# A New Class of Function with Finitely Many Fixed Points

Authors: Matthew O. Oluwayemi; Olubunmi A. Fadipe-Joseph
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

The study introduced a generalized multiplier operator used as a tool to define and investigate a new class of function,TSQσ,ξ,ω,εμ,λ,ηϖ,n and its subclass TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi. Various properties of the class of functions were investigated. The results extend some known results in literature.


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## Body

## 1. Introduction and Preliminaries

Let(1)fz=z+∑k=2∞akzk,which are analytic in the open unit disk U=z∈ℂ:z<1 and normalized by f0=0 and f′0=1 be denoted by S.We denote byT the subclass of S which are normalized univalent function of the form
(2)fz=z−∑k=2∞akzk,ak>0,in the open unit disk U. The function of the form (2) was first introduced in [1].


### 1.1. Differential Operators (Multiplier Transformations)

In 1975, Ruscheweyh with the use of convolution introduced a differential operatorDn as a tool to investigate a certain class of univalent functions. The continuous extension and generalization of the tool have since then continue to gain attention among the function theorists. The differential operator and some of such extensions are discussed as follows.


#### 1.1.1. Ruscheweyh Differential Operator (1975)

Letn∈ℕ0=0,1,2,⋯. The Ruscheweyh derivative of nthorder denoted by Rnfz is defined by
(3)Rnfz=z+∑k=2∞Bknakzk,where
(4)Bk=Bn,k=n+k−1n=n+1n+2⋯n+k−1k−1!=n+1n+2⋯n+k−1k−1!=n+1k−11k−1.Hence,(5)B0,k=k−10=n+1k−11k−1=1.So thatR0fz is of the form (1).


#### 1.1.2. Salagean Differential Operator (1983)

Salagean defined an operatorDn as follows.Letfz∈A, the differential operator Dnfz is defined as
(6)Dnfz=z+∑k=2∞knakzk,n∈ℕ0.Since the introduction of differential operators in geometric function theory by Ruscheweyh in [2] and Salagean in [3], several attempts have been made to extend and generalize the operators by various authors and researchers in the field. According to Aldawish et al. in [4], the differential operator forms the link between function theory and mathematical physics. Perhaps, this is the reason why the study of differential operators is growing interest in Geometric Functions Theory and Application (GFTA). Some of the numerous differential operators (multiplier transformations) established by some authors are found in [5–11].The Salagean differential operator (6) defined in [3] appears to be the most famous and most referred to in general.


