EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3801-3814
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Family of Bi-Univalent Functions Defined by
(p, q)-Derivative Operator Subordinate to a Generalized

Bivariate Fibonacci Polynomials

Basem Aref Frasin1,2, Sondekola Rudra Swamy3, Ala Amourah4,5,∗,
Jamal Salah6,∗, Ranjitha Hebbar Maheshwarappa2

1 Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq - Jordan
2 Jadara Research Center, Jadara University, Irbid 21110, Jordan
3 Department of Information Science and Engineering, Acharya Institute of Technology,
Bengaluru- 560 107, Karnataka, India
4 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar
3111, Oman
5 Applied Science Private University, Amman, Jordan
6 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42,
Post Code No. 400 Ibra, Sultanate of Oman

Abstract. Making use of a generalized bivariate Fibonacci polynomials, we propose a family of
normalized regular functions ψ(ζ) = ζ + d2ζ

2 + d3ζ
3 + · · · , which are bi-univalent in the disc

{ζ ∈ C : |ζ| < 1} involving (p, q)-derivative operator. We find estimates on the coefficients |d2|,
|d3| and the Fekete-Szegö inequality for members of this family. New implications of the primary
result as well as pertinent links to previously published findings are also provided.

2020 Mathematics Subject Classifications: 30C45, 11B39

Key Words and Phrases: (p, q)-derivative operator, Regular function, Fekete - Szegö functional,
Bi-univalent function, Bivariate Fibonacci Polynomials

1. Introduction

The quantum (or q-) calculus is essential because it is applied in many different
branches of mathematics, computer science, physics, and other related fields. The ex-
tension of the q-calculus to the (p, q)-calculus, was taken into consideration by the re-
searchers. The (p, q)-calculus, which includes the (p, q)-number, is first examined around

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5526

Email addresses: bafrasin@yahoo.com (B. Frasin),
swamy2704@acharya.ac.in (S. Swamy), AAmourah@su.edu.om (A. Amourah),
damous73@yahoo.com (J. Salah), ranjithah.m@acharya.ac.in (R. Maheshwarappa)

https://www.ejpam.com 3801 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3802

the same time (1991) and subsequently on its own by [12, 15, 21, 46]. Fibonacci oscillators
were studied with the presentation of the (p, q)-number in [12]. The investigation of the
(p, q)-number in [15] allows for the construction of a (p, q)-Harmonic oscillator. In[21],
the (p, q)-number was explored to unify or generalize various forms of q-oscillator alge-
bras. The (p, q)-numbers are investigated in [46] to calculate the (p, q)-Stirling numbers.
Consequently, many mathematical, computer science, physical, chemical and other related
problems require knowledge of (p, q)-calculus. Expanding upon the previously mentioned
papers, numerous scientists have studied the (p, q)-calculus in a variety of research fields
since 1991. A syntax for embedding the q-series into a (p, q)-series was given by the results
in [31]. Additionally, they looked into (p, q)-hypergeometric series and discovered some
outcomes that matched (p, q)-extensions of the well-known q-identities. The q-identities
are extended correspondingly to yield the (p, q)-series (see, e.g., [11]). We provide some
elementary definitions of the (p, q)-calculus concepts. The (p, q)-bracket number is given

by [j]p,q = pj−1+pj−2q+ ...+p2qj−3+pqj−2+qj−1 = pj−qj

p−q (p ̸= q), which is an extension

of q-number (see [30]), that is [j]q = 1−qj

1−q (q ̸= 1). Note that [j]p,q is symmetric and if
p=1, then [j]p,q=[j]q.

let D = {ζ ∈ C : |ζ| < 1}, where C is the complex plane. Let R be the family of real
numbers and N = N0\{0} := {1, 2, 3, ...}.

Definition 1. [1] Let ψ be a function defined on C and 0 < q < p ≤ 1. Then the
(p, q)-derivative of θ is defined by

Dp,qψ(ζ) =
ψ(pζ)− ψ(qζ)

(p− q)ζ
(ζ ̸= 0),

and Dp,qψ(0) = ψ′(0), provided ψ′(0) exists.

