# A New Theoretical Investigation of the Generalized Formulations of Spirallike Functions

Authors: Geetha Balachandar; Mohammed K. A. Kaabar; Charles Robert Kenneth; Kins Yenoke
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, the idea ofq-calculus is applied to define new classes of spirallike functions α−UCSPTq,ϕ,δ and α−SPpTq,ϕ,δ. The coefficient inequalities, rotational invariance, and containment results are proved for these classes. Further, by introducing a parameter β, two more subclasses are defined and their properties are investigated. Finally, we have probed into these classes by fixing finitely several coefficients.


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## Body

## 1. Introduction

The class of functions(1)hz=z+∑m=2∞dmzm,which are analytic and univalent in U=z∈C:z<1 and are normalized by h0=0, h′0=1, will be denoted by S.UST, the class of uniformly starlike functions and UCV, the class of uniformly convex functions, were initiated by Goodman in [1, 2]. These were further extended as α−UCVϕ and α−STϕ by Kanas and Wisniowska [3, 4]. The classes of uniformly convex spirallike and uniformly spirallike were introduced in [5].Definition 1.
[5] A functionhzgiven by (1) is inSPpϕiff(2)Ree−iϕzh′zhz≥zh′zhz−1,z∈U.Definition 2.
[5] A functionhzgiven by (1) is inUCSPϕiff(3)Ree−iϕ1+zh′′zh′z≥zh′′zh′z,z∈U.Definition 3.
[6] The functionhzgiven by (1) is uniformly convexϕ-spiral of orderδif the image of every circular arcΓzwith center atξlying inUis convexϕ-spirallike of orderδ.
The class of such functions is denoted byUCSPϕ,δ. The single variable characterization of the classUCSPϕ,δis given by(4)Ree−iϕ1+zh′′zh′z≥zh′′zh′z+δ,z∈U.Definition 4.
[6] A functionhzdefined by (1) is inSPpϕ,δiff(5)Ree−iϕzh′zhz≥zh′zhz−1+δ,z∈U.
The subclassT of S, launched by Herb Silverman [7] consists of functions(6)hz=z−∑m=2∞dmzm,dm>0andz∈S.
Influenced by the studies in [8] UCSPTϕ,δ and SPpTϕ,δ were defined and analyzed in [9]. Generalizing and fixing the coefficients in the above classes, new class of functions α−UCSPTβϕ,δ and α−SPpTβϕ,δ were defined in [10].
q-calculus is in the limelight of research in geometric function theory. Jackson [11, 12], the pioneer in the systematic initiation of q-calculus defined q-derivative as(7)∂qhz=h′0,z=0,hz−hzq1−qz,z≠0,with q∈0,1.
It follows thatlimq⟶1−∂qhz=h′z, if h is differentiable in the domain. Also, ∂2qhz=∂q∂qhz. hz is defined by (1).(8)∂qhz=1+∑m=2∞m,qdmzm−1,where m,q=1−qm/1−q=1+∑i=1m−1qi and limq⟶1m,q=m.
Motivated by the current developments inq-calculus, the q-analogue of α−UCSPTϕ,δ and α−SPpTϕ,δ are defined and examined.