### 1.2. A Generalized Multiplier Transformation

In this work, a generalized operator is defined and used to investigate the classTSQσ,ξ,ω,εμ,λ,ηϖ,n and its subclass TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi with their geometric properties. The results improve some existing results in literature as pointed out in the work. The work was motivated by [12].Letμ,λ,η,σ,ξ,ε≥0 with ω>0 such that ξ<ω and n∈ℕ0=ℕ∪0. For f∈S, we define the following differential operator as follows:
(7)Dμ,λ,η0σ,ξ,ω,εfz=fz,(8)Dμ,λ,η1σ,ξ,ω,εfz=1−λω−ξσμ+ωfz+λω−ξσμ+ωzf′z+η+εμ+ωz2f′′z,(9)1−λω−ξσμ+ωfz=1−λω−ξσμ+ηz+∑k=2∞1−λω−ξσμ+ωakzk,(10)λω−ξσμ+ωzf′z=λω−ξσμ+ωz+∑k=2∞kλω−ξσμ+ωakzk,(11)η+εμ+ωz2f′′z=∑k=2∞kk−1δ+εμ+ωakzk.Substituting (9), (10), and (11) in (8), we have
(12)Dμ,λ,η1σ,ξ,ω,εfz=z+∑k=2∞1−λω−ξσμ+ω+kλω−ξσμ+ω+kk−1η+εμ+ωakzk,Dμ,λ,η1σ,ξ,ω,εfz=z+∑k=2∞1+k−1λω−ξσ+kη+εμ+ωakzk.Similarly,(13)Dμ,λ,ηnσ,ξ,ω,εfz=zDμ,λ,ηn−1σ,ξ,ω,εfz.Thus,(14)Dμ,λ,ηnσ,ξ,ω,εfz=z+∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnakzk.For convenience, forf∈A, we let
(15)Dμ,λ,ηnσ,ξ,ω,εfz=z+∑k=2∞Φn,kakzk,where Φn,k=1+k−1λω−ξσ+kη+ε/μ+ωn.Similarly, iff∈T as defined in (2), then
(16)Dμ,λ,ηnσ,ξ,ω,εfz=z−∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnakzk,(17)Dμ,λ,ηnσ,ξ,ω,εfz=z−∑k=2∞Φn,kakzk,where Φn,k=1+k−1λω−ξσ+kη+ε/μ+ωn.The operator defined in (14) reduces to (1) when n=0. Furthermore, it reduces to some existing operators in literature. Setting some parameters as follows:
(i)
Forλ=ω=1 and σ=μ=ε=η=0, we obtain the Salagean operator in [3](ii)
Whenλ=1, μ=1, σ=0, ω=1, ε=0, and η=0, we get the Uralegaddi and Somanatha differential operator in [13](iii)
The differential operator reduces to the Cho and Kim operator in [10] with λ=1, μ=1, σ=0, ε=0, and η=0(iv)
Whenμ=0, ω=1, σ=0, ε=0, and η=0, we get the Al-Oboudi differential operator in [14](v)
Whenμ=1, σ=0, ε=0, and η=0, we get multiplier transformation in [9](vi)
Supposeμ=0, ω=1, σ=0, ε=0, η=0, and λ=α+δ/γ+ρ for α,δ,γ,ρ≥0 and provided that γ+ρ≠0, then (14) reduces to the Alamri and Darus operator in [5](vii)
Whenσ=1 and ε=0, we get the Amourah and Darus differential operator in [7](viii)
Letλ=β−ξ, σ=1, ε=0, and η=0, we have the Amourah and Yousef differential operator in [8] provided β>ξIn [15–17], the authors defined and studied different classes of univalent functions. Motivated by the work of Varma and Rosy in [12], we define the following class of univalent functions.Definition 1.
Letϖ≥0, a function f defined by (2) belongs to the class SQσ,ξ,ω,εμ,λ,ηϖ,n if it satisfies the following condition:
(18)ReDμ,λ,ηnσ,ξ,ω,εfzz≥ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz,ϖ≥0.Investigation of univalent function is based on various classes and their geometric properties some of which are given in the following theorems and definitions.LetT∩SQσ,ξ,ω,εμ,λ,ηϖ,n=TSQσ,ξ,ω,εμ,λ,ηϖ,n. The necessary and sufficient condition for the functions in S to be in TSQσ,ξ,ω,εμ,λ,ηϖ,n is investigated.Remark 2.
LetTSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi⊂SQσ,ξ,ω,εμ,λ,ηϖ,n. Then,
(i)
SQσ,ξ,ω,εμ,λ,ηϖ,0≡SDα and TSQσ,ξ,ω,εμ,λ,ηϖ,0;Mi≡TSDα studied by both Varma and Rosy and [12, 18]. The class of functions investigated in the work generalized some known classes of functions in literature(ii)
Similar classes of functions were also investigated in [19–21](iii)
Different functions with finitely many fixed points were also considered in [22–24]


## 1.1. Differential Operators (Multiplier Transformations)

In 1975, Ruscheweyh with the use of convolution introduced a differential operatorDn as a tool to investigate a certain class of univalent functions. The continuous extension and generalization of the tool have since then continue to gain attention among the function theorists. The differential operator and some of such extensions are discussed as follows.


