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https://doi.org/10.22199/issn.0717-6279-2019-05-0071
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1138 Sahsene Altınkaya and Sibel Yalın

1. Introduction and definitions

Fibonacci polynomials, Lucas polynomials, Lucas-Lehmer polynomials, Cheby-
chev polynomials, Pell polynomials, Morgan-Voyce polynomials, Orthogo-
nal polynomials and the other special polynomials and their generalizations
are of wide spectra in a variety of branches such as Physics, Engineering,
Architecture, Nature, Art, Number Theory, Combinatorics and Numerical
analysis (see, for example, [8], [10], [11], [12], [14], [15], [16] and [17]).

The well-known (p, q)-Lucas polynomials are defined by the following
definition:

Definition 1.1. (see [7]) Let p(x) and q(x) be polynomials with real co-
efficients. The (p, q)-Lucas polynomials Lp,q,n(x) are established 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)

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) =
∞X
n=0

Lp,q,n(x)z
n =

2− p(x)z

1− p(x)z − q(x)z2
.

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(p, q)-Lucas polynomials and their applications to bi-univalent... 1139

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 condition f(0) = f 0(0) − 1 = 0. Further, by S we
represent the class of all functions in A which are univalent in ∆.

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 show 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(∆).

According to the Koebe-One Quarter Theorem [4], it ensures that the
image of ∆ under every univalent function f ∈ A contains a disc 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

³
|w| < r0 (f) , r0 (f) ≥ 1

4

´
,

where

f−1 (w) = w − a2w
2 +

³
2a22 − a3

´
w3 −

³
5a32 − 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 indicate the class of bi-univalent functions in ∆ given

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1140 Sahsene Altınkaya and Sibel Yalın

by (1.2). For a brief history and interesting examples in the class Σ, see
[13] (see also [1], [2], [3], [6] and [9]).

In the present paper, by using the Lp,q,n(x) functions, our methodology
intertwine to yield the Theory of Geometric Functions and that of Special
Functions, which are usually considered as very different fields. Thus, we
aim at introducing a new class of bi-univalent functions defined through the
(p, q)-Lucas polynomials. Furthermore, we derive coefficient inequalities
and obtain Fekete-Szeg problem for this new function class.

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

WΣ (τ, µ, η;x) (τ ∈ C\{0}, µ ≥ 0, η ≥ 0; z, w ∈ ∆)
if the following subordinations are satisfied:

∙
1 +

1

τ

µ
(1− µ+ 2η)

f(z)

z
+ (µ− 2η)f 0(z) + ηzf 00(z)− 1

¶¸
≺ G{Lp,q,n(x)}(z)−1

and

∙
1 +

1

τ

µ
(1− µ+ 2η)

g(w)

w
+ (µ− 2η)g0(w) + ηwg00(w)− 1

¶¸
≺ G{Lp,q,n(x)}(w)−1

where the function g is given by (1.3).

It is interesting to note that the special values of τ, µ and η lead the class
WΣ (τ, µ, η;x) to various subclasses, we illustrate the following subclasses:

1. For µ = 1+2η, we get the class WΣ (τ, 1 + 2η, η;x) =WΣ (τ, η;x). A
function f ∈ Σ is said to be in the class

WΣ (τ, η;x) (τ ∈ C\{0}, µ ≥ 0, z, w ∈ ∆)

if the following subordinations are satisfied:

∙
1 +

1

τ

¡
f 0(z) + ηzf 00(z)− 1

¢¸
≺ G{Lp,q,n(x)}(z)− 1

and

∙
1 +

1

τ

¡
g0(w) + ηwg00(w)− 1

¢¸
≺ G{Lp,q,n(x)}(w)− 1

where the function g is given by (1.3).

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(p, q)-Lucas polynomials and their applications to bi-univalent... 1141

2. For η = 0, we obtain the class WΣ (τ, µ, 0;x) = WΣ (τ, µ;x). A func-
tion f ∈ Σ is said to be in the class

WΣ (τ, µ;x) (τ ∈ C\{0}, µ ≥ 0; z, w ∈ ∆)

if the following subordinations are satisfied:

∙
1 +

1

τ

µ
(1− µ)

f(z)

z
+ µzf 0(z)− 1

¶¸
≺ G{Lp,q,n(x)}(z)− 1

and

∙
1 +

1

τ

µ
(1− µ)

g(w)

w
+ µg0(w)− 1

¶¸
≺ G{Lp,q,n(x)}(w)− 1

where the function g is given by (1.3).

