MATHEMATICAL SCIENCES AND APPLICATIONS E-NOTES
8 (1) 134-141 (2020) c©MSAEN

HTTPS://DOI.ORG/10.36753/MATHENOT.650271

Some Applications of the (p, q)-Lucas Polynomials to
the bi-univalent Function Class Σ

Şahsene Altınkaya* and Sibel Yalçın

Abstract
In this present investigation, based on the (p, q)-Lucas polynomials, we want to build a bridge between
the Theory of Geometric Functions and that of Special Functions, which are usually considered as very
different fields.

Keywords: (p, q)-Lucas polynomials; coefficient bounds; bi-univalent functions; Fekete-Szegö inequalities.

AMS Subject Classification (2020): Primary: 30C45; Secondary: 05A15; 33D15.

*Corresponding author

1. Introduction, preliminaries and known results

In modern science there is a huge interest in the theory and application of the the Dickson polynomials,
Chebyshev polynomials, Fibonacci polynomials, Lucas polynomials and Lucas-Lehmer polynomials. These
polynomials play a fundamental role in mathematics, and have numerous important applications in combinatorics,
number theory, numerical analysis, etc. Therefore, they have been studied extensively, and various generalizations
of them have been introduced (see, for example, [9, 12, 13, 18, 19]).

The classical Lucas polynomials Ln(x) studied by M. Bicknell in 1970 are defined by

Ln(x) = xLn−1(x) + Ln−2(x) (n ≥ 2),

with the initial condition L0(x) = 2 and L1(x) = x.
Since the above classical Lucas polynomials appeared, some authors have explored their different extensions.

For example, the (p, q)-Lucas polynomials with some properties introduced by Lee and Aşçı [7] as follows.

Definition 1.1. (see [7]) Let p(x) and q(x) be polynomials with real coefficients. The (p, q)-Lucas polynomials
Lp,q,n(x) are defined by the recurrence relation

Lp,q,n(x) = p(x)Lp,q,n−1(x) + q(x)Lp,q,n−2(x) (n ≥ 2),

from which the first few Lucas polynomials can be found as

Lp,q,0(x) = 2,

Lp,q,1(x) = p(x),

Lp,q,2(x) = p2(x) + 2q(x),

Lp,q,3(x) = p3(x) + 3p(x)q(x),
...

(1.1)

Received : 24–01–2019, Accepted : 27–09–2019

https://doi.org/10.36753/mathenot.650271


Some Applications of the (p, q)-Lucas Polynomials... 135

For the special cases of p(x) and q(x), we can get the polynomials given in Table 1.

Table 1: Special cases of the Lp,q,n(x) with given initial conditions are given.
p(x) q(x) Lp,q,n(x)
x 1 Lucas polynomials Ln(x)
2x 1 Pell-Lucas polynomials Dn(x)
1 2x Jacobsthal-Lucas polynomials jn(x)
3x -2 Fermat-Lucas polynomials fn(x)
2x -1 Chebyshev polynomials first kind Tn(x)

Theorem 1.1. (see [7]) Let G{Lp,q,n(x)}(z) be the generating function of the (p, q)-Lucas polynomial sequence Lp,q,n(x).
Then

G{Lp,q,n(x)}(z) =

∞∑
n=0

Lp,q,n(x)zn =
2− p(x)z

1− p(x)z − q(x)z2
.

Let A be the class of functions f of the form

f(z) = z + a2z
2 + a3z

3 + · · · , (1.2)

which are analytic in the open unit disk ∆ = {z : z ∈ C and |z| < 1} and normalized under the conditions

f(0) = 0,

f ′(0) = 1.

Further, by S we shall denote the class of all functions in A which are univalent in ∆.
Here, we recall some definitions and concepts of classes of analytic functions. Denote by S∗ the subclass of S of

starlike functions, so that f ∈ S∗ if and only if

<
(
zf ′(z)

f(z)

)
> 0 (z ∈ ∆).

Let f be given by (1.2). Then f ∈ RT if it satisfies the inequality

< (f ′(z)) > 0 (z ∈ ∆).