### 1.1.1. Ruscheweyh Differential Operator (1975)

Letn∈ℕ0=0,1,2,⋯. The Ruscheweyh derivative of nthorder denoted by Rnfz is defined by
(3)Rnfz=z+∑k=2∞Bknakzk,where
(4)Bk=Bn,k=n+k−1n=n+1n+2⋯n+k−1k−1!=n+1n+2⋯n+k−1k−1!=n+1k−11k−1.Hence,(5)B0,k=k−10=n+1k−11k−1=1.So thatR0fz is of the form (1).


### 1.1.2. Salagean Differential Operator (1983)

Salagean defined an operatorDn as follows.Letfz∈A, the differential operator Dnfz is defined as
(6)Dnfz=z+∑k=2∞knakzk,n∈ℕ0.Since the introduction of differential operators in geometric function theory by Ruscheweyh in [2] and Salagean in [3], several attempts have been made to extend and generalize the operators by various authors and researchers in the field. According to Aldawish et al. in [4], the differential operator forms the link between function theory and mathematical physics. Perhaps, this is the reason why the study of differential operators is growing interest in Geometric Functions Theory and Application (GFTA). Some of the numerous differential operators (multiplier transformations) established by some authors are found in [5–11].The Salagean differential operator (6) defined in [3] appears to be the most famous and most referred to in general.


## 1.1.1. Ruscheweyh Differential Operator (1975)

Letn∈ℕ0=0,1,2,⋯. The Ruscheweyh derivative of nthorder denoted by Rnfz is defined by
(3)Rnfz=z+∑k=2∞Bknakzk,where
(4)Bk=Bn,k=n+k−1n=n+1n+2⋯n+k−1k−1!=n+1n+2⋯n+k−1k−1!=n+1k−11k−1.Hence,(5)B0,k=k−10=n+1k−11k−1=1.So thatR0fz is of the form (1).


## 1.1.2. Salagean Differential Operator (1983)

Salagean defined an operatorDn as follows.Letfz∈A, the differential operator Dnfz is defined as
(6)Dnfz=z+∑k=2∞knakzk,n∈ℕ0.Since the introduction of differential operators in geometric function theory by Ruscheweyh in [2] and Salagean in [3], several attempts have been made to extend and generalize the operators by various authors and researchers in the field. According to Aldawish et al. in [4], the differential operator forms the link between function theory and mathematical physics. Perhaps, this is the reason why the study of differential operators is growing interest in Geometric Functions Theory and Application (GFTA). Some of the numerous differential operators (multiplier transformations) established by some authors are found in [5–11].The Salagean differential operator (6) defined in [3] appears to be the most famous and most referred to in general.