3. For η = 0 and µ = 1, we get the class WΣ (τ, 1, 0;x) = WΣ (τ ;x). A
function f ∈ Σ is said to be in the class

WΣ (τ, µ;x) (τ ∈ C\{0}; z, w ∈ ∆)

if the following subordinations are satisfied:

∙
1 +

1

τ

¡
f 0(z)− 1

¢¸
≺ G{Lp,q,n(x)}(z)− 1

and

∙
1 +

1

τ

¡
g0(w)− 1

¢¸
≺ G{Lp,q,n(x)}(w)− 1

where the function g is given by (1.3).

2. Coefficient bounds

In this section, we shall make use of the (p, q)-Lucas polynomials to get
the estimates on the coefficients |a2| and |a3| for functions in the class
WΣ (τ, µ, η;x) proposed by Definition 1.2.

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1142 Sahsene Altınkaya and Sibel Yalın

Theorem 2.1. Let f given by (1.2) be in the class WΣ (τ, µ, η;x) . Then

|a2| ≤
|τ | |p(x)|

p
|p(x)|r¯̄̄h

(1 + 2µ+ 2η)τ − (1 + µ)2
i
p2(x)− 2 (1 + µ)2 q(x)

¯̄̄
and

|a3| ≤
|τ |2 p2(x)
(1 + µ)2

+
|τ | |p(x)|
1 + 2µ+ 2η

.

Proof. Let f ∈ WΣ (τ, µ, η;x) . From Definition 1.2, for some analytic
functions Φ,Ψ such that Φ(0) = Ψ(0) = 0 and |Φ(z)| < 1, |Ψ(w)| < 1 for
all z, w ∈ ∆, we can write

1+
1

τ

µ
(1− µ+ 2η)

f(z)

z
+ (µ− 2η)f 0(z) + ηzf 00(z)− 1

¶
= G{Lp,q,n(x)}(Φ(z))−1,

1+
1

τ

µ
(1− µ+ 2η)

g(w)

w
+ (µ− 2η)g0(w) + ηwg00(w)− 1

¶
= G{Lp,q,n(x)}(Ψ(w))−1,

or equivalently

1 +
1

τ

³
(1− µ+ 2η)f(z)z + (µ− 2η)f 0(z) + ηzf 00(z)− 1

´
= −1 + Lp,q,0(x) + Lp,q,1(x)Φ(z) + Lp,q,2(x)Φ

2(z) + · · · ,
(2.1)

1 +
1

τ

³
(1− µ+ 2η)g(w)w + (µ− 2η)g0(w) + ηwg00(w)− 1

´
= −1 + Lp,q,0(x) + Lp,q,1(x)Ψ(w) + Lp,q,2(x)Ψ

2(w) + · · · .
(2.2)

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

1 +
1

τ

³
(1− µ+ 2η)f(z)z + (µ− 2η)f 0(z) + ηzf 00(z)− 1

´
= 1 + Lp,q,1(x)t1z +

£
Lp,q,1(x)t2 + Lp,q,2(x)t

2
1

¤
z2 + · · · ,

(2.3)

and

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(p, q)-Lucas polynomials and their applications to bi-univalent... 1143

1 +
1

τ

³
(1− µ+ 2η)g(w)w + (µ− 2η)g0(w) + ηwg00(w)− 1

´
= 1 + Lp,q,1(x)s1w +

£
Lp,q,1(x)s2 + Lp,q,2(x)s

2
1

¤
w2 + · · · .

(2.4)

It is fairly well-known that if

|Φ(z)| =
¯̄̄
t1z + t2z

2 + t3z
3 + · · ·

¯̄̄
< 1 (z ∈ ∆)

and

|Ψ(w)| =
¯̄̄
s1w + s2w

2 + s3w
3 + · · ·

¯̄̄
< 1 (w ∈ ∆),

then

|tk| ≤ 1 and |sk| ≤ 1 (k ∈N).(2.5)

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

1

τ
(1 + µ)a2 = Lp,q,1(x)t1,(2.6)

1

τ
(1 + 2µ+ 2η)a3 = Lp,q,1(x)t2 + Lp,q,2(x)t

2
1,(2.7)

