# On Fekete–Szegö Problems for Certain Subclasses of Analytic Functions Defined by Differential Operator Involvingq-Ruscheweyh Operator

Authors: Abdullah Alsoboh; Maslina Darus
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we define a new derivative operator involvingq-Ruscheweyh differential operator using convolution. Using this new operator, we introduce two new classes of analytic functions and obtain the Fekete–Szegö inequalities.


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## Body

## 1. Introduction

The applications ofq-calculus are important and pivotal as they contributed worthy of noticed expansion in geometric function theory. In 1908, Jackson, was the first mathematician to develop the application of q-calculus in a systematic way [1, 2]. Then, Aral and Gupta [3] proposed q-analogue of Baskakov and Durrmeyer operator depending on q-calculus; Mohammed and Darus [4] defined a new operator involving the q-hypergeometric function. Some other applications of q-calculus are studied by the authors [5, 6] and Elhaddad et al. [7]. Recently, many mathematicians have worked intensively in this field (see [8–12]) and obtained various results.LetA be the class of functions f
z of the form(1)
f
z
=
z
+
∑
i
=
2
∞
a
i
z
i
,
z
∈
E
,which are analytic inE
=
z
:
z
∈
C
,
|
z
|
<
1 and we denote by S the subclass of A that are consisted of one-to-one (univalent) functions in E.The convolution of functionsf as in (1.1) and Γ
z
=
z
+
∑
i
=
2
∞
γ
i
z
i will be:(2)
f
∗
Γ
z
=
Γ
∗
f
z
=
z
+
∑
i
=
2
∞
a
i
γ
i
z
i
.Letf
z and τ
z are analytic functions in E, then we say that f is subordinate to τ denoted by f
z
≺
τ
z in E, if there exists a Schwarz function ξ
z which is analytic in E with ξ
0
=
0 and |
ξ
z
|
<
1 (z
∈
E) such that f
z
=
τ
ξ
z
∀
z
∈
E.Now, we give some notations and definitions of the principal terms ofq-calculus by assuming 0
<
q
<
1, as follows:(1) Theq-number i
q is defined by:(3)
i
q
=
1
−
q
i
1
−
q
i
∈
C
\
N
,
1
+
q
+
q
2
+
⋯
+
q
i
−
1
i
∈
N
.(2) Theq-factorial i
! is defined by:(4)
i
q
!
=
i
q
i
−
1
q
…
2
q
1
q
,
i
=
2
,
3
,
4
,
…
1
,
i
=
1
.(3) Theq-derivative of a function f is defined by:(5)
∂
q
f
z
=
f
z
−
f
qz
z
−
q
z
,
z
∈
C
∗
,
f
′
0
,
z
=
0
,whereC
∗
=
C
\
0 will be used throughout the article.Asf given by (1), then(6)
∂
q
f
z
=
1
+
∑
i
=
2
∞
i
q
a
i
z
i
−
1
.The authors in [6] introduced a q-differential operator D
q
,
μ
,
δ
,
κ
,
λ
m
f
z by:(7)
D
q
,
μ
,
δ
,
κ
,
λ
m
f
z
=
z
+
∑
i
=
2
∞
k
−
λ
δ
−
μ
i
q
−
1
+
1
m
a
i
z
i
,whereδ
,
κ
,
λ
,
μ
≥
0, κ
>
λ
,