We note that Dp,qζ
j = [j]p,qζ

j−1 and Dp,qln(ζ) = ln(p/q)
(p−q)ζ . Also, we observe that

[j]p,q → j, if q → 1− and p = 1.Therefore, Dp,qψ(ζ) → ψ′(ζ) as q → 1− and p = 1. Any
function’s (p, q)-derivative is a linear operator.More accurately Dp,q(aψ1(ζ) + bψ2(ζ)) =
aDp,qψ1(ζ) + bDp,qψ2(ζ), for any constants a and b. The product rules and quotient rules
are satisfied by the (p, q)-derivative (see [37]). The exponential functions are used to define
the (p, q)-analogs of many functions, including sine, cosine, and tangent in the same way
as their Euler expressions. Duran et al.[23] have examined the (p, q)-derivatives of these
functions. To learn more about (p, q)-calculus, see [1, 8, 16, 24].

Let us take a normalized regular function ψ in D given by

ψ(ζ) = ζ +

∞∑
j=2

djζ
j , (1)

and let A be the set of all such functions. Let S = {ψ ∈ A : ψ is univalent inD}. If ψ ∈A
is of the form (1), then

Dp,qψ(ζ) = 1 +

∞∑
j=2

[j]p,qdjζ
j−1, (ζ ∈ D), (2)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3803

The renowned Koebe theorem (see[25]) states that, each function ψ ∈ S has an inverse
given by

ψ−1(ω) = Ψ(ω) = ω − d2ω
2 + (2d22 − d3)ω

3 − (5d32 − 5d2d3 + d4)ω
4 + ... (3)

satisfying ζ = ψ−1(ψ(ζ)) and ω = ψ(ψ−1(ω)), |ω| < r0(ψ), r0(ψ) ≥ 1/4, ζ, ω ∈ D.
The notion of bi-univalent functions was first presented by Levin in his work [33]. These

are analytic functions, denoted by ψ, where both ψ and ψ−1are univalent in D. The set of

all bi-univalent functions of the type (1) is symbolized by Σ. 1
2 log

(
1+ζ
1−ζ

)
, −log(1− ζ) and

ζ
1−ζ are some of the functions in the Σ family.However, ζ− ζ2

2 ,
ζ

1−ζ2
, and the Koebe function

do not belong in Σ, even though they are in S. For a concise analysis and to discover some
of the characteristics of the family Σ, see [3, 5, 13, 14, 29, 44] and the citation provided
in these papers. The article by Srivastava et al.[40] gave rise to the recent momentum
of studies of the bi-univalent function family.Numerous scholars have looked into several
fascinating special families of Σ since this article brought the subject back to life (see
[9, 10, 17, 18, 27].

The (p, q)-calculus was used to study several subfamilies of the family S and the
family Σ. In [41], the subordination principle is used to define the (p, q)-starlike and
(p, q)-convex functions families. Novel subfamilies of the family σ associated with (p, q)-
differential operators have also been presented and examined in a number of studies (refer
to [6, 7, 22, 35, 45]).

Let s(κ, y) and t(κ, y) be polynomials with real coefficients. For, j ≥ 2, the generalized
bivariate Fibonacci polynomials(GBFP) are defined by the recurrence relation:

Fj(κ, y) = s(κ, y)Fj−1(κ, y) + t(κ, y)Fj−2(κ, y), (4)

where F0(κ, y) = 0, F1(κ, y) = 1 and s2(κ, y) + 4t(κ, y) > 0. The generating function of
GBFP is (see [32])

F(κ, y, z) =
∞∑
j=2

Fj(κ, y)zj =
z

1− s(κ, y)z − t(κ, y)z2
. (5)

For specific selections of s(κ, y) and t(κ, y), GBFP leads to various known polynomials
(see [47]). Readers with an interest in GBFP can find a brief history and extensive
information in [19] and its references. For members of specific subclasses of σ associated
with GBFP, interesting results have been obtained in [2, 28] regarding coefficient estimates
and Fekete-Szegö functional.

For brevity, we write hereafter that s(κ, y) = s and t(κ, y) = t. F2(κ, y) = s,
F3(κ, y) = s2 + t,..., are evident from (4).

