Bi-univalent Function Subfamilies Defined by q- Analogue of Bessel
Functions Subordinate to (p, q)- Lucas Polynomials
Sondekola Rudra Swamy1and Alina Alb Lupas2,∗
1Department of Computer Science and Engineering
RV College of Engineering
8th Mile, Mysuru Road, Bengaluru-560 059, INDIA
2Department of Mathematics and Computer Science
University of Oradea
1 Universitatii street, 410087, Oradea, ROMANIA
Abstract: - In the theory of bi-univalent functions,variety of special polynomials and special functions have been
used. Using the q- analogue of Bessel functions, two families of regular and bi-univalent functions subordinate to
(p, q)- Lucas polynomials are introduced in this paper. For elements in these defined families, we derive estimates
for |a2|,|a3|and for δa real number we consider Fekete-Szegö problem |a3−δa2
2|. We also provide relevent
connection to existing result and discuss few interesting observations of the results investigated.
Key-Words: - Fekete-Szegö inequality, Bessel function, Bi-univalent functions, q- derivative operator, (p,
q)-Lucas polynomials.
Received: May 18, 2021. Revised: January 21, 2022. Accepted: February 15, 2022. Published: March 14, 2022.
1 Introduction
Let Cbe the set of complex numbers and ∆ = {ς:
ς∈Cand |ς|<1}be the unit disk. Let Rand N:=
{1,2,3, ...}=N0\{0}be the sets of real numbers
and natural numbers, respectively. Let Adenote the
family of functions of the form
f(ς) = ς+
∞
X
n=2
anςn(1)
which are holomorphic in ∆.Further, we denote the
subfamily of Awhich are univalent in ∆by S.Ac-
cording to the well-known theorem of Koebe, every
function f∈ S contains a disk of radius 1
4.Thus, ev-
ery f∈ S has an inverse f−1satisfying
f−1(f(ς)) = ς(ς∈∆) and f(f−1(w)) = w
where
|w|< r0(f), r0(f)≧1
4
and is in fact given by
f−1(w) = w−a2w2+ (2a2
2−a3)w3−
(5a3
2−5a2a3+a4)w4+··· := g(w).(2)
If a function fand its inverse f−1are both univa-
lent in ∆, then a member fof Ais called bi-univalent
(or bi-schlicht) in ∆. The family of bi-univalent (or
bi-schlicht) functions in ∆given by (1) is indicated by
σ. The functions −log(1−ς),1
2log 1 + ς
1−ς,ς
1−ς
and so on are members of the class σ. Howewer, the
familiar Koebe function as well as ς−ς2
2,ς
1−ς2
(members of S) are not members the class σ.
Lewin [18] examined the family σand proved that
|a2|<1.51 for elements in the family σ. Later,
Brannan and Clunie[6] claimed that |a2| ≤ √2for
f∈σ. Subsequently, Tan [30] found the initial
coefficient bounds of bi-univalent functions. Bran-
nan and Taha [5] proposed bi-convex and bi-starlike
functions which are similar to well-known subfam-
ilies of S. The momentum on investigation of the
family σwas gained in recent years, which is due
to the paper of Srivastava et al.][23] and that has
led to a large number of papers in recent times.
Some interesting results concerning initial bounds for
certain special sets of σhave been examined in (
[2],[7],[10],[11],[16],[17],[21] and [24]) on subfam-
ilies of σ. They have obtained estimates on |a2|,
|a3|and |a3−δa2
2|,δ∈R, a2and a3being the
first two coefficients of Taylor-Maclaurin’s expan-
sion. However, the problem of finding the bounds on
|an|(n= 3,4,···) for members of σis still open.
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Let Γbe the Gamma function. The first kind
Bessel function of order νis defined by (see [20])
Jν(z) :=
∞
X
n=0
(−1)n(ς/2)2n+ν
n!Γ(n+ν+ 1) ,(ς∈C, ν ∈R).(3)
Recently, Szasz and Kupan [25] found the univa-
lence of the first kind Bessel function κν: ∆ →C
defined by
κν(z) := 2νΓ(ν+ 1)ς1−ν/2Jν(ς1/2)
=z+
∞
X
n=2
(−1)n−1Γ(ν+ 1)
4n−1(n−1)!Γ(n+ν)ςn,(4)
where ς∈∆, ν ∈R.