## 1.2. A Generalized Multiplier Transformation

In this work, a generalized operator is defined and used to investigate the classTSQσ,ξ,ω,εμ,λ,ηϖ,n and its subclass TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi with their geometric properties. The results improve some existing results in literature as pointed out in the work. The work was motivated by [12].Letμ,λ,η,σ,ξ,ε≥0 with ω>0 such that ξ<ω and n∈ℕ0=ℕ∪0. For f∈S, we define the following differential operator as follows:
(7)Dμ,λ,η0σ,ξ,ω,εfz=fz,(8)Dμ,λ,η1σ,ξ,ω,εfz=1−λω−ξσμ+ωfz+λω−ξσμ+ωzf′z+η+εμ+ωz2f′′z,(9)1−λω−ξσμ+ωfz=1−λω−ξσμ+ηz+∑k=2∞1−λω−ξσμ+ωakzk,(10)λω−ξσμ+ωzf′z=λω−ξσμ+ωz+∑k=2∞kλω−ξσμ+ωakzk,(11)η+εμ+ωz2f′′z=∑k=2∞kk−1δ+εμ+ωakzk.Substituting (9), (10), and (11) in (8), we have
(12)Dμ,λ,η1σ,ξ,ω,εfz=z+∑k=2∞1−λω−ξσμ+ω+kλω−ξσμ+ω+kk−1η+εμ+ωakzk,Dμ,λ,η1σ,ξ,ω,εfz=z+∑k=2∞1+k−1λω−ξσ+kη+εμ+ωakzk.Similarly,(13)Dμ,λ,ηnσ,ξ,ω,εfz=zDμ,λ,ηn−1σ,ξ,ω,εfz.Thus,(14)Dμ,λ,ηnσ,ξ,ω,εfz=z+∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnakzk.For convenience, forf∈A, we let
(15)Dμ,λ,ηnσ,ξ,ω,εfz=z+∑k=2∞Φn,kakzk,where Φn,k=1+k−1λω−ξσ+kη+ε/μ+ωn.Similarly, iff∈T as defined in (2), then
(16)Dμ,λ,ηnσ,ξ,ω,εfz=z−∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnakzk,(17)Dμ,λ,ηnσ,ξ,ω,εfz=z−∑k=2∞Φn,kakzk,where Φn,k=1+k−1λω−ξσ+kη+ε/μ+ωn.The operator defined in (14) reduces to (1) when n=0. Furthermore, it reduces to some existing operators in literature. Setting some parameters as follows:
(i)
Forλ=ω=1 and σ=μ=ε=η=0, we obtain the Salagean operator in [3](ii)
Whenλ=1, μ=1, σ=0, ω=1, ε=0, and η=0, we get the Uralegaddi and Somanatha differential operator in [13](iii)
The differential operator reduces to the Cho and Kim operator in [10] with λ=1, μ=1, σ=0, ε=0, and η=0(iv)
Whenμ=0, ω=1, σ=0, ε=0, and η=0, we get the Al-Oboudi differential operator in [14](v)
Whenμ=1, σ=0, ε=0, and η=0, we get multiplier transformation in [9](vi)
Supposeμ=0, ω=1, σ=0, ε=0, η=0, and λ=α+δ/γ+ρ for α,δ,γ,ρ≥0 and provided that γ+ρ≠0, then (14) reduces to the Alamri and Darus operator in [5](vii)
Whenσ=1 and ε=0, we get the Amourah and Darus differential operator in [7](viii)
Letλ=β−ξ, σ=1, ε=0, and η=0, we have the Amourah and Yousef differential operator in [8] provided β>ξIn [15–17], the authors defined and studied different classes of univalent functions. Motivated by the work of Varma and Rosy in [12], we define the following class of univalent functions.Definition 1.
Letϖ≥0, a function f defined by (2) belongs to the class SQσ,ξ,ω,εμ,λ,ηϖ,n if it satisfies the following condition:
(18)ReDμ,λ,ηnσ,ξ,ω,εfzz≥ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz,ϖ≥0.Investigation of univalent function is based on various classes and their geometric properties some of which are given in the following theorems and definitions.LetT∩SQσ,ξ,ω,εμ,λ,ηϖ,n=TSQσ,ξ,ω,εμ,λ,ηϖ,n. The necessary and sufficient condition for the functions in S to be in TSQσ,ξ,ω,εμ,λ,ηϖ,n is investigated.Remark 2.
LetTSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi⊂SQσ,ξ,ω,εμ,λ,ηϖ,n. Then,
(i)
SQσ,ξ,ω,εμ,λ,ηϖ,0≡SDα and TSQσ,ξ,ω,εμ,λ,ηϖ,0;Mi≡TSDα studied by both Varma and Rosy and [12, 18]. The class of functions investigated in the work generalized some known classes of functions in literature(ii)
Similar classes of functions were also investigated in [19–21](iii)
Different functions with finitely many fixed points were also considered in [22–24]