For functions θ1, θ2∈ A, we say that θ1 is subordinate to θ2, if there is κ(ζ), a Schwarz
function in D with κ(0) = 0 and |κ(ζ)| < 1 (ζ ∈ D), such that θ1(ζ) = θ2(κ(ζ)), ζ ∈ D.
This is indicated as θ1 ≺ θ2 or θ1(ζ) ≺ θ2(ζ) (ζ ∈ D). In particular, if θ2 ∈ S, then
θ1(ζ) ≺ θ2(ζ) ⇔ θ1(0) = θ2(0) and θ1(D) ⊂ θ2(D).



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3804

Definition 2. The (p, q)-analogue of Swamy differential operator for ψ ∈ A is defined as
follows:

Ων,µ,0
p,q ψ(ζ) = ψ(ζ),

Ων,µ,1
p,q ψ(ζ) =

νψ(ζ) + µzDp,qψ(ζ)

ν + µ
,

... ,
Ων,µ,k
p,q ψ(ζ) = Ων,µ

p,q (Ω
ν,µ,k−1
p,q θ(ζ)),

where k ∈ N, µ ≥ 0, ν a real number with ν + µ > 0, 0 < q < p ≤ 1 and ζ ∈ D.

Remark 1. i). Ων,µ,k
p,q : A → A is a linear operator, as we can see, and for ψ(ζ), as

provided by (1), we have

Ων,µ,k
p,q ψ(ζ) = ζ +

∞∑
j=2

(
ν + µ[j]p,q
ν + µ

)k

djζ
j , (6)

ii). If we let ν = 0 and µ = 1, then Ων,µ,k
p,q ψ(ζ) reduces to the (p, q)-analogue of

Salagean operator discussed in[39].

iii). If we take ν = 1 − µ, µ ≥ 0, then Aµ,k
p,q (= Ω1−µ,µ,k

p,q ) : A → A is a linear operator
and for ψ(ζ) given by (1), we have

Aµ,k
p,q ψ(ζ) = ζ +

∞∑
j=2

(1 + µ([j]p,q − 1))k djζ
j , (7)

which is (p, q)-analogue of Al-Oboudi differential operator.

iv). If we put ν = l + 1 − µ, l > −1, µ ≥ 0, then C l,µ,k
p,q (= Ωl+1−µ,µ,k

p,q ) : A → A is a
linear operator and for ψ(ζ) given by (1), we have

= C l,µ,k
p,q ψ(ζ) = ζ +

∞∑
j=2

(
l + 1 + µ([j]p,q − 1)

l + 1

)k

djζ
j , (8)

which is (p, q)-analogue of Catas differential operator.
v). Swamy operator[42, 43], Al-Oboudi operator[4], and Cătaş operator [20] are ob-

tained by taking q → 1− and p = 1 in (6), (7), and (8), respectively.

With the generating function F(κ, y, z) as in (5), we introduce a new family of σ sub-
ordinate to GBFP Fj(κ, y) as in (4). The Fekete-Szegö functional[26] on some subclasses
of σ associated with GBFP and the previously mentioned trends on coefficient-related
problems serve as inspiration for the defined family. The inverse function ϕ−1(ω) = ψ(ω)
is as in (3), and F(κ, y, z) is as in (5) are assumed throughout this paper unless otherwise
noted.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3805

Definition 3. A function ψ ∈ Σ is said to be in the family Eλ,k
Σ,p,q(F, ν, µ), if

1

2

ζ(Ων,µ.k
p,q ψ(ζ))′

ψ(ζ)
+

(
ζ(Ων,µ.k

p,q ψ(ζ))′

ψ(ζ)

) 1
λ

 ≺ F (ζ) =
F(κ, y, ζ)

ζ
, ζ ∈ D

and

1

2

ω(Ων,µ.k
p,q Ψ(ω))′

Ψ(ω)
+

(
ω(Ων,µ.k

p,q Ψ(ω))′

Ψ(ω)

) 1
λ

 ≺ F (ω) =
F(κ, y, ω)