For κν, the q−derivative operator (0< q < 1) is
defined by
∂qκν(ς) := κν(qς)−κν(ς)
ς(q−1)
=∂q"ς+
∞
X
n=2
(−1)n−1Γ(ν+ 1)
4n−1(n−1)!Γ(n+ν)ςn#
= 1 +
∞
X
n=2
(−1)n−1Γ(ν+ 1)
4n−1(n−1)!Γ(n+ν)[n, q]ςn−1,(5)
where ς∈∆,
[n, q] := 1−qn
1−q= 1+
n−1
X
j=1
qj,[0, q] := 0.(6)
Using (6), we will define the following:
1. For any n∈N0,
[n, q]! := 1,if n= 0
[1, q][2, q] …[n, q] if n∈N.
(7)
is the q- shifted factorial.
2. For any n∈N0,
[r, q]n:= 1,if n= 0
[r, q][r+1, q] …[r+n-1, q] if n∈N.
(8)
is the q- generalized Pochhammer symbol.
For 0< q < 1, ν > 0and λ > −1, El-Deeb and
Bulboacǎ [9] defined the function Jλ
ν, q : ∆ →Cby
Jλ
ν, q(ς) := ς+(9)
∞
X
n=2
(−1)n−1Γ(ν+ 1)
4n−1(n−1)!Γ(n+ν)
[n, q]!
[λ+ 1, q]n−1
ςn, ς ∈∆.
A computation shows that
Jλ
ν, q(ς)∗ Mq, λ+1(ς) = z∂qκν(ς), ς ∈∆,(10)
where Mq, λ+1(ς)is given by
Mq, λ+1(ς) := ς+
∞
X
n=2
[λ+ 1, q]n−1
[n−1, q]! ςn, ς ∈∆.
(11)
Using the idea of convolutions and the definition of q-
derivative, El-Deeb and Bulboacǎ [9] examined the
operator Nλ
ν, q :A → A defined by
Nλ
ν, qf(ς) := Jλ
ν, q(ς)∗f(z)
=ς+
∞
X
n=2
ψnanςn,(12)
where 0< q < 1, ν > 0, λ > −1, ς ∈∆and
ψn:= (−1)n−1Γ(ν+ 1)
4n−1(n−1)!Γ(n+ν)
[n, q]!
[λ+ 1, q]n−1
(13)
Remark 1.1.One can verify from (12) that the follow-
ing identity hold for all f∈ A :
[λ+ 1, q]Nλ
ν, qf(ς) = [λ, q]Nλ+1
ν, q f(ς)+ (14)
qλς∂q[λ+ 1, q]Nλ+1
ν, q f(ς), ς ∈∆
and
lim
q→1−Nλ
ν, qf(ς) = Jλ
ν, 1f(ς) =: Jλ
νf(ς) = ς(15)
+
∞
X
n=2
(−1)n−1Γ(ν+ 1)
4n−1(n−1)!Γ(n+ν)
n!
(λ+ 1)n−1
an
n, z ∈∆.
The (p, q)-Lucas polynomials Ln(p(κ), q(κ),κ),
(or Ln(κ)) are given by the recurrence relation (see
[13, 14]):
Ln(κ) = p(κ)Ln−1(κ)+q(κ)Ln−2(κ)(n∈N\{1}),
(16)
with
L0(κ) = 2 and L1(κ) = p(κ),
where p(κ)and q(κ)be polynomials with real coeffi-
cients. One can find from (16) that L2(κ) = p2(κ) +
2q(κ),L3(κ) = p3(κ)+3p(κ)q(κ). Note that for
particular choices of p(κ)and q(κ), the (p, q)- Lucas
polynomials Ln(p(κ), q(κ),κ),leads to the follow-
ing polynomials:
1. Ln(2κ,1,κ) = Pn(κ)the Pell-Lucas polyno-
mials.