The subclass RT was studied systematically by MacGregor [10] who indeed referred to numerous earlier investiga-
tions involving functions whose derivative has a positive real part.

For α > 0, let B(α) denote the class of Bazilevic̆ functions defined in the open unit disk ∆ such that

<

(
f ′(z)

(
f(z)

z

)α−1
)
> 0 (z ∈ ∆).

This class of functions was studied first by Singh [15] and considered subsequently by London and Thomas [8].
With a view to recalling the principle of subordination between analytic functions, let the functions f and g be

analytic in ∆. Given functions f, g ∈ A, f is subordinate to g if there exists a Schwarz function w ∈ Λ, where

Λ = {w : w (0) = 0, |w (z)| < 1, z ∈ ∆} ,

such that
f (z) = g (w (z)) (z ∈ ∆) .

We denote this subordination by
f ≺ g or f (z) ≺ g (z) (z ∈ ∆) .

In particular, if the function g is univalent in ∆, the above subordination is equivalent to

f(0) = g(0), f(∆) ⊂ g(∆).



136 Ş. Altınkaya & S. Yalçın

According to the Koebe-One Quarter Theorem [4], it ensures that the image of ∆ under every univalent function f ∈
A contains a disk of radius 1/4. Thus every univalent function f ∈ A has an inverse f−1 satisfying f−1 (f (z)) = z
and f

(
f−1 (w)

)
= w

(
|w| < r0 (f) , r0 (f) ≥ 1

4

)
, where

g(w) = f−1 (w) = w − a2w
2 +

(
2a2

2 − a3

)
w3 −

(
5a3

2 − 5a2a3 + a4

)
w4 + · · · . (1.3)

A function f ∈ A is said to be bi-univalent in ∆ if both f and f−1 are univalent in ∆. Let Σ denote the class of
bi-univalent functions in ∆ given by (1.2). For a brief history and interesting examples in the class Σ, see [16] (see
also [1, 3, 6, 11, 14]).

We want to remark explicitly that, in our article, by using the Lp,q,n(x), functions, our methodology builds a
bridge, to our knowledge not previously well known, between the Theory of Geometric Functions and that of
Special Functions, which are usually considered as very different fields.

Definition 1.2. A function f ∈ Σ is said to be in the class

BΣ (β;x) (0 ≤ β ≤ 1, z, w ∈ ∆)

if the following subordinations are satisfied:(
z

f(z)

)1−β

f ′(z) ≺ G{Lp,q,n(x)}(z)− 1

and (
w

g(w)

)1−β

g′(w) ≺ G{Lp,q,n(x)}(w)− 1,

where the function g is given by (1.3).

Remark 1.1. Here, and in what follows, the argument x ∈ R is independent of the argument z ∈ C.

Example 1.1. Upon setting β = 0 in Definition 1.2, it is readily seen that a function f ∈ Σ is in the class

BΣ (x) (z, w ∈ ∆)

if the following conditions are satisfied:

zf ′(z)

f(z)
≺ G{Lp,q,n(x)}(z)− 1

wg′(w)

g(w)
≺ G{Lp,q,n(x)}(w)− 1,

where the function g is given by (1.3).

Example 1.2. Upon setting β = 1 in Definition 1.2, it is readily seen that a function f ∈ Σ is in the class

HΣ (x) (z, w ∈ ∆)

if the following conditions are satisfied:

f ′(z) ≺ G{Lp,q,n(x)}(z)− 1

and
g′(w) ≺ G{Lp,q,n(x)}(w)− 1,

where the function g is given by (1.3).



Some Applications of the (p, q)-Lucas Polynomials... 137

2. Coefficient Estimates
We begin this section by finding the estimates on the coefficients |a2| and |a3| for functions in the class BΣ (β;x)

proposed by Definition 1.2.

Theorem 2.1. Let f given by (1.2) be in the class BΣ (β;x) . Then

|a2| ≤
|p(x)|

√
2 |p(x)|√

(β + 1) |βp2(x) + 4(β + 1)q(x)|

and

|a3| ≤
p2(x)

(β + 1)2
+
|p(x)|
β + 2

.