ω
, ω ∈ D,

where 0 < λ ≤ 1, µ ≥ 0, ν a real number with ν + µ > 0, k ∈ N, and

F (z) =
1

1− sz − tz2
, s2 + 4t > 0. (9)

For particular chioces of p, q, λ, and ν, the family Eλ,k
Σ,p,q(F, ν, µ) includes many new

subfamilies of Σ as mentioned below:
Example 1.1. Fλ,k

Σ,p,q(F, µ) ≡ Eλ,k
Σ,p,q(F, 1−µ, µ), 0 < λ ≤ 1, µ ≥ 0, and k ∈ N is the set

of members ψ in Σ that satisfy

1

2

ζ(Aµ.k
p,q ψ(ζ))′

ψ(ζ)
+

(
ζ(Aµ.k

p,q ψ(ζ))′

ψ(ζ)

) 1
λ

 ≺ F (ζ) =
F(κ, y, ζ)

ζ
, ζ ∈ D,

and

1

2

ω(Aµ.k
p,qΨ(ω))′

Ψ(ω)
+

(
ω(Aµ.k

p,qΨ(ω))′

Ψ(ω)

) 1
λ

 ≺ F (ω) =
F(κ, y, ω)

ω
, ω ∈ D.

where F (z) is as mentioned in (9).

Example 1.2. Gλ,k
Σ,p,q(F, l, µ) ≡ Eλ,k

Σ,p,q(F, l+1−µ, µ), 0 < λ ≤ 1, l > −1, µ ≥ 0, and k ∈
N is the set of members ψ ∈ Σ that satisfy

1

2

ζ(C l,µ.k
p,q ψ(ζ))′

ψ(ζ)
+

(
ζ(C l,µ.k

p,q ψ(ζ))′

ψ(ζ)

) 1
λ

 ≺ F (ζ) =
F(κ, y, ζ)

ζ
, ζ ∈ D,

and

1

2

ω(C l,µ.k
p,q Ψ(ω))′

Ψ(ω)
+

(
ω(C l,µ.k

p,q Ψ(ω))′

Ψ(ω)

) 1
λ

 ≺ F (ω) =
F(κ, y, ω)

ω
, ω ∈ D.

where F (z) is as mentioned in (9).



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3806

Example 1.3. Hk
Σ,p,q(F, ν, µ) ≡ E1,k

Σ,p,q(F, ν, µ) is the collection of elements ψ ∈ Σ that
satisfy

ζ(Ων,µ.k
p,q ψ(ζ))′

ψ(ζ)
≺ F (ζ) =

F(κ, y, ζ)
ζ

, ζ ∈ D

and
ω(Ων,µ.k

p,q Ψ(ω))′

Ψ(ω)
≺ F (ω) =

F(κ, y, ω)
ω

, ω ∈ D,

where µ ≥ 0, ν a real number with ν + µ > 0, k ∈ N and F (z) is as mentioned in (9).

Example 1.4. If q → 1− and p = 1 in the set Eλ,k
Σ,p=1,q→1−(F, ν, µ), then we obtain a

subset Kλ,k
Σ (F, ν, µ), which is a collection of functions ψ ∈ Σ that satisfy

1

2

{
ζ(Γν,µ.kψ(ζ))′

ψ(ζ)
+

(
ζ(Γν,µ.kψ(ζ))′

ψ(ζ)

) 1
λ

}
≺ F (ζ) =

F(κ, y, ζ)
ζ

, ζ ∈ D,

and

1

2

{
ω(Γν,µ.kΨ(ω))′

Ψ(ω)
+

(
ω(Γν,µ.kΨ(ω))′

Ψ(ω)

) 1
λ

}
≺ F (ω) =

F(κ, y, ω)
ω

, ω ∈ D,

where Γν,µ.k ≡ Ων,µ,k
p=1,q→1− , 0 < λ ≤ 1, µ ≥ 0, ν a real number with ν + µ > 0, k ∈

N andF (z) is as mentioned in (9)..
Fekete-Szegö inequality[26] and estimates for |d2| and |d3| are found in Section 2 for

functions ∈ Sλ,k
Σ,p,q(F, ν, µ). A few intriguing ramifications of the main result as well as

pertinent links to the previous results are also provided.