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2. Ln(κ,1,κ) = Ln(κ)the Lucas polynomials.
3. Ln(2κ,−1,κ) = Tn(κ)the first kind Cheby-
shev polynomials.
4. Ln(3κ,−2,κ) = Fn(κ)the Fermat-Lucas
polynomials.
5. Ln(1,2κ,κ) = Qn(κ)the Jacobsthal-Lucas
polynomials.
It is known from [19] that
GLn(κ)(ς) :=
∞
X
n=0 Ln(κ)zn=2−p(κ)ς
1−p(κ)ς−q(κ)ς2.
(17)
is the generating function of the (p, q)-Lucas polyno-
mials Ln(κ).
It is well-known that these polynomials have po-
tential applications in branches such as approximation
theory, architecture, engineering sciences , statistics,
mathematical and physical sciences. For more details
about the above mentioned polynomials one can refer
[13], [14],[15] and [19]. The recent research trends
on functions ∈σlinked with (p, q)- Lucas polyno-
mial can be seen in [1], [3], [4], [27], [28] and [29].
Motivated essentially by the fruitful usages of
above mentioned polynomials and Bassel functions
in Geometric function theory and the recent papers
[8], [12] and [26], we present two subfamilies of
bi-univalent functions defined by making use of q-
analogue of Bessel functions subordinate to (p, q)-
Lucas polynomials. Throughout this paper, the func-
tion f−1(w) = g(w)is as in (2) and the generating
function Gis as in (17).
The subordination principle for holomorphic func-
tions fand gin ∆, is due to Miller and Mocanu (see
[22]). fis said to be subordinate to g, if there exists
a Schwarz function ωsuch that f(ς) = g(ω(ς)) (ς∈
∆), ω(0) = 0 and |ω(ς)|<1.This subordination
will be indicated by f≺g(ς∈∆) (or f(ς)≺
g(ς)) (ς∈∆). Further, if gis univalent in ∆, f ≺
g(ς∈∆) ⇔f(0) = g(0) and f(∆) ⊂g(∆).
Definition 1.1.For τ≥1, µ ≥0,0≤γ≤1,0<
q < 1, ν > 0, λ > −1a function f∈σof the form
(1) is said to be in the class S∗
σ(τ, γ, µ, λ, ν, q, κ),
if
ςNλ
ν, qf(ς)′τ+µς2Nλ
ν, qf(ς)′′
(1 −γ)ς+γNλ
ν, qf(ς)≺
GLn(κ)(ς)−1
and
wNλ
ν, qg(w)′τ+µw2Nλ
ν, qg(w)′′
(1 −γ)w+γNλ
ν, qg(w)≺
GLn(κ)(w)−1,
where ς, w ∈∆.
Remark 1.2.Putting q→1−,we obtain
lim
q→1−S∗
σ(τ, γ, µ, λ, ν, q, κ) =: S∗
σ(τ, γ, µ, λ, ν, κ),
the class of f∈σsatisfying the following two con-
ditions
ςJλ
νf(ς)′τ+µς2Jλ
νf(ς)′′
(1 −γ)ς+γJλ
νf(ς)≺ GLn(κ)(ς)−1,
and
wJλ
νg(w)′τ+µw2Jλ
νg(w)′′
(1 −γ)w+γJλ
ν, g(w)≺
GLn(κ)(w)−1,
where
τ≥1, µ ≥0,0≤γ≤1, ν > 0, λ > −1, ς, w ∈∆.
The family S∗
σ(τ, γ, µ, λ, ν, q, κ),is of special
interest for it contains many new subfamilies of σfor
particular choices of γand µ, as illustrated below:
1. S∗
σ(τ, 0, µ, λ, ν, q, κ)≡J∗
∑(τ, µ, λ, ν, q, κ)is
the collection of functions f∈σsatisfying
Nλ
ν, qf(ς)′τ
+µς Nλ
ν, qf(ς)′′
≺ GLn(κ)(ς)−1
and
Nλ
ν, qg(w)′τ
+µw Nλ
ν, qg(w)′′
≺ GLn(κ)(w)−1,
where ς, w ∈∆.