## 2. Main Results

Theorem 3.
A functionfz defined by (16) is said to be in the class TSQσ,ξ,ω,εμ,λ,ηϖ,n if and only if
(19)∑k=2∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak≤1.Proof.
Supposefz defined by (16) satisfies (19). Then,
(20)ReDμ,λ,ηnσ,ξ,ω,εfzz−ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz=1−1−∑k=2∞Φn,kakzk−1−−ϖ1−∑k=2∞kΦn,kakzk−1−1−∑k=2∞Φn,kakzk−1=1−∑k=2∞Φn,kakzk−1−ϖ1−∑k=2∞kΦn,kakzk−1−1+∑k=2∞Φn,kakzk−1=1−∑k=2∞Φn,kakzk−1−ϖ∑k=2∞k−1Φn,kakzk−1,z=r<1⇒≤1−∑k=2∞Φn,kak−ϖ∑k=2∞k−1Φn,kak=1−∑k=2∞Φn,k1+ϖk−1ak≥0.☐Hence,f∈TSQσ,ξ,ω,εμ,λ,ηϖ,Mj.Conversely,(21)ReDμ,λ,ηnσ,ξ,ω,εfzz−ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz>0,which implies that Re1−∑k=2∞Φn,kak−ϖ∑k=2∞k−1Φn,kak>0.Lettingz take real values and as z⟶1, we get
(22)1−∑k=2∞Φn,kak−ϖ∑k=2∞Φn,kk−1ak≥0,which implies ∑k=2∞Φn,k1+ϖk−1ak≤1 following (17).Remark 4.
For any functionf∈SQσ,ξ,ω,εμ,λ,ηϖ,n,
(23)ak≤11+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1,k≥2.
The equality holds for(24)fz=z−∑k=2∞11+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk.
The class is the generalization of the classTSDα investigated in [12, 18, 25].Corollary 5.
A functionfz defined by (16) belongs to the class f∈SQσ,ξ,ω,εμ,λ,ηϖ,0 if
(25)∑k=2∞1+ϖk−1ak≤1,ϖ≥0.
The classSQσ,ξ,ω,εμ,λ,ηϖ,0≡SDα was investigated in [12, 18, 25].Remark 6.
Letn=0. Then, f∈SQσ,ξ,ω,εμ,λ,ηϖ,0 with
(26)ak≤11+ϖk−1,k≥2.
The equality holds for(27)fz=z−∑k=2∞11+ϖk−1zk.
See [12, 18, 25] and the articles cited therein for details.