2. Main results

We first determine the bounds for |d2|, |d3| and an inequality of Fekete-Szegö for

elements in Sλ,k
Σ,p,q(F, ν, µ).

Theorem 1. Let 0 < λ ≤ 1, µ ≥ 0, ν a real number such that ν + µ > 0, and k ∈ N. If a
function ψ ∈ Eλ,k

Σ,p,q(F, ν, µ), then

i).|d2| ≤
2λs

√
s√

|(2λ(λ+ 1)(N −M) + (1− λ)M2)s2 − (1 + λ)2M2(s2 + t)|
, (10)

ii). |d3| ≤
2λs

(1 + λ)N
+

4λ2s2

(1 + λ)2M2
, (11)

and for ξ ∈ R

iii). |d3 − ξd22| ≤

{
2λs

(1+λ)N ; |1− ξ| ≤ J
4λ2s3 |1−ξ|

|(2λ(λ+1)(N−M)+(1−λ)M2)s2−(1+λ)2M2(s2+t)| ; |1− ξ| ≥ J ,
(12)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3807

where

J =
|(2λ(λ+ 1)(N −M) + (1− λ)M2)s2 − (1 + λ)2M2(s2 + t)|

2λ(1 + λ)N s2
, (13)

M =

(
2

(
ν + µ[2]p,q
ν + µ

)k

− 1

)
, (14)

and

N =

(
3

(
ν + µ[3]p,q
ν + µ

)k

− 1

)
. (15)

Proof. Let ψ ∈ Eλ,k
Σ,p,q(F, ν, µ). Then, based on Definition 3, we can write

1

2

ζ(Ων,µ.k
p,q ψ(ζ))′

ψ(ζ)
+

(
ζ(Ων,µ.k

p,q ψ(ζ))′

ψ(ζ)

) 1
λ

 = F (u(ς)), ς ∈ U (16)

and

1

2

ω(Ων,µ.k
p,q Ψ(ω))′

Ψ(ω)
+

(
ω(Ων,µ.k

p,q Ψ(ω))′

Ψ(ω)

) 1
λ

 = F (v(w)), w ∈ U. (17)

where u(ς) =
∞∑
j=1

ujς
j , and v(w) =

∞∑
j=1

vjw
j , ς, w ∈ U are Schwarz functions with the

property (See[25])
|uj | ≤ 1, and |vj | ≤ 1 (j ∈ N). (18)

By using few fundamental mathematical technics we can write equations (16) and (17) as

1

2

ζ(Ων,µ.k
p,q ψ(ζ))′

ψ(ζ)
+

(
ζ(Ων,µ.k

p,q ψ(ζ))′

ψ(ζ)

) 1
δ

 =

1 +

(
1 + λ

2λ

)
Md2ζ +

((
1 + λ

2λ

)
(Nd3 −Md22) +

(
1− λ

4λ2

)
M2d22

)
ζ2 + ..., (19)

F (u(ς)) = 1 + F2(κ, y)u1ς +
[
F2(κ, y)u2 + F3(κ, y)u21

]
ς2 + ..., (20)

and

1

2

ω(Ων,µ.k
p,q Ψ(ω))′

Ψ(ω)
+

(
ω(Ων,µ.k

p,q Ψ(ω))′

Ψ(ω)

) 1
δ

 =

1+

(
1 + λ

2λ

)
Md2ω+

((
1 + λ

2λ

)
(N (2d22 − d3)−Md22) +

(
1− λ

4δ2

)
M2d22

)
ω2+ ..., (21)

F (v(w)) = 1 + F2(κ, y)v1w +
[
F2(κ, y)v2 + F3(κ, y)v21

]
w2 + ... . (22)

where M and N are as mentioned in (14), and (15), respectively.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3808

Comparing the terms with the same degree in (19) and (20), we conclude due to
equality (16) (