## 2. Coefficient Inequalities

Definition 5.
The function(9)hz=z−∑m=2∞dmzm,
is inα−UCSPTq,ϕ,δ if(10)Ree−iϕ1+z∂q∂qhz∂qhz≥αz∂q∂qhz∂qhz+δ,
ϕ<π/2, 0≤δ<1, 0<α≤1, and 0<q<1.Theorem 1.
Lethzbe defined by (9). If(11)∑m=2∞m,qα+1−cosϕ−δm,qdm≤cosϕ−δ.
Then,hz∈α−UCSPTq,ϕ,δ.Proof.
From Definition5, it is enough if we prove(12)αz∂q∂qhz∂qhz≤Ree−iϕ1+z∂q∂qhz∂qhz−δ.
However, we have(13)αz∂q∂qhz∂qhz−Ree−iϕz∂q∂qhz∂qhz=α−1z∂q∂qhz∂qhz+z∂q∂qhz∂qhz−Ree−iϕz∂q∂qhz∂qhz≤α−1z∂q∂qhz∂qhz+2z∂q∂qhz∂qhz≤α+1z∂q∂qhz∂qhz≤α+1∑m=2∞m,qm−1,qdm1−∑m=2∞m,qdm≤∑m=2∞m,qα+1−cosϕ−δm,qdm1−∑m=2∞m,qdm.
This is bounded bycosϕ−δ, only when(14)∑m=2∞m,qα+1−cosϕ−δm,qdm≤cosϕ−δ.Definition 6.
The functionhzdefined by (9) is inα−SPpTq,ϕ,δif(15)Ree−iϕ1+z∂qhzhz≥αz∂qhzhz+δ.
ϕ<π/2, 0≤δ<1, 0<α≤1 and 0<q<1.Theorem 2.
Lethzbe defined by (9). If(16)∑m=2∞m,qα+1−cosϕ−δdm≤cosϕ−δ.
Then,hz∈α−SPpTq,ϕ,δ.Proof.
The proof is analogous to that of Theorem1.Theorem 3.
The classα−UCSPTq,ϕ,δis rotationally invariant.Proof.
Forh∈α−UCSPTq,ϕ,δ, we prove that Gz=t−1hz also belongs to α−UCSPTq,ϕ,δ and t=1.
Consider(17)Ree−iϕ1+z∂q∂qGz∂qGz=Ree−iϕ1+tz∂q∂qhtz∂qhtz=Ree−iϕ1+ζ∂q∂qhζ∂qhζ≥αζ∂q∂qhζ∂qhζ+δ,by Definition 5 and ζ=tz∈S whenever z∈S.(18)∴Ree−iϕ1+z∂q∂qGz∂qGz≥αtz∂q∂qhtz∂qhtz+δ=αz∂q∂qGz∂qGz+δ,sincet=1,⇒Gz=t−1hz∈α−UCSPTq,ϕ,δ.
The same result holds true forα−SPpTq,ϕ,δ.Theorem4.
α−UCSPTq,ϕ,δ⊆α−UCSPTq,ϕ′,δ, whenever0<ϕ′≤ϕ.Proof.
Leth∈α−UCSPTq,ϕ,δ and ϕ′≤ϕ.
This impliescosϕ≤cosϕ′ and(19)∑m=2∞m,qα+1−cosϕ′−δm,qdm≤∑m=2∞m,qα+1−cosϕ−δm,qdm,≤cosϕ−δ,by Theorem2.1≤cosϕ′−δ,⇒h∈α−UCSPTq,ϕ′,δ,⇒α−UCSPTq,ϕ,δ⊆α−UCSPTq,ϕ′,δ.Definition 7.
Letα−UCSPTβq,ϕ,δbe the class of functions inα−UCSPTq,ϕ,δof the form(20)hz=z−βcosϕ−δz22,qα+12,q−cosϕ−δ∑m=3∞dmzm,dm≥0,0≤β≤1,0<α≤1.Definition 8.
Letα−SPpTβq,ϕ,δbe the class of functions inα−SPpTq,ϕ,δof the form(21)hz=z−βcosϕ−δz2α+12,q−cosϕ−δ−∑m=3∞dmzm,dm≥0,0≤β≤1,0<α≤1.Theorem 5.
The functionhzwhich is defined by (20) belongs toα−UCSPTβq,ϕ,δ, if and only if(22)∑m=3∞α+1m,q−cosϕ−δm,qdm≤cosϕ−δ1−β.
The result is sharp.Proof.
Settingd2=βcosϕ−δ/2,qα+1m,q−cosϕ−δ in (11), we get the required result. The result is sharp for(23)hz=z−βcosϕ−δz22,qα+12,q−cosϕ−δ−1−βcosϕ−δzmm,qα+1m,q−cosϕ−δ,m≥3.
We now state the coefficient inequality forα−SPpTβq,ϕ,δ. The proof is similar to that of Theorem 5 and hence, it is omitted.Theorem 6.
The functionhzdefined by (21) belongs toα−SPpTβq,ϕ,δif and only if(24)∑m=3∞α+1m,q−cosϕ−δdm≤cosϕ−δ1−β.


## 3. Some Properties ofα−UCSPTβq,ϕ,δ and α−SPpTβq,ϕ,δ

We consider a few integral operators and prove preserving properties of these operators on the two classes.


### 3.1. Bernadi Integral Operator

(25)Iϕ1:S⟶S,ϕ1=1,2,3,…,Iϕ1Gz=hz=ϕ1+1/zϕ1∫0zGttϕ1−1dt,Id+λ:S⟶S,0<t≤1,1≤λ<∞,0<d<∞,Id+λGz=hz=d+λ∫01td+λ−2Gtzdt.Withd=1,2,… and λ=1, Id+λ reduces to the Bernadi integral operator.