### 2.1. Subclass of Analytic Functions with Finitely Many Fixed Coefficients

We now introduce the class ofTSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi as a subclass of TSQσ,ξ,ω,εμ,λ,ηϖ,n with
(28)fz=z−∑i=2tMi1+i−1λω−ξσ+iη+ε/μ+ωn1+ϖi−1zi−∑k=t+1∞akzk.In recent times, the authors in [12, 19–24, 26] have also investigated classes of univalent functions with finitely many fixed coefficients.Theorem 7.
A functionfz defined by (28) is said to be in the class TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi if
(29)∑k=t+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak≤1−∑i=2tMi,where ϖ≥0, 0≤Mi≤1, and 0≤∑i=2tMi≤1.Proof.
Supposef∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi. Then, from (23),
(30)ai≤Mi1+i−1λω−ξσ+jη+ε/μ+ωn1+ϖi−1,i=2,3,⋯,t,0≤Mi≤1,0≤∑i=2tMi≤1,which implies ∑i=2tMi+∑k=j+1∞1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1ak≤1.
Conversely,(31)ReDμ,λ,ηnσ,ξ,ω,εfzz−ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz≥1−Dμ,λ,ηnσ,ξ,ω,εfzz−1−ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz=1−∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnak−ϖ∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnk−1ak=1−∑i=2t1+i−1λω−ξσ+kη+εμ+ωn1+ϖi−1aj−∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak=1−∑i=2tMi−∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak≥0,by (29) and (30). Thus, f∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi.
The result is sharp for(32)fz=z−∑i=2tMi1+i−1λω−ξσ+jη+ε/μ+ωn1+ϖi−1zi−1−∑i=2tMj1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk.☐Corollary 8.
LetTSQσ,ξ,ω,εμ,0,ηϖ,n;Mi. Then,
(33)ak≤1−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1,k≥t+1.
The result is sharp forfz given by (32).Corollary 9.
ForTSQσ,ξ,ω,εμ,λ,ηϖ,0;Mi,
(34)ak≤1−∑i=2tMi1+ϖk−1,k≥t+1.
The result is sharp forfz given by
(35)fz=z−∑i=2tMi1+ϖi−1zi−1−∑i=2tMi1+ϖk−1zk,k≥1.
The classTSQσ,ξ,ω,εμ,0,ηϖ,0;Mi≡TSDα was investigated in [12].Theorem 10.
The classTSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi is convex.Proof.
Suppose there exist functionsf,g∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi such that
(36)fz=z−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi−∑i=t+1∞akzk,(37)gz=z−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi−∑i=t+1∞bkzk,where 0≤Mi≤1 and 0≤∑i=2tMi≤1. Then, let 0≤ζ≤1 and
(38)hz=ζfz+1−ζgz.
So that by substituting (36) and (37) in (38), we have
(39)hz=z−∑i=2∞Mi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi−∑i=t+1∞ζak+1−ζbkzk.
Such that ifζ=1 and ζ=0, equation (39) reduces to (36) and (37), respectively. Now,
(40)∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ζak+1−ζbk=ζ∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak+1−ζ∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1bk≤ζ1−∑i=2tMi+1−ζ1−∑i=2tMi=1−∑i=2tMi.
Thus,hz∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi.☐Theorem 11.
Let(41)ftz=z−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi,fkz=z−∑i=2tMi1+i−1λω−ξσ+iη+ε/μ+ωn1+ϖi−1zi−∑k=t+1∞1−∑k=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk,k≥i+1.
Then,f∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi if and only if f can be expressed in the form
(42)fz=∑k=t∞ζkfkz,where ζk≥0 with k≥t and ∑k=t∞ζkfkz=1.Proof.
Iff∈T can be expressed in form of (42). Then,
(43)fz=z−∑i=2tMi1+i−1λω−ξσ+iη+ε/μ+ωn1+ϖi−1zi−∑k=t+1∞ζk1−∑k=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk.
Now,(44)∑k=t+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ζk1−∑k=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1=∑k=t+1∞ζk1−∑k=2tMi=1−∑k=2tMi∑k=t+1∞ζk=1−∑k=2tMi1−ζk≤1−∑k=2tMi,which implies f∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi.
Conversely, fork≥t+1, set
(45)ζk=1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1ak1−∑i=2∞Mi,k≥t+1,(46)ζt=1−∑k=t+1∞ζk.
Then,f can be represented as fz=∑k=t∞ζkfkz.☐