1 + λ

2λ

)
Md2 = F2(κ, y)u1, (23)(

1 + λ

2λ

)
(Nd3 −Md22) +

(
1− λ

4λ2

)
M2d22 = F2(κ, y)u2 + F3(κ, y)m2

1. (24)

Similarly, due to equality (17), we draw our conclusion by comparing the terms of the
same degree in (21) and (22)

−
(
1 + λ

2λ

)
Md2 = F2(κ, y)v1, (25)

(
1 + λ

2λ

)
(N (2d22 − d3)−Md22) +

(
1− λ

4λ2

)
M2d22 = F2(κ, y)v2 + F3(κ, y)v21. (26)

From (23) and (25), we can easily obtain

u1 = −v1, (27)(
(1 + λ)2

2λ2

)
M2d22 = (u21 + v21)F2

2 (κ, y). (28)

When (24) and (26) are added, we get

2

[(
1 + λ

λ

)
(N −M) +

(
1− λ

2λ2

)
M2

]
d22 = F2(κ, y)(u2 + v2) + F3(κ, y)(u21 + v21). (29)

Substituting the value of u21 + v21 from (28) in (29), we get

d22 =
2λ2F3

2 (κ, y)(u2 + v2)[
(2λ(λ+ 1)(N −M) + (1− λ)M2)F2

2 (κ, y)− (1 + λ)2M2F3(κ, y)
] , (30)

which produces (10), when applied (18).
After deducting (26) from (24) and using (27), we arrive at

d3 = d22 +
λF2(κ, y)(u2 − v2)

(1 + λ)N
. (31)

This results in the inequality that follows:

|d3| ≤ |d2|2 +
|F2(κ, y)||u2 − v2|(

λ+1
λ

)
U [3]p,q

. (32)

From (10) and (32) we obtain (11), applying (18) for u2 and v2.
Clearly, for ξ ∈ R we get from (30) and (31) that,

|d3 − ξd22| = |F2(κ, y)|
∣∣∣∣(G(ξ, F ) + λ

(1 + λ)N

)
u2 +

(
G(ξ, F )− λ

(1 + λ)N

)
v2

∣∣∣∣ ,



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3809

where

G(ξ, F ) =
2λ2(1− ξ)F2

2 (κ, y)[
(2λ(λ+ 1)(N −M) + (1− λ)M2)F2

2 (κ, y)− (1 + λ)2M2F3(κ, y)
] .

Then, using (18) we can deduce that

|d3 − ξd22| ≤

{
2λ|F2(κ,y)|
(1+λ)N ; 0 ≤ |G(ξ, F )| ≤ λ

(1+λ)N
2|F2(κ, y)||G(ξ, F )| ; |G(ξ, F )| ≥ λ

(1+λ)N ,

which leads us to the conclusion (12), with J as in (13), considering F2(κ, y) = s,
F3(κ, y) = s2 + t. This completes the proof of Theorem 1.

If we take ξ = 1 in the part iii) of Theorem 1, we obtain the following result:

Corollary 1. Let 0 < λ ≤ 1, µ ≥ 0, ν a real number such that ν+µ > 0, k ∈ N andψ(ζ) =

ζ +
∞∑
j=2

djζ
j be in the class Eλ,k

Σ,p,q(F, ν, µ). Then |d3 − d22| ≤ 2λs
(1+λ)N .

Corollary 2. Let us assume that ν = 1 − µ in Theorem 1. Then the upper bounds of
|d2|, |d3|, and |d3 − ξd22|, ξ ∈R, for a function ψ ∈ Fλ,k

Σ,p,q(F, µ) are given by (10), (11), and

(12), respectively, with M = M1 = 2(1+µ([2]p,q−1)k−1), and N = N1 = 3(1+µ([3]p,q−
1)k − 1). For J in (13), M, andN are to be substituted with M1, andN1, respectively.

Corollary 3. Let us assume that ν = l + 1 − µ in Theorem 1. Then the upper bounds
of |d2|, |d3|, and |d3 − ξd22|, ξ ∈R, for a function ψ ∈ Gλ,k

Σ,p,q(F, l, µ) are given by (10), (11),

and (12), respectively, with M = M2 =

(
2
(
l+1+µ([2]p,q−1)

l+1

)k
− 1

)
, and N = N2 =(

3
(
l+1+µ([3]p,q−1)

l+1

)k
− 1

)
. For J in (13), M, andN are to be substituted with M2, andN2,

respectively.