2. S∗
σ(τ, 1, µ, λ, ν, q, κ)≡K∗
∑(τ, µ, λ, ν, q, κ)is
the set of functions f∈σsatisfying
ςNλ
ν, qf(ς)′τ
Nλ
ν, qf(ς)+µ ς2(Nλ
ν, qf(ς))′′
Nλ
ν, qf(ς)!≺
GLn(κ)(ς)−1
and
wNλ
ν, qg(w)′τ
Nλ
ν, qf(w)+µ w2(Nλ
ν, qg(w))′′
Nλ
ν, qg(w)!≺
GLn(κ)(w)−1,
where ς, w ∈∆.
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3. S∗
σ(τ, γ, 1, λ, ν, q, κ)≡L∗
∑(τ, γ, λ, ν, q, κ)is
the collection of functions ∈σsatisfying
ςNλ
ν, qf(ς)′τ+ς2Nλ
ν, qf(ς)′′
(1 −γ)ς+γNλ
ν, qf(ς)≺ GLn(κ)(ς)−1
and
wNλ
ν, qg(w)′τ+w2Nλ
ν, qg(w)′′
(1 −γ)w+γNλ
ν, qg(w)≺
GLn(κ)(w)−1,
where ς, w ∈∆.
Definition 1.2.For ξ≥1, τ ≥1,0< q < 1, ν >
0, and λ > −1a function f∈σof the form (1) is
said to be in the class M∗
σ(ξ, τ, λ, ν, q, κ),if
ξhς(Nλ
ν, qf(ς))′′iτ+ (1 −ξ)
Nλ
ν, qf(ς)′≺ GLn(κ)(ς)−1
and
ξhw(Nλ
ν, qg(w))′′iτ+ (1 −ξ)
Nλ
ν, qg(w)′≺ GLn(κ)(w)−1,
where ς, w ∈∆.
Remark 1.3.Putting q→1−,we obtain
lim
q→1−M∗
σ(ξ, τ, λ, ν, q, κ) =: M∗
σ(ξ, τ, λ, ν, κ),
the class of f∈σsatisfying the following two con-
ditions
ξhς(Jλ
νf(z))′′iτ+ (1 −ξ)
(Jλ
νf(ς))′≺ GLn(κ)(ς)−1
and
ξhw(Jλ
νg(w))′′iτ+ (1 −ξ)
(Jλ
νg(w))′≺ GLn(κ)(w)−1,
where ς, w ∈∆.
We note that M∗
σ(1, τ, λ, ν, q, κ)≡
U∗
σ(τ, λ, ν, q, κ)is the family investigated in [26].
Mσ(1, τ, λ, ν, q, κ)≡ T ∗
σ(τ, λ, ν, q, κ)is the
collection of functions f∈σsatisfying
hς(Nλ
ν, qf(ς))′′iτ
Nλ
ν, qf(ς)′≺ GLn(κ)(ς)−1, ς ∈∆
and
hw(Nλ
ν, qg(w))′′iτ
Nλ
ν, qg(w)′≺ GLn(κ)(w)−1, w ∈∆.
2 The set of main results
In this section, we propose to find bounds on |a2|,|a3|
and |a3−δa2
2|(δ∈R) for functions in the classes
S∗
σ(τ, γ, µ, λ, ν, q, x)and M∗
σ(ξ, τ, λ, ν, q, x),intro-
duced in Definition1.1 and Definition 1.2 , respec-
tively.
Theorem 2.1. Let τ≥1, µ ≥0,0≤γ≤1,0<
q < 1, ν > 0, λ > −1and f(ς) = ς+
∞
P
n=2
anςnbe
in the class S∗
σ(τ, γ, µ, λ, ν, q, κ).Then
|a2| ≤ |p(κ)|p|p(κ)|
p|tp2(κ)−2sq(κ)|,
|a3| ≤ |p(κ)|
(3(η+µ)−γ)ψ3
+p2(κ)
s
and for δ∈R
a3−δa2
2≤
|p(κ)|
(3(η+µ)−γ)ψ3
,if |δ−1| ≤ J
|p(κ)|3|δ−1|
|tp2(κ)−2sq(κ)|,if |δ−1| ≥ J
.