δ
>
μ, m
∈
N
0.In 2014, Aldweby and Darus in [9] introduced q-Ruscheweyh operator R
q
ϑ
f
z by:(8)
R
q
ϑ
f
z
=
z
+
∑
i
=
2
∞
ϑ
−
1
+
i
q
!
ϑ
q
!
i
−
1
q
!
a
i
z
i
,
ϑ
≥
0
.Using convolution onf
∈
A, we define new differential operator D
q
,
μ
,
δ
,
κ
,
λ
m
,
ϑ
:
A
→
A by:(9)
D
q
,
μ
,
δ
,
κ
,
λ
m
,
ϑ
f
z
=
D
q
,
μ
,
δ
,
κ
,
λ
m
f
z
∗
R
q
ϑ
f
z
=
z
+
∑
i
=
2
∞
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
i
q
a
i
z
i
,where(10)
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
i
q
=
k
−
λ
δ
−
μ
i
q
−
1
+
1
m
ϑ
−
1
+
i
q
!
ϑ
q
!
i
−
1
q
!
.Remarks.
(i)
Whenϑ
=
0, κ
=
1, λ
=
0, δ
=
1, and μ
=
0, we get q-Sãlãgean differential operator introduced in [13].
(ii)
Whenϑ
=
0, κ
=
1, λ
=
0, δ
=
1, μ
=
0, and q
→
1
−, we get Sãlãgean differential operator introduced in [14].
(iii)
Whenϑ
=
0, β
=
1, κ
=
0, δ
=
1, μ
=
0, and q
→
1
−, then we get Al-Oboudi differential operator introduced in [15].
(iv)
Whenϑ
=
0 and q
→
1
−, we get Ramadan and Darus operator introduced in [16].
(v)
Whenϑ
=
0, we get Alsoboh and Darus operator introduced in [6].
(vi)
Whenm
=
0, then we get q-Ruscheweyh operator introduced in [9].
(vii)
Whenm
=
0 and q
→
1
−, then we have Ruscheweyh operator introduced in [17].
(viii)
Whenm
=
0, δ
=
1, μ
=
0, λ
=
0, and q
→
1
−, we get D
λ
n
f
z [18].Many sublasses of analytic functions have been introduced by many different authors, for example, Ma and Minda [19], Ravichandran et al. [20], Seoudy and Aouf [12] and others. These works have inspired our introduction of the new subclasses S
q
,
ϱ
m
,
ϑ
Φ and C
q
,
ϱ
m
,
ϱ
Φ of A, involving the differential operator D
q
,
μ
,
δ
,
κ
,
λ
m
,
ϑ
f
z and the principle of subordination as:Definition 1.
Let℘ be the subclass of functions Φ which are analytic and univalent in E and for which Φ
E is convex with Φ
0
=
1 and R
Φ
z
>
0 for z
∈
E. A function f
∈
A is said to be in the class f
∈
S
q
,
ϱ
m
,
ϑ
Φ if it satisfies the subordination condition:(11)
1
+
1
ϱ
z
∂
q
D
q
m
,
ϑ
f
z
D
q
m
,
ϑ
f
z
−
1
≺
Φ
z
,
ϱ
∈
C
∗
;
Φ
∈
P
.Definition 2.
A functionf
∈
A is said to be in the class f
∈
C
q
,
ϱ
m
,
ϑ
Φ if it satisfies the subordination condition:(12)
1
+
1
ϱ
∂
q
z
∂
q
D
q
m
,
ϑ
f
z
∂
q
D
q
m
,
ϑ
f
z
−
1
≺
Φ
z
,
ϱ
∈
C
∗
;
Φ
∈
℘
.Note that:(i)
Whenm
=
0 and ϑ
=
0, then S
q
,
ϱ
m
,
ϑ
Φ
=
:
ℓ
q
,
ϱ
ϕ and C
q
,
ϱ
m
,
ϑ
Φ
=
:
C
q
,
ϱ
ϕ [12].
(ii)
Whenm
=
0, ϑ
=
0, and q
→
1
−, then S
q
,
ϱ
m
,
ϑ
Φ
=
:
ℓ
ϱ
ϕ and C
q
,
ϱ
m
,
ϑ
Φ
=
:
C
ϱ
ϕ [20].
(iii)
Whenm
=
0, ϑ
=
0, and q
→
1
−, then S
q
,
1
m
,
ϑ
Φ
=
:
S
∗
ϕ, and C
q
,
1
m
,
ϑ
Φ
=
:
C
ϕ [19].