− 1
τ
(1 + µ)a2 = Lp,q,1(x)s1(2.8)

and

1

τ
(1 + 2µ+ 2η)

³
2a22 − a3

´
= Lp,q,1(x)s2 + Lp,q,2(x)s

2
1.(2.9)

From the equations (2.6) and (2.8), we can easily see that

t1 = −s1,(2.10)

2

τ2
(1 + µ)2 a22 = L2p,q,1(x)

³
t21 + s21

´
.(2.11)

If we add (2.7) to (2.9), we get

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1144 Sahsene Altınkaya and Sibel Yalın

2

τ
(1 + 2µ+ 2η)a22 = Lp,q,1(x) (t2 + s2) + Lp,q,2(x)

³
t21 + s21

´
.(2.12)

Clearly, by using (2.11) in the equality (2.12), we have

2
h
(1 + 2µ+ 2η)τL2p,q,1(x)− (1 + µ)2 Lp,q,2(x)

i
τ2L2p,q,1(x)

a22 = Lp,q,1(x) (t2 + s2) .

(2.13)

which gives

|a2| ≤
|τ | |p(x)|

p
|p(x)|r¯̄̄h

(1 + 2µ+ 2η)τ − (1 + µ)2
i
p2(x)− 2 (1 + µ)2 q(x)

¯̄̄ .
Moreover, if we subtract (2.9) from (2.7), we obtain

2

τ
(1 + 2µ+ 2η)(a3 − a22) = Lp,q,1(x) (t2 − s2) + Lp,q,2(x)

³
t21 − s21

´
.

(2.14)

Then, in view of (2.10) and (2.11), (2.14) becomes

a3 =
τ2L2p,q,1(x)

¡
t21 + s21

¢
2 (1 + µ)2

+
τLp,q,1(x) (t2 − s2)

2(1 + 2µ+ 2η)
.

It is seen from (1.1) and (2.5) that

|a3| ≤
|τ |2 p2(x)
(1 + µ)2

+
|τ | |p(x)|
1 + 2µ+ 2η

.

2

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(p, q)-Lucas polynomials and their applications to bi-univalent... 1145

Corollary 2.1. Let f given by (1.2) be in the class WΣ (τ, η;x) . Then

|a2| ≤
|τ | |p(x)|

p
|p(x)|r¯̄̄h

3(1 + 2η)τ − 4 (1 + η)2
i
p2(x)− 8 (1 + η)2 q(x)

¯̄̄
and

|a3| ≤
|τ |2 p2(x)
4 (1 + η)2

+
|τ | |p(x)|
3(1 + 2η)

.

Corollary 2.2. Let f given by (1.2) be in the class WΣ (τ, µ;x) . Then

|a2| ≤
|τ | |p(x)|

p
|p(x)|r¯̄̄h

(1 + 2µ)τ − (1 + µ)2
i
p2(x)− 2 (1 + µ)2 q(x)

¯̄̄
and

|a3| ≤
|τ |2 p2(x)
(1 + µ)2

+
|τ | |p(x)|
1 + 2µ

.

Corollary 2.3. Let f given by (1.2) be in the class WΣ (τ ;x) . Then

|a2| ≤
|τ | |p(x)|

p
|p(x)|p

|(3τ − 4)p2(x)− 8q(x)|
and

|a3| ≤
|τ |2 p2(x)

4
+
|τ | |p(x)|

3
.

3. Fekete-Szeg problem

The classical Fekete-Szeg inequality, presented by means of Loewner’s method,
for the coefficients of f ∈ S is¯̄̄

a3 − ξa22

¯̄̄
≤ 1 + 2 exp(−2ξ/(1− ξ)) for ξ ∈ [0, 1) .

As ξ → 1−, we have the elementary inequality
¯̄
a3 − a22

¯̄
≤ 1. Moreover,

the coefficient functional

Γξ(f) = a3 − ξa22

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1146 Sahsene Altınkaya and Sibel Yalın

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 Γξ(f) is called the Fekete-Szeg problem, see [5].

In this section, we aim to provide Fekete-Szeg inequalities for functions
in the class Tn

Σ (τ ;x). These inequalities are given in the following theorem.