where
η=τ+µ, (18)
t= [(3(η+µ)−γ)ψ3−2(τ(τ+1)+(2µ−γ)η+2µτ )ψ2
2],
(19)
s= (2η−γ)2ψ2
2(20)
and
J=tp2(κ)−2sq(κ)
(3(η+µ)−γ)ψ3p2(κ).(21)
Proof. Let f∈ S∗
σ(τ, γ, µ, λ, ν, q, κ),be given
by (1). Then, for holomorphic functions uand vwith
u(0) = 0, v(0) = 0,|u(ς)|=u1ς+u2ς2+. . .<1,
and
|v(w)|=v1w+v2w2+. . .<1, ς, w ∈∆.
Therefore, on account of Definition 1.1, we can write
ςNλ
ν, qf(ς)′τ+µς2Nλ
ν, qf(ς)′′
(1 −γ)ς+γNλ
ν, qf(ς)=
GLn(κ)(u(ς)) −1
and
wNλ
ν, qg(w)′τ+µw2Nλ
ν, qg(w)′′
(1 −γ)w+γNλ
ν, qg(w)=
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GLn(κ)(v(w)) −1.
Or, equivalently,
ςNλ
ν, qf(ς)′τ+µς2Nλ
ν, qf(ς)′′
(1 −γ)ς+γNλ
ν, qf(ς)=
−1+L0(κ)+L1(κ)u(ς)+L2(κ)[u(ς)]2+. . .
and
wNλ
ν, qg(w)′τ+µw2Nλ
ν, qg(w)′′
(1 −γ)w+γNλ
ν, qg(w)=
−1 + L0(κ) + L1(κ)v(w) + L2(κ)[v(w)]2+. . . .
We obtain, from the above equalities
ςNλ
ν, qf(ς)′τ+µς2Nλ
ν, qf(ς)′′
(1 −γ)ς+γNλ
ν, qf(ς)=
1 + L1(κ)u1ς+ [L1(κ)u2+L2(κ)u2
1]ς2+. . . (22)
and
wJλ
ν, qg(w)′τ+µw2Jλ
ν, qg(w)′′
(1 −γ)w+γJλ
ν, qg(w)=
1+L1(κ)v1w+[L1(κ)v2+L2(κ)v2
1]w2+. . . . (23)
It is known that
|uk| ≤ 1,|vk| ≤ 1 (k∈N).(24)
Comparing (22) and (23), we have
(2η−γ)ψ2a2=L1(κ)u1(25)
(3(η+µ)−γ)ψ3a3+γ2+ 2τ(τ−1) −2γηψ2
2a2
2=
L1(κ)u2+L2(κ)u2
1(26)
−(2η−γ)ψ2a2=L1(κ)v1(27)
and
(3(η+µ)−γ)ψ32a2
2−a3+
γ2+ 2τ(τ−1) −2γηψ2
2a2
2=L1(κ)v2+L2(κ)v2
1,
(28)
where ηis as in (18).
From (25) and (27), we get
u1=−v1(29)
and
2sa2
2= [L1(κ)]2(u2
1+v2
1)(30)
where sis given by (20).
If we add (26) to (28), we obtain
2b a2
2=L1(κ)(u2+v2) + L2(κ)(u2
1+v2
1),(31)
where
b= [(3(η+µ)−γ)ψ3+γ2+ 2τ(τ−1) −2γηψ2
2].
(32)
From (30) and (31), we deduce that
2a2
2=[L1(κ)]3(u2+v2)
bL2
1(κ)−sL2(κ).(33)
Putting the values of L1(κ),L2(κ)and applying (24)
for |u2|and |v2|, we get
|a2| ≤ |p(κ)|p|p(κ)|
p|tp2(κ)−2sq(κ)|.
where tis as mentioned in (19).