## 2. Main Results

In this section, we obtain the Fekete-Szegö inequalities for the subclassesS
q
,
ϱ
m
,
ϑ
Φ and C
q
,
ϱ
m
,
ϑ
Φ, by assuming δ
,
κ
,
λ
,
μ
≥
0, κ
>
λ
,
δ
>
μ, m
∈
N
0, ϱ
∈
C
∗ and Φ
∈
℘. In order to prove our results, we use the following lemmas of Ma and Minda [19].Lemma 3.
IfP
z
=
1
+
c
1
z
+
c
2
z
2
+
c
3
z
3
+
… is a function with positive real part in the open unit disk E and σ
∈
C, then(13)
c
2
−
σ
c
1
2
≤
2
max
1
;
2
σ
−
1
.Lemma 4.
IfP
z
=
1
+
c
1
z
+
c
2
z
2
+
c
3
z
3
+
.
.
. is a function with positive real part in E and σ
∈
C, then(14)
c
2
−
σ
c
1
2
≤
−
4
σ
+
2
,
σ
≤
0
,
2
,
0
≤
σ
≤
1
,
4
σ
−
2
,
σ
≥
1
.To get our results, we use the similar methods studied by Alsoboh and Darus [5], Elhaddad and Darus [11], and Seoudy and Aouf [12].Theorem 5.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯ with B
1
≠
0, and f given by (1) belongs to S
q
,
ϱ
m
,
ϑ
Φ, then(15)
a
3
−
σ
a
2
2
≤
ϱ
B
1
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
max
1
;
B
2
B
1
+
ϱ
B
1
2
q
−
1
1
−
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
.Proof.
Letf
∈
S
q
,
n
∗
Φ, then there is a Schwarz function ξ which is analytic in E with ξ
0
=
0, and |
ξ
z
|
<
1 in E, such that(16)
z
∂
q
D
q
m
,
ϑ
f
z
D
q
m
,
ϑ
f
z
=
Φ
ξ
z
.
Now, we define the functionP
z with R
e
P
z
>
0 and P
0
=
1 by:(17)
P
z
=
1
+
ξ
z
1
−
ξ
z
=
1
+
p
1
z
+
p
2
z
2
+
p
3
z
3
+
⋯
.
Sinceξ
z is a Schwarz function, therefore,(18)
Φ
ξ
z
=
Φ
P
z
−
1
P
z
+
1
=
1
+
1
2
B
1
p
1
z
+
1
2
B
1
p
2
−
p
1
2
2
+
1
2
B
2
p
1
2
z
2
+
⋯
.
Now, substitute (18) in (16), we get(19)
1
+
1
ϱ
z
∂
q
D
q
m
,
ϑ
f
z
D
q
m
,
ϑ
f
z
−
1
=
1
+
1
2
B
1
p
1
z
+
1
2
B
1
p
2
−
p
1
2
2
+
1
4
B
2
p
1
2
z
2
+
⋯
.
From Equation (19), we get(20)
a
2
=
B
1
p
1
ϱ
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
,
and(21)
a
3
=
ϱ
B
1
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
p
2
−
1
2
1
−
B
2
B
1
−
B
1
ϱ
2
q
−
1
p
1
2
.
Therefore,(22)
a
3
−
σ
a
2
2
=
ϱ
B
1
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
p
2
−
ν
p
1
2
,
where(23)
ν
=
1
2
1
−
B
2
B
1
−
B
1
ϱ
2
q
−
1
1
−
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
,
by applying Lemma3, we get our result. This completes the proof.Next, we prove the following theorem for the subclassC
q
,
ϱ
m
,
ϑ
Φ.Theorem 6.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
.
.
. with B
1
≠
0, and f given by (1) belongs to C
q
,
ϱ
m
,
ϑ
Φ, then (24)
a
3
−
σ
a
2
2
≤
ϱ
B
1
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
·
max
1
;
B
2
B
1
+
ϱ
B
1
2
q
−
1
1
−
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
.Proof.
Letf
∈
C
q
,
n
∗
Φ, then there is a Schwarz function ξ
z, analytic in E with ξ
0
=
0, and  |
ξ
z
|
<
1 in E such that(25)
∂
q
z
∂
q
D
q
m
,
ϑ
f
z
∂
q
D
q
m
,
ϑ
f
z
=
Φ
ξ
z
.
From the equations (17), (18), and last equality, we have(26)
1
+
1
ϱ
∂
q
z
∂
q
D
q
m
,
ϑ
f
z
∂
q
D
q
m
,
ϑ
f
z
−
1
=
1
+
1
2
B
1
p
1
z
+
1
2
B
1
p
2
−
p
1
2
2
+
1
4
B
2
p
1
2
z
2
+
⋯
.
From Equation (26), we can find(27)
a
2
=
B
1
p
1
ϱ
2
2
q
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
and(28)
a
3
=
ϱ
B
1
2
3
q
3
q
−
1
Ω
k
,
λ
,
δ
,
μ
m
,
ϑ
3
q
p
2
−
1
2
1
−
B
2
B
1
−
B
1
ρ
2
q
−
1
p
1
2
.
Therefore,(29)
a
3
−
σ
a
2
2
=
ϱ
B
1
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
p
2
−
ν
1
p
1
2
,
where(30)
ν
1
=
1
2
1
−
B
2
B
1
−
B
1
ϱ
2
q
−
1
1
−
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
,
by applying Lemma3, we get our result and this completes the proof.Whenm
=
0
and
ϑ
=