## 2.1. Subclass of Analytic Functions with Finitely Many Fixed Coefficients

We now introduce the class ofTSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi as a subclass of TSQσ,ξ,ω,εμ,λ,ηϖ,n with
(28)fz=z−∑i=2tMi1+i−1λω−ξσ+iη+ε/μ+ωn1+ϖi−1zi−∑k=t+1∞akzk.In recent times, the authors in [12, 19–24, 26] have also investigated classes of univalent functions with finitely many fixed coefficients.Theorem 7.
A functionfz defined by (28) is said to be in the class TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi if
(29)∑k=t+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak≤1−∑i=2tMi,where ϖ≥0, 0≤Mi≤1, and 0≤∑i=2tMi≤1.Proof.
Supposef∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi. Then, from (23),
(30)ai≤Mi1+i−1λω−ξσ+jη+ε/μ+ωn1+ϖi−1,i=2,3,⋯,t,0≤Mi≤1,0≤∑i=2tMi≤1,which implies ∑i=2tMi+∑k=j+1∞1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1ak≤1.
Conversely,(31)ReDμ,λ,ηnσ,ξ,ω,εfzz−ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz≥1−Dμ,λ,ηnσ,ξ,ω,εfzz−1−ϖDμ,λ,ηnσ,ξ,ω,εfz′−Dμ,λ,ηnσ,ξ,ω,εfzz=1−∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnak−ϖ∑k=2∞1+k−1λω−ξσ+kη+εμ+ωnk−1ak=1−∑i=2t1+i−1λω−ξσ+kη+εμ+ωn1+ϖi−1aj−∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak=1−∑i=2tMi−∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak≥0,by (29) and (30). Thus, f∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi.
The result is sharp for(32)fz=z−∑i=2tMi1+i−1λω−ξσ+jη+ε/μ+ωn1+ϖi−1zi−1−∑i=2tMj1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk.☐Corollary 8.
LetTSQσ,ξ,ω,εμ,0,ηϖ,n;Mi. Then,
(33)ak≤1−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1,k≥t+1.
The result is sharp forfz given by (32).Corollary 9.
ForTSQσ,ξ,ω,εμ,λ,ηϖ,0;Mi,
(34)ak≤1−∑i=2tMi1+ϖk−1,k≥t+1.
The result is sharp forfz given by
(35)fz=z−∑i=2tMi1+ϖi−1zi−1−∑i=2tMi1+ϖk−1zk,k≥1.
The classTSQσ,ξ,ω,εμ,0,ηϖ,0;Mi≡TSDα was investigated in [12].Theorem 10.
The classTSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi is convex.Proof.
Suppose there exist functionsf,g∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi such that
(36)fz=z−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi−∑i=t+1∞akzk,(37)gz=z−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi−∑i=t+1∞bkzk,where 0≤Mi≤1 and 0≤∑i=2tMi≤1. Then, let 0≤ζ≤1 and
(38)hz=ζfz+1−ζgz.
So that by substituting (36) and (37) in (38), we have
(39)hz=z−∑i=2∞Mi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi−∑i=t+1∞ζak+1−ζbkzk.
Such that ifζ=1 and ζ=0, equation (39) reduces to (36) and (37), respectively. Now,
(40)∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ζak+1−ζbk=ζ∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ak+1−ζ∑k=i+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1bk≤ζ1−∑i=2tMi+1−ζ1−∑i=2tMi=1−∑i=2tMi.
Thus,hz∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi.☐Theorem 11.
Let(41)ftz=z−∑i=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zi,fkz=z−∑i=2tMi1+i−1λω−ξσ+iη+ε/μ+ωn1+ϖi−1zi−∑k=t+1∞1−∑k=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk,k≥i+1.
Then,f∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi if and only if f can be expressed in the form
(42)fz=∑k=t∞ζkfkz,where ζk≥0 with k≥t and ∑k=t∞ζkfkz=1.Proof.
Iff∈T can be expressed in form of (42). Then,
(43)fz=z−∑i=2tMi1+i−1λω−ξσ+iη+ε/μ+ωn1+ϖi−1zi−∑k=t+1∞ζk1−∑k=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1zk.
Now,(44)∑k=t+1∞1+k−1λω−ξσ+kη+εμ+ωn1+ϖk−1ζk1−∑k=2tMi1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1=∑k=t+1∞ζk1−∑k=2tMi=1−∑k=2tMi∑k=t+1∞ζk=1−∑k=2tMi1−ζk≤1−∑k=2tMi,which implies f∈TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi.
Conversely, fork≥t+1, set
(45)ζk=1+k−1λω−ξσ+kη+ε/μ+ωn1+ϖk−1ak1−∑i=2∞Mi,k≥t+1,(46)ζt=1−∑k=t+1∞ζk.
Then,f can be represented as fz=∑k=t∞ζkfkz.☐


## 3. Conclusion

In this work, a generalized multiplier operator was defined and used to define a class of univalent function. Geometric properties of the classTSQσ,ξ,ω,εμ,λ,ηϖ,n and its subclass TSQσ,ξ,ω,εμ,λ,ηϖ,n;Mi were investigated. The results obtained are the generalization of many known results in literature. See [12, 18–21].



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