### 3.2. Alexander Operator

HGz=∫0zGu/udu andHϕ1Gz=∫0zGu/uϕ1du for G∈S.Theorem 7.
The Alexander operator preserves the classesα−UCSPTβq,ϕ,δandα−SPpTβq,ϕ,δ.Proof.
LetGz=z−∑m=2∞dmzm be in α−UCSPTβq,ϕ,δ.
We prove thatHGz=hz∈α−UCSPTβq,ϕ,δ.
Now considering the following,(26)HGz=hz=∫0zGuudu,=∫0z1uu−∑m=2∞dmumdu,=z−∑m=2∞dmmzm,=z−∑m=2∞βmzm,whereβm=dmm.
dm≥0 gives βm≥0⇒f∈T.
AsF∈α−UCSPTβq,ϕ,δ and by Theorem 5,(27)∑m=3∞m,qα+1−cosϕ−δm,qdm≤cosϕ−δ1−β.
Also, whenm≥3, dm/m≤dm.
Hence,(28)∑m=3∞m,qα+1−cosϕ−δm,qdmm≤cosϕ−δ1−β,⇒∑m=3∞m,qα+1−cosϕ−δm,qβm≤cosϕ−δ1−β,⇒hz=HGz∈α−UCSPTβq,ϕ,δ.
Similarly, we can prove the result forα−SPpTβq,ϕ,δ.Theorem8.
The classesα−UCSPTβq,ϕ,δandα−SPpTβq,ϕ,δare preserved byId+λ.Proof.
ConsideringGz=z−∑m=2∞bmzm, bm≥0, in α−UCSPTβq,ϕ,δ, using(29)∑m=3∞m,qα+1−cosϕ−δm,qbm≤cosϕ−δ1−β,
and the definition ofId+λ, we get(30)hz=Id+λGz=d+λ∫01td+λ−2Gtzdt,=d+λz1d+λ−∑m=2∞bmzm−1d+λ+m−1,=z−∑m=2∞d+λd+λ+m−1bmzm,
with0<d<∞ and 1≤λ<∞.
Form≥3,(31)0<d+λd+λ+m−1<1,⇒h∈T.to prove h∈α−UCSPTβq,ϕ,δ. From (31), d+λ/d+λ+m−1bm<bm.
Using (29), we get(32)∑m=3∞m,qα+1−cosϕ−δm,qd+λd+λ+m−1bm≤cosϕ−δ1−β,⇒h∈α−UCSPTβq,ϕ,δ.
We can prove the result forα−SPpTβq,ϕ,δ in a similar way.Theorem 9.
The Bernadi integral operator and the integral operatorHϕ1preserves the classesα−SPpTβq,ϕ,δandα−UCSPTβq,ϕ,δ.
The proof of this theorem is analogous to that of the above theorem.
By fixing finitely many coefficients, we define new subclasses of functions in the following section.


## 3.1. Bernadi Integral Operator

(25)Iϕ1:S⟶S,ϕ1=1,2,3,…,Iϕ1Gz=hz=ϕ1+1/zϕ1∫0zGttϕ1−1dt,Id+λ:S⟶S,0<t≤1,1≤λ<∞,0<d<∞,Id+λGz=hz=d+λ∫01td+λ−2Gtzdt.Withd=1,2,… and λ=1, Id+λ reduces to the Bernadi integral operator.


## 3.2. Alexander Operator

HGz=∫0zGu/udu andHϕ1Gz=∫0zGu/uϕ1du for G∈S.Theorem 7.
The Alexander operator preserves the classesα−UCSPTβq,ϕ,δandα−SPpTβq,ϕ,δ.Proof.
LetGz=z−∑m=2∞dmzm be in α−UCSPTβq,ϕ,δ.
We prove thatHGz=hz∈α−UCSPTβq,ϕ,δ.
Now considering the following,(26)HGz=hz=∫0zGuudu,=∫0z1uu−∑m=2∞dmumdu,=z−∑m=2∞dmmzm,=z−∑m=2∞βmzm,whereβm=dmm.
dm≥0 gives βm≥0⇒f∈T.
AsF∈α−UCSPTβq,ϕ,δ and by Theorem 5,(27)∑m=3∞m,qα+1−cosϕ−δm,qdm≤cosϕ−δ1−β.
Also, whenm≥3, dm/m≤dm.
Hence,(28)∑m=3∞m,qα+1−cosϕ−δm,qdmm≤cosϕ−δ1−β,⇒∑m=3∞m,qα+1−cosϕ−δm,qβm≤cosϕ−δ1−β,⇒hz=HGz∈α−UCSPTβq,ϕ,δ.
Similarly, we can prove the result forα−SPpTβq,ϕ,δ.Theorem8.
The classesα−UCSPTβq,ϕ,δandα−SPpTβq,ϕ,δare preserved byId+λ.Proof.
ConsideringGz=z−∑m=2∞bmzm, bm≥0, in α−UCSPTβq,ϕ,δ, using(29)∑m=3∞m,qα+1−cosϕ−δm,qbm≤cosϕ−δ1−β,
and the definition ofId+λ, we get(30)hz=Id+λGz=d+λ∫01td+λ−2Gtzdt,=d+λz1d+λ−∑m=2∞bmzm−1d+λ+m−1,=z−∑m=2∞d+λd+λ+m−1bmzm,
with0<d<∞ and 1≤λ<∞.
Form≥3,(31)0<d+λd+λ+m−1<1,⇒h∈T.to prove h∈α−UCSPTβq,ϕ,δ. From (31), d+λ/d+λ+m−1bm<bm.
Using (29), we get(32)∑m=3∞m,qα+1−cosϕ−δm,qd+λd+λ+m−1bm≤cosϕ−δ1−β,⇒h∈α−UCSPTβq,ϕ,δ.
We can prove the result forα−SPpTβq,ϕ,δ in a similar way.Theorem 9.
The Bernadi integral operator and the integral operatorHϕ1preserves the classesα−SPpTβq,ϕ,δandα−UCSPTβq,ϕ,δ.
The proof of this theorem is analogous to that of the above theorem.
By fixing finitely many coefficients, we define new subclasses of functions in the following section.