0 in previous two theorems, we obtain the result for functions in the classes ℓ
q
,
ϱ
(
ϕ
) and C
q
,
ϱ
ϕ, respectively, which improved by Seoudy and Aouf [12]:Corollary 7.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯ with B
1
≠
0, and fgiven by (1) belong to ℓ
q
,
ϱ
(
ϕ
), then(31)
a
3
−
σ
a
2
2
≤
B
1
ϱ
3
q
−
1
m
a
x
1
;
B
2
B
1
+
ϱ
B
1
2
q
−
1
1
−
3
q
−
1
2
q
−
1
σ
.
The result is sharp.Corollary 8.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯with B
1
≠
0, and fgiven by (1) belong to ℓ
q
,
ϱ
(
ϕ
), then(32)
a
3
−
σ
a
2
2
≤
B
1
ϱ
3
q
3
q
−
1
m
a
x
1
;
B
2
B
1
+
ϱ
B
1
2
q
−
1
1
−
3
q
3
q
−
1
2
q
2
2
q
−
1
σ
.The result is sharp.Form
=
0
,
ϑ
=
0 and taking q
→
1
− in Theorem 5, we obtain the following corollary.Corollary 9. (see [20]).
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯with B
1
≠
0, and fgiven by (1)belong to ℓ
ϱ
ϕ, then(33)
a
3
−
σ
a
2
2
≤
B
1
ϱ
2
m
a
x
1
;
B
2
B
1
+
ϱ
B
1
1
−
2
σ
.
The result is sharp.Next, by using Lemma4, we obtain the following theorems:Theorem 10.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯ with B
1
>
0, B
2
≥
0and fgiven by (1) belong to S
q
,
ϱ
m
,
ϑ
Φ, then(34)
a
3
−
σ
a
2
2
≤
ϱ
B
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
+
ϱ
2
B
1
2
2
q
−
1
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
σ
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
,
σ
≤
χ
1
,
ϱ
B
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
,
χ
1
≤
σ
≤
χ
2
,
ϱ
2
B
1
2
2
q
−
1
σ
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
−
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
ϱ
B
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
,
σ
≤
χ
2
, where(35)
χ
1
=
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
−
1
2
B
1
+
B
2
+
2
q
−
1
ϱ
B
1
2
3
q
−
1
ϱ
B
1
2
,
(36)
χ
2
=
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
B
2
−
B
1
2
q
−
1
2
+
ϱ
B
1
2
2
q
−
1
ϱ
B
1
2
3
q
−
1
.Proof.
Applying Lemma4 to the Equations (22) and (23), we have three cases:
Case (1): Ifσ
≤
χ
1
(37)
a
3
−
σ
a
2
2
≤
ϱ
B
1
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
−
4
ν
≤
ϱ
B
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
B
2
B
1
+
B
1
ϱ
2
q
−
1
1
−
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
.
Therefore,(38)
a
3
−
σ
a
2
2
≤
ϱ
B
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
+
ϱ
2
B
1
2
2
q
−
1
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
σ
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
.
Case (2): Ifχ
1
≤
σ
≤
χ
2
(39)
a
3
−
σ
a
2
2
≤
ϱ
B
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
.
Case (3): Ifσ
≥
χ
2
(40)
a
3
−
σ
a
2
2
≤
ϱ
B
1
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
4
ν
−
2
≤
−
ϱ
B
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
B
2
B
1
+
B
1
ϱ
2
q
−
1
1
−
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
,
(41)
a
3
−
σ
a
2
2
≤
ϱ
2
B
1
2
2
q
−
1
σ
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
−
1
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
ϱ
B
2
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
.
This completes our proof.Theorem 11.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯ with B
1
>
0, B
2
≥
0 and f given by (1) belong to C
q
,
ϱ
m
,
ϑ
Φ, then(42)
a
3
−
σ
a
2
2
≤
ϱ
B
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
+
ϱ
2
B
1
2
2
q
−
1
2
2
q
−
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
σ
2
q
2
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
,
σ
≤
ψ
1
,
ϱ
B
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
,
ψ
1
≤
σ
≤
ψ
2
,
ϱ
2
B
