, formal analysis,
investigation, resources, data duration,
writingoriginal draft preparation, writing–review
and editing, visualization, supervision, project
administration, funding acquisition,
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.87
Arzu Akgül
E-ISSN: 2224-2880
797
Volume 22, 2023