To find the estimate on |a3|, first we subtract (28)
from (26) and then in view of (29), we obtain
2(3(η+µ)−γ)ψ3a3−2(3(η+µ)−γ)ψ3a2
2=
L1(κ) (u2−v2) + L2(κ)u2
1−v2
1
a3=L1(κ) (u2−v2)
2(3(η+µ)−γ)ψ3
+a2
2.(34)
Then in view of (30), (34) becomes
a3=L1(κ) (u2−v2)
2(3(η+µ)−γ)ψ3
+[L1(κ)]2(u2
1+v2
1)
2s.
Applying (24), we deduce that
|a3| ≤ |p(κ)|
(3(η+µ)−γ)ψ3
+p2(κ)
s.
From (34), for δ∈R,we write
a3−δa2
2=L1(κ) (u2−v2)
2(3(η+µ)−γ)ψ3
+ (1 −δ)a2
2.(35)
Substituting the value of a2
2from (33) in (35), we have
a3−δa2
2=
L1(x)Ω(δ, κ) + 1
βu2+Ω(δ, κ)−1
βv2,
(36)
where
β= 2(3(η+µ)−γ)ψ3
Ω(δ, κ) = (1 −δ) [L1(κ)]2
2(b[L1(κ)]2−sL2(κ))
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and bis given by (32). Then
a3−δa2
2≤
|L1(κ)|
(3(η+µ)−γ)ψ3
;
0≤ |Ω(δ, κ)| ≤ 1
β
2|L1(κ)||Ω(δ, κ)|;
|Ω(δ, κ)| ≥ 1
β,
which evidently completes the proof of Theorem 2.1.
In the next theorem, we determine the bounds
for |a2|,|a3|and |a3−δa2
2|for function f∈
M∗
σ(ξ, τ, λ, ν, q, x),the proof of which is omitted
as it is similar to that of Theorem 2.1.
Theorem 2.2. Let ξ≥1, τ ≥1,0< q < 1, λ >
−1, ν > 0and f(ς) = ς+
∞
P
n=2
anςnbe in the class
M∗
σ(ξ, τ, λ, ν, q, κ).Then
|a2| ≤ |p(x)|p|p(x)|
p|yp2(x)−2zq(x)|,
|a3| ≤ |p(x)|
3(3ξτ −1)ψ3
+p2(x)
z
and for δ∈R
a3−δa2
2≤
|p(x)|
3(3ξτ −1)ψ3
,if |δ−1| ≤ H
|p(x)|3|δ−1|
|yp2(x)−2zq(x)|,
if |δ−1| ≥ H
where
y= [(3(3ξτ −1)ψ3−8ξτ 2(2ξ−1)ψ2
2],
z= 4(2ξτ −1)2ψ2
2
and
H=yp2(x)−2zq(x)
3 (3ξτ −1) ψ3p2(x).
3 Outcome of main results
We arrive at the following outcome when γ= 0 in
Theorem 2.1.
Corollary 3.1.Let τ≥1, µ ≥0,0< q < 1, λ > −1,
ν > 0and f(ς) = ς+
∞
P
n=2
anςnbe in the family
J∗
∑(τ, µ, λ, ν, q, κ).Then
|a2| ≤ |p(κ)|p|p(κ)|
p|t1p2(κ)−2s1q(κ)|,
|a3| ≤ |p(κ)|
3(η+µ)ψ3
+p2(κ)
s1
and for δ∈R
a3−δa2
2≤
|p(κ)|
3(η+µ)ψ3
,if |δ−1| ≤ J1
|p(x)|3|δ−1|
|t1p2(κ)−2s1q(κ)|,
if |δ−1| ≥ J3
.
where ηis as in (18),
t1= [3(η+µ)ψ3−2(τ(τ+1)+2µ(η+τ))ψ2
2],
s1= 4η2ψ2
2
and
J1=t1p2(κ)−2s1q(κ)
3(η+µ)ψ3p2(κ)
We arrive at the following outcome by taking γ=
1in Theorem 2.1.