Proof. Let f ∈ BΣ (β;x) . From Definition 1.2, for some analytic functions Φ,Ψ such that

Φ(0) = 0, |Φ(z)| =
∣∣t1z + t2z

2 + t3z
3 + · · ·

∣∣ < 1 (z ∈ ∆),

|tk| ≤ 1 (k ∈ N)

and
Ψ(0) = 0, |Ψ(w)| =

∣∣s1w + s2w
2 + s3w

3 + · · ·
∣∣ < 1 (w ∈ ∆),

|sk| ≤ 1 (k ∈ N),

we can write (
z

f(z)

)1−β

f ′(z) = G{Lp,q,n(x)}(Φ(z))− 1

and (
w

g(w)

)1−β

g′(w) = G{Lp,q,n(x)}(Ψ(w))− 1,

or equivalently (
z

f(z)

)1−β

f ′(z) = −1 + Lp,q,0(x) + Lp,q,1(x)Φ(z) + Lp,q,2(x)Φ2(z) + · · · (2.1)

and (
w

g(w)

)1−β

g′(w) = −1 + Lp,q,0(x) + Lp,q,1(x)Ψ(w) + Lp,q,2(x)Ψ2(w) + · · · . (2.2)

From the equalities (2.1) and (2.2), we obtain that(
z

f(z)

)1−β

f ′(z) = 1 + Lp,q,1(x)t1z +
[
Lp,q,1(x)t2 + Lp,q,2(x)t21

]
z2 + · · · (2.3)

and (
w

g(w)

)1−β

g′(w) = 1 + Lp,q,1(x)s1w +
[
Lp,q,1(x)s2 + Lp,q,2(x)s2

1

]
w2 + · · · . (2.4)

Thus, upon comparing the corresponding coefficients in (2.3) and (2.4), we have

(β + 1)a2 = Lp,q,1(x)t1, (2.5)

(β − 1)(β + 2)

2
a2

2 + (β + 2)a3 = Lp,q,1(x)t2 + Lp,q,2(x)t21, (2.6)

− (β + 1)a2 = Lp,q,1(x)s1 (2.7)

and
(β + 2)(β + 3)

2
a2

2 − (β + 2)a3 = Lp,q,1(x)s2 + Lp,q,2(x)s2
1. (2.8)



138 Ş. Altınkaya & S. Yalçın

From the equations (2.5) and (2.7), we can easily see that

t1 = −s1, (2.9)

2(β + 1)2a2
2 = L2

p,q,1(x)
(
t21 + s2

1

)
. (2.10)

If we add (2.6) to (2.8), we get

(β + 1)(β + 2)a2
2 = Lp,q,1(x) (t2 + s2) + Lp,q,2(x)

(
t21 + s2

1

)
. (2.11)

By using (2.10) in the equality (2.11), we have[
(β + 1)(β + 2)L2

p,q,1(x)− 2(β + 1)2Lp,q,2(x)
]
a2

2 = L3
p,q,1(x) (t2 + s2) . (2.12)

which gives

|a2| ≤
|p(x)|

√
2 |p(x)|√

(β + 1) |βp2(x) + 4(β + 1)q(x)|
.

Moreover, if we subtract (2.8) from (2.6), we obtain

2(β + 2)(a3 − a2
2) = Lp,q,1(x) (t2 − s2) + Lp,q,2(x)

(
t21 − s2

1

)
. (2.13)

Then, in view of (2.9) and (2.10), (2.13) becomes

a3 =
L2
p,q,1(x)

2(β + 1)2

(
t21 + s2

1

)
+
Lp,q,1(x)

2(β + 2)
(t2 − s2) .

Then, with the help of (1.1), we finally deduce

|a3| ≤
p2(x)

(β + 1)2
+
|p(x)|
β + 2

.

Corollary 2.1. Let f given by (1.2) be in the class BΣ (x) . Then

|a2| ≤ |p(x)|

√∣∣∣∣ p(x)

2q(x)

∣∣∣∣
and

|a3| ≤ p2(x) +
|p(x)|

2
.