1
2
2
q
−
1
2
3
q
−
1
σ
3
q
−
1
2
q
2
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
−
2
q
−
1
3
q
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
ϱ
B
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
,
σ
≥
ψ
2
,
where(43)
ψ
1
=
2
q
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
ϱ
B
1
2
+
B
2
−
B
1
2
q
−
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
ϱ
B
1
2
,
(44)
ψ
2
=
2
q
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
ϱ
B
1
2
+
B
2
+
B
1
2
q
−
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
ϱ
B
1
2
.Proof.
Applying Lemma4 to the Equations (29) and (30), then we have
Case (1): Ifσ
≤
ψ
1
(45)
a
3
−
σ
a
2
2
≤
ϱ
B
1
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
−
4
ν
,
≤
ϱ
B
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
B
2
B
1
+
B
1
ϱ
2
q
−
1
1
−
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
,
≤
ϱ
B
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
+
ϱ
2
B
1
2
2
q
−
1
2
2
q
−
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
σ
2
q
2
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
.
Case (2): Ifψ
1
≤
σ
≤
ψ
2
(46)
a
3
−
σ
a
2
2
≤
ϱ
B
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
.
Case (3): Ifσ
≥
ψ
2
(47)
a
3
−
σ
a
2
2
≤
ϱ
B
1
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
4
ν
−
2
,
a
3
−
σ
a
2
2
≤
−
ϱ
B
1
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
B
2
B
1
+
B
1
ϱ
2
q
−
1
1
−
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
2
q
2
2
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
σ
,
≤
ϱ
2
B
1
2
2
q
−
1
2
3
q
−
1
σ
3
q
−
1
2
q
2
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
2
q
2
−
2
q
−
1
3
q
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
−
ϱ
B
2
3
q
3
q
−
1
Ω
κ
,
λ
,
δ
,
μ
m
,
ϑ
3
q
.
This completes the proof.Whenm
=
0
,
ϑ
=
0 in Theorems 5 and 6, we obtain the Fekete-Szegö inequalities for the subclasses ℓ
q
,
ϱ
(
ϕ
) and C
q
,
ϱ
ϕ, respectively [12].Corollary 12.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
.
.
. with B
1
>
0, B
2
≥
0 and fgiven by (1) belong to ℓ
q
,
ϱ
(
ϕ
), with ϱ
>
0 then(48)
a
3
−
σ
a
2
2
≤
ϱ
B
2
3
q
−
1
+
ϱ
2
B
1
2
2
q
−
1
1
3
q
−
1
−
σ
2
q
−
1
i
f
σ
≤
χ
1
ϱ
B
1
3
q
−
1
i
f
χ
1
≤
σ
≤
χ
2
ϱ
2
B
1
2
2
q
−
1
σ
2
q
−
1
−
1
3
q
−
1
−
ϱ
B
2
3
q
−
1
i
f
σ
≥
χ
2
.
where(49)
χ
1
=
2
q
−
1
2
B
1
+
B
2
+
2
q
−
1
ϱ
B
1
2
3
q
−
1
ϱ
B
1
2
,
(50)
χ
2
=
B
2
−
B
1
2
q
−
1
2
+
ϱ
B
1
2
2
q
−
1
ϱ
B
1
2
3
q
−
1
.
The result is sharp.Corollary 13.
Letϕ
z
=
1
+
B
1
z
+
B
2
z
2
+
B
3
z
3
+
⋯ with B
1
>
0, B
2
≥
0 and fgiven by (1) belong to C
q
,
ϱ
ϕ, with ϱ
>
0 then(51)
a
3
−
σ
a
2
2
≤
ϱ
B
2
3
q
3
q
−
1
+
ϱ
2
B
1
2
3
q
2
q
−
1
3
q
−
1
−
ϱ
2
B
1
2
σ
2
q
2
2
q
−
1
2
i
f
σ
≤
ψ
1
ϱ
B
1
3
q
3
q
−
1
i
f
ψ
1
≤
σ
≤
ψ
2
ϱ
2
B
1
2
σ
2
q
2
2
q
−
1
2
−
ϱ
B
2
3
q
3
q
−
1
−
ϱ
2
B
1
2
3
q
2
q
−
1
3
q
−
1
i
f
σ
≥
ψ
2
,
where(52)
ψ
1
=
2
q
2
2
q
−
1
ϱ
B
1
2
+
B
2
−
B
1
2
q
−
1
3
q
3
q
−
1
ϱ
B
1
2
,
(53)
ψ
2
=
2
q
2
2
q
−
1
ϱ
B
1
2
+
B
2
+
B
1
2
q
−
1
3
q
3
q
−
1
ϱ
B
1
2
.
The result is sharp.


## 3. Conclusion

Fekete-Szegö problems have always been the main interests of many researchers in geometric function theory. Many studies related to Fekete-Szegö revolved around classes of analytic normalised univalent functions. In this particular work, the Fekete-Szegö inequality is obtained for functions in more general classes denoted byS
q
,
ϱ
m
,
ϑ
Φ and C
q
,
ϱ
m
,
ϑ
Φ, respectively, using a new differential operator associated with q-calculus.



---

*Source: 8459405-2020-01-08.xml*