## 4. The New Classα−UCSPTβm,Mq,ϕ,δ

The classα−UCSPTβm,Mq,ϕ,δ denotes functions in α−UCSPTβq,ϕ,δ and is of the form(33)hz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞dmzm,where 0≤∑m=2Mβm=β≤1.α−UCSPTβm,Mq,ϕ,δ reduces to α−UCSPTβq,ϕ,δ, where M=2.The following theorem gives the coefficient estimate for the functions in this class.Theorem 10.
A functionhzdefined by (33) will be inα−UCSPTβm,Mq,ϕ,δif and only if(34)∑m=M+1∞α+1m,q−cosϕ−δm,qdm≤1−∑m=2Mβmcosϕ−δ.Proof.
Assumingdm as cosϕ−δβm/m,qm,qα+1−cosϕ−δ for m=2,3,…M and 0≤∑m=2Mβm≤1 in Theorem 1, we get (34).
By taking the function(35)hz=z−∑m=2∞βmcosϕ−δzmm,qm,qα+1−cosϕ−δ−1−βcosϕ−δzmm,qm,qα+1−cosϕ−δform≥M+1.
The sharpness of the result follows.


### 4.1. Closure Theorems

Theorem 11.
α−UCSPTβm,Mq,ϕ,δis closed under convex linear combination.Proof.
Lethz and fz belong to α−UCSPTβm,Mq,ϕ,δ and be defined as(36)hz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞dmzm,fz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞bmzm,with dm,bm≥0, 0≤βi≤1, and 0≤∑m=2Mβm≤1.
TakingGz=1−μhz+μfz, 0≤μ≤1, we prove.
Gz∈α−UCSPTβm,Mq,ϕ,δ. Considering(37)Gz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞1−μdm+μbmzm,
and as(38)∑m=M+1∞m,qm,qα+1−cosϕ−δ1−μdm+μbm=1−μ∑m=M+1∞m,qα+1−cosϕ−δm,qdm+μ∑m=M+1∞α+1m,q−cosϕ−δm,qbm≤1−μcosϕ−δ1−∑m=2Mβm+μcosϕ−δ1−∑m=2Mβm=cosϕ−δ1−∑m=2Mβm.which shows that Gz is in α−UCSPTβm,Mq,ϕ,δ, by Theorem 10.
Hence, the classα−UCSPTβm,Mq,ϕ,δ is convex.Theorem 12.
LethMz=z−∑j=2Mβjcosϕ−δzj/j,qj,qα+1−cosϕ−δand(39)hmz=z−∑j=2Mβjcosϕ−δzjj,qj,qα+1−cosϕ−δ−1−∑j=2Mβjcosϕ−δzmm,qm,qα+1−cosϕ−δ,with m≥M+1.
Then, a functionhwill belong to the classα−UCSPTβm,Mq,ϕ,δif and only ifhz=∑m=M∞μmhmz, whereμm≥0and∑m=M∞μm=1.Proof.
Lethz=∑m=M∞μmhmz. Then,(40)hz=z−∑j=2Mβjcosϕ−δzjj,qj,qα+1−cosϕ−δ−∑m=M+1∞1−∑j=2Mβjcosϕ−δμmzmm,qm,qα+1−cosϕ−δ.
From the above,(41)∑m=M+1∞1−∑j=2Mβjμmcosϕ−δm,qm,qα+1−cosϕ−δm,qm,qα+1−cosϕ−δ=∑m=M+1∞μmcosϕ−δ1−∑j=2Mβj=cosϕ−δ1−μM1−∑j=2Mβj≤1−∑j=2Mβjcosϕ−δ,which implies that f∈α−UCSPTβm,Mq,ϕ,δ by using Theorem 10.
To prove the converse part let us takeμm=m,qm,qα+1−cosϕ−δdm/cosϕ−δ1−∑j=2Mβj, for m≥M+1 and μM=1−∑m=M+1∞μm, then, we obtain hz=∑m=M∞μmhmz.Corollary 1.
For the classα−UCSPTβm,Mq,ϕ,δ, the extreme points are the