Corollary 3.2.Let τ≥1, µ ≥0,0< q < 1, λ >
−1, ν > 0and f(ς) = ς+
∞
P
n=2
anςnbe in the set
K∗
∑(τ, µ, λ, ν, q, κ).Then
|a2| ≤ |p(κ)|p|p(x)|
p|t2p2(κ)−2s2q(κ)|,
|a3| ≤ |p(κ)|
3(η+µ)−1)ψ3
+p2(κ)
s2
and for δ∈R
a3−δa2
2≤
|p(x)|
(3(η+µ)−1)ψ3
,if |δ−1| ≤ J2
|p(κ)|3|δ−1|
|t2p2(κ)−2s2q(κ)|,
if |δ−1| ≥ J2
.
where ηis as in (18),
t2= [(3(η+µ)−1)ψ3−2(τ(τ+1)+2µ(η+τ)−η)ψ2
2],
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DOI: 10.37394/23206.2022.21.15
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s2= (2η−1)2ψ2
2
and
J2=t2p2(κ)−2s2q(κ)
(3(η+µ)−1) ψ3p2(κ).
We arrive at the following outcome by taking µ=
1in Theorem 2.1.
Corollary 3.3.Let τ≥1,0≤γ≤1,0< q < 1,
λ > −1, ν > 0and f(ς) = ς+
∞
P
n=2
anςnbe in the
family L∗
∑(τ, γ, λ, ν, q, κ).Then
|a2| ≤ |p(κ)|p|p(κ)|
p|t3p2(κ)−2s3q(κ)|,
|a3| ≤ |p(κ)|
(3τ+ 6 −γ)ψ3
+p2(κ)
s3
and for δ∈R
a3−δa2
2≤
|p(κ)|
(3τ+ 6 −γ)ψ3
,if |δ−1| ≤ J3
|p(κ)|3|δ−1|
|t3p2(κ)−2s3q(κ)|,
if |δ−1| ≥ J3
.
where
t3= [(3τ+6−γ)ψ3−2(τ2+5τ+2−γ(τ+1))ψ2
2],
s3= (2τ+ 2 −γ)2ψ2
2
and
J3=t3p2(κ)−2s3q(κ)
(3τ+ 6 −γ)ψ3p2(κ).
Setting ξ= 1 in Theorem 2.2,we obtain
Corollary 3.4.Let τ≥1,0< q < 1, λ > −1,
ν > 0and f(ς) = ς+
∞
P
n=2
anςnbe in the set
T∗
σ(τ, λ, ν, q, κ).Then
|a2| ≤ |p(κ)|p|p(κ)|
p|y1p2(κ)−2z1q(κ)|,
|a3| ≤ |p(κ)|
3(3τ−1)ψ3
+p2(κ)
z1
and for δ∈R
a3−δa2
2≤
|p(κ)|
3(3τ−1)ψ3
,if |δ−1| ≤ H1
|p(κ)|3|δ−1|
|y1p2(κ)−2z1q(κ)|,
if |δ−1| ≥ H1
where
y1= [(3(3τ−1)ψ3−8τ2ψ2
2],
z1= 4(2τ−1)2ψ2
2
and
H1=y1p2(κ)−2z1q(κ)
3 (3τ−1) ψ3p2(κ).
4 Conclusions
Our investigation is motivated by the fruitful usage of
certain special polynomials and Bassel functions, in
the theory of bi-univalent functions. Making use of
the q-analoguue of Bassel functions, we have intro-
duced two subfamilies of bi-univalent (or bi-schlicht)
functions subordinate to (p, q)-Lucas polynomials.
For functions belonging to these subfamilies, we have
found the upper bounds of |a2|,|a3|and for δa real
number the Fekete- Szegö functional |a3−δa2
2|is
considered. The special cases and implications of the
main results have been identified. Finding estimate
on the bound of |an|, n ∈R− {1,2,3}is an open
problem.
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Sondekola Rudra Swamy, Alina Alb Lupas
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Volume 21, 2022
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Contribution of individual authors to
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(Ghostwriting Policy)
Author Contributions:
Sondekol Rudra Swamy did conceptualization, the
initial investigation of work and also prepared the
original draft. Alina Alb Lupas reviewed the work
after the initial draft to ascertain the correctness of
the work. Both the authors worked on the method-
ology and validation of the study. Both the authors
approved the final draft and agreed upon the submis-
sion for publication in SWEAS transactions on math-
ematics.
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