If λ = 1 in Theorem 1, we get

Corollary 4. Let µ ≥ 0, ν a real number such that ν + µ > 0, and k ∈ N. If a function
ψ ∈ Hk

Σ,p,q(F, ν, µ), then

i). |d2| ≤
s
√
s√

|(N −M)s2 −M2(s2 + t)|
, ii). |d3| ≤

s2

M2
+

s

N

and for ξ ∈ R

iii). |d3 − ξd22| ≤


s
N ; |1− ξ| ≤

∣∣∣∣(N −M)s2 −M2(s2 + t)

N s2

∣∣∣∣
s3 |1−ξ|

|(N−M)s2−M2(s2+t)| ; |1− ξ| ≥
∣∣∣∣(N −M)s2 −M2(s2 + t)

N s2

∣∣∣∣ ,
where M, andN are given by (14) and (15), respectively.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3801-3814 3810

Remark 2. Taking k = 0 in Corollary 4, we get two results of Yilmaz and Aktaş[47,
Corollaries 2 and 6].

Corollary 5. Let us assume that q → 1− and p = 1 in Theorem 1. Then the upper
bounds of |d2|, |d3|, and |d3 − ξd22|, ξ ∈R, for any function ψ ∈ Yλ,k

Σ (F, ν, µ), are given by

(10), (11), and (12), respectively, with M = M3 =

(
2
(
ν+2µ
ν+µ

)k
− 1

)
, and N = N3 =(

3
(
ν+3µ
ν+µ

)k
− 1

)
. For J in (13), M, andN are to be substituted with M3, andN3,

respectively.

Remark 3. If k = 0 in the set Yλ,k
Σ (F, ν, µ), then we obtain a subset Qλ

Σ(F ), 0 < λ ≤ 1,
which is the collection of members of ψ ∈ Σ that satisfy

1

2

{
ζψ′(ζ)

ψ(ζ)
+

(
ζψ′(ζ)

ψ(ζ)

) 1
λ

}
≺ F (ζ) =

F(κ, y, ζ)
ζ

, ζ ∈ D,

and
1

2

{
ωΨ′(ω)

Ψ(ω)
+

(
ωΨ′(ω)

Ψ(ω)

) 1
λ

}
≺ F (ω) =

F(κ, y, ω)
ω

, ζ ∈ D.

Corollary 6. Let 0 < λ ≤ 1. If a function θ ∈ Qλ
Σ(F ), then

i). |d2| ≤
2λs

√
s√

|λ(λ− 1)s2 − (1 + λ)2t|
, ii). |d3| ≤

4λ2s2

(1 + λ)2
+

λs

1 + λ
,

and for ξ ∈ R

iii). |d3 − ξd22| ≤

{
λs
1+λ ; |1− ξ| ≤ |λ(λ−1)s2−(1+λ)2t)|

4λ(1+λ)s2

4λ2s3 |1−ξ|
|λ(λ−1)s2−(1+λ)2t| ; |1− ξ| ≥ |λ(λ−1)s2−(1+λ)2t|

4λ(1+λ)s2
.

Remark 4. We derive two results in[47, Corollaries 2 and 6] by taking λ = 1 in Corollary
6.

3. Conclusions

This study establishes upper bounds on |d2| and |d3| for functions in subfamily of σ
related to (m,n)−Lucas polynomials. Moreover, the Fekete-Szegö functional |d3−µd22|, µ ∈
R has been identified for functions in these subfamilies. Through adjusting the parameters
in Theorem 1, few implications have been brought to light. Relevant connections to the
current research are also discovered. Nevertheless, this paper does not address all of the
significant subclasses of Σ that exist in the literature. For example, authors[34, 36, 38] have
examined various subclasses involving (p, q)-operators introduced in (p, q)-calculus. It is
recommended that the interested reader review these papers and the associated references.



REFERENCES 3811

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