Theorem 3.1. Let f given by (1.2) be in the class WΣ (τ, µ, η;x) and
ξ ∈ R. Then

¯̄
a3 − ξa22

¯̄
≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

|p(x)|
(1 + 2µ+ 2η) |τ | ,

|1− ξ| ≤
¯̄̄̄
¯ 1τ2 − (1 + µ)2

(1 + 2µ+ 2η)τ3

µ
1 +

2q(x)

p2(x)

¶¯̄̄̄
¯

|τ |2
¯̄
p3(x)

¯̄
|1− ξ|¯̄̄h

(1 + 2µ+ 2η)τ − (1 + µ)2
i
p2(x)− 2 (1 + µ)2 q(x)

¯̄̄ ,

|1− ξ| ≥
¯̄̄̄
¯ 1τ2 − (1 + µ)2

(1 + 2µ+ 2η)τ3

µ
1 +

2q(x)

p2(x)

¶¯̄̄̄
¯

Proof. From (2.13) and (2.14)

a3 − ξa22 =
τ2L3p,q,1(x) (1− ξ) (t2 + s2)

2
h
(1 + 2µ+ 2η)τL2p,q,1(x)− (1 + µ)2 Lp,q,2(x)

i
+
τLp,q,1(x) (t2 − s2)

2(1 + 2µ+ 2η)

= Lp,q,1(x)

∙µ
K(ξ, x) +

1

2(1 + 2µ+ 2η)τ

¶
t2

+

µ
K(ξ, x)− 1

2(1 + 2µ+ 2η)τ

¶
s2

where

K (ξ, x) =
τ2L2p,q,1(x) (1− ξ)

2
h
(1 + 2µ+ 2η)τL2p,q,1(x)− (1 + µ)2 Lp,q,2(x)

i .
Along the way, in view of (1.1), we conclude that

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(p, q)-Lucas polynomials and their applications to bi-univalent... 1147

¯̄
a3 − ξa22

¯̄
≤

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
|p(x)|

(1 + 2µ+ 2η) |τ | , 0 ≤ |K(ξ, x)| ≤ 1

2(1 + 2µ+ 2η) |τ |

2 |p(x)| |K(ξ, x)| , |K(ξ, x)| ≥ 1

2(1 + 2µ+ 2η) |τ |
2

Corollary 3.1. Let f given by (1.2) be in the classWΣ (τ, η;x) and ξ ∈ R.
Then

¯̄
a3 − ξa22

¯̄
≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

|p(x)|
3(1 + 2η) |τ | ,

|1− ξ| ≤
¯̄̄̄
¯ 1τ2 − 4 (1 + η)2

3(1 + 2η)τ3

µ
1 +

2q(x)

p2(x)

¶¯̄̄̄
¯

|τ |2
¯̄
p3(x)

¯̄
|1− ξ|¯̄̄h

3(1 + 2η)τ − 4 (1 + η)2
i
p2(x)− 8 (1 + η)2 q(x)

¯̄̄ ,

|1− ξ| ≥
¯̄̄̄
¯ 1τ2 − 4 (1 + η)2

3(1 + 2η)τ3

µ
1 +

2q(x)

p2(x)

¶¯̄̄̄
¯

.

Corollary 3.2. Let f given by (1.2) be in the classWΣ (τ, µ;x) and ξ ∈ R.
Then

¯̄
a3 − ξa22

¯̄
≤

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

|p(x)|
(1 + 2µ) |τ | ,

|1− ξ| ≤
¯̄̄̄
¯ 1τ2 − (1 + µ)2

(1 + 2µ)τ3

µ
1 +

2q(x)

p2(x)

¶¯̄̄̄
¯

|τ |2
¯̄
p3(x)

¯̄
|1− ξ|¯̄̄h

(1 + 2µ)τ − (1 + µ)2
i
p2(x)− 2 (1 + µ)2 q(x)

¯̄̄ ,

|1− ξ| ≥
¯̄̄̄
¯ 1τ2 − (1 + µ)2

(1 + 2µ)τ3

µ
1 +

2q(x)

p2(x)

¶¯̄̄̄
¯

Corollary 3.3. Let f given by (1.2) be in the class WΣ (τ ;x) and ξ ∈ R.
Then

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https://doi.org/10.1016/j.crma.2015.09.003


https://doi.org/10.1112/jlms/s1-8.2.85
https://doi.org/10.1090/S0002-9939-1967-0206255-1
https://doi.org/10.1155/2012/264842
https://doi.org/10.1007/BF00247676
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https://doi.org/10.1007/s11139-016-9862-5
https://bit.ly/36TMDIJ


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