Corollary 2.2. Let f given by (1.2) be in the class HΣ (x) . Then

|a2| ≤
|p(x)|

√
|p(x)|√

|p2(x) + 8q(x)|

and

|a3| ≤
p2(x)

4
+
|p(x)|

3
.

3. Fekete-Szegö Problem

The classical Fekete-Szegö inequality, presented by means of Loewner’s method, for the coefficients of f ∈ S is∣∣a3 − ϑa2
2

∣∣ ≤ 1 + 2 exp(−2ϑ/(1− ϑ)) for ϑ ∈ [0, 1) .

As ϑ→ 1−, we have the elementary inequality
∣∣a3 − a2

2

∣∣ ≤ 1. Moreover, the coefficient functional

zϑ(f) = a3 − ϑa2
2

on the normalized analytic functions f in the unit disk ∆ plays an important role in function theory. The problem
of maximizing the absolute value of the functional zϑ(f) is called the Fekete-Szegö problem, see [5]. Many other
recent works on the Fekete-Szegö problem include, for example, [2, 17, 20].

In this section, we aim to provide Fekete-Szegö inequalities for functions in the classBΣ (β;x). These inequalities
are given in the following theorem.



Some Applications of the (p, q)-Lucas Polynomials... 139

Theorem 3.1. Let f given by (1.2) be in the class BΣ (β;x) and ϑ ∈ R. Then

∣∣a3 − ϑa2
2

∣∣ ≤

|p(x)|
β + 2

, |ϑ− 1| ≤ β + 1

2(β + 2)

∣∣∣∣β + 4(β + 1)
q(x)

p2(x)

∣∣∣∣
2 |1− ϑ|

∣∣p3(x)
∣∣

(β + 1) |βp2(x) + 4(β + 1)q(x)|
, |ϑ− 1| ≥ β + 1

2(β + 2)

∣∣∣∣β + 4(β + 1)
q(x)

p2(x)

∣∣∣∣
.

Proof. From (2.12) and (2.13), we have

a3 − ϑa2
2 =

L3
p,q,1(x) (1− ϑ) (t2 + s2)

(β + 1)(β + 2)L2
p,q,1(x)− 2(β + 1)2Lp,q,2(x)

+
Lp,q,1(x) (t2 − s2)

2(β + 2)

= Lp,q,1(x)

[(
h (ϑ, x) +

1

2(β + 2)

)
t2 +

(
h (ϑ, x)− 1

2(β + 2)

)
s2

]
,

where

h (ϑ, x) =
L2
p,q,1(x) (1− ϑ)

(β + 1)(β + 2)L2
p,q,1(x)− 2(β + 1)2Lp,q,2(x)

.

Then, in view of (1.1), we conclude that

∣∣a3 − ϑa2
2

∣∣ ≤

|p(x)|
β + 2

, 0 ≤ |h (ϑ, x)| ≤ 1

2(β + 2)

2 |p(x)| |h (ϑ, x)| , |h (ϑ, x)| ≥ 1

2(β + 2)

.

Corollary 3.1. Let f given by (1.2) be in the class BΣ (x) and ϑ ∈ R. Then

∣∣a3 − ϑa2
2

∣∣ ≤

|p(x)|

2
, |ϑ− 1| ≤ |q(x)|

p2(x)

|1− ϑ|
∣∣p3(x)

∣∣
2 |q(x)|

, |ϑ− 1| ≥ |q(x)|
p2(x)

.

Corollary 3.2. Let f given by (1.2) be in the class HΣ (x) and ϑ ∈ R. Then

∣∣a3 − ϑa2
2

∣∣ ≤

|p(x)|

3
, |ϑ− 1| ≤

∣∣p2(x) + 8q(x)
∣∣

3p2(x)

|1− ϑ|
∣∣p3(x)

∣∣
|p2(x) + 8q(x)|

, |ϑ− 1| ≥
∣∣p2(x) + 8q(x)

∣∣
3p2(x)

.

If we choose ϑ = 1, we get the next corollaries.

Corollary 3.3. If f ∈ BΣ (β;x) , then ∣∣a3 − a2
2

∣∣ ≤ |p(x)|
β + 2

.

Corollary 3.4. If f ∈ BΣ (x) , then ∣∣a3 − a2
2

∣∣ ≤ |p(x)|
2

.

Corollary 3.5. If f ∈ HΣ (x) , then ∣∣a3 − a2
2

∣∣ ≤ |p(x)|
3

.



140 Ş. Altınkaya & S. Yalçın

References
[1] Altınkaya, Ş., Yalçın, S.: Faber polynomial coefficient bounds for a subclass of bi-univalent functions. C.R. Acad. Sci.

Paris, Ser. I 353 (12), 1075-1080 (2015).

[2] Altınkaya, Ş., Yalçın, S.: Fekete-Szegö inequalities for certain classes of bi-univalent functions. Internat. Scholar. Res.
Notices 2014, 1-6 (2014). Article ID 327962.

[3] Brannan, D. A., Taha, T. S.: On some classes of bi-univalent functions. Stud. Univ. Babeş-Bolyai Math. 31, 70-77
(1986).

[4] Duren, P. L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, Bd. 259, Springer-Verlag,
Berlin, Heidelberg, New York and Tokyo, (1983).

[5] Fekete, M., Szegö, G.: Eine Bemerkung Über Ungerade Schlichte Funktionen. J. London Math. Soc. [s1-8(2)], 85-89
(1933).

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

[7] Lee, G. Y., Aşcı, M.: Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials. J. Appl. Math. 2012, 1-18
(2012). Article ID 264842.

[8] London, R. R., Thomas, D. K.: The derivative of Bazilevic̆ functions. Proc. Amer. Math. Soc. 104, 235-238 (1988).

[9] Lupas, A.: A guide of Fibonacci and Lucas polynomials. Octagon Mathematics Magazine 7, 2-12 (1999).

[10] MacGregor, T. H.: Functions whose derivative has a positive real part. Trans. Amer. Math. Soc. 104, 532-537 (1962).

[11] 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. 32, 100-112 (1969)..

[12] Özkoç, A., Porsuk, A.: A note for the (p, q)-Fibonacci and Lucas quaternion polynomials. Konuralp J. Math. 5 (2),
36-46 (2017).

[13] Filipponi, P., Horadam, A. F.: Derivative sequences of Fibonacci and Lucas polynomials. In: Applications of Fibonacci
Numbers, Springer, Dordrecht, (1991).

[14] Srivastava, H. M., Murugusundaramoorthy, G., Magesh, N.: Certain subclasses of bi-univalent functions associated
with the Hohlov operator. Appl. Math. Lett. 1, 67-73 (2013).

[15] Singh, R.: On Bazilevic̆ functions. Proc. Amer. Math. Soc. 38, 261-271 (1973).

[16] Srivastava, H. M., Mishra, A., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math.
Lett. 23, 1188-1192 (2010).

[17] Srivastava, H. M., Mishra, A. K., Das, M. K.: The Fekete-Szegö problem for a subclass of close-to-convex functions.
Complex Variables Theory Appl. 44 (2), 145-163 (2001).

[18] Vellucci, P., Bersani, A.M.: The class of Lucas-Lehmer polynomials. Rendiconti di Matematica 37 (1-2), 43-62 (2016).

[19] Wang, T., Zhang, W.: Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc.
Sci. Math. Roum. 55 (1), 95-103 (2012).

[20] Zaprawa, Z.: On Fekete-Szegö problem for classes of bi-univalent functions. Bull. Belg. Math. Soc. Simon Stevin 21,
169–178 (2014).



Some Applications of the (p, q)-Lucas Polynomials... 141

Affiliations

ŞAHSENE ALTINKAYA
ADDRESS: Bursa Uludag University, Dept. of Mathematics, 16059, Bursa-Turkey.
E-MAIL: sahsenealtinkaya@gmail.com
ORCID ID: 0000-0002-7950-8450

SIBEL YALÇIN
ADDRESS: Bursa Uludag University, Dept. of Mathematics, 16059, Bursa-Turkey.
E-MAIL: syalcin@uludag.edu.tr
ORCID ID: 0000-0002-0243-8263


	Introduction, preliminaries and known results
	Coefficient Estimates
	Fekete-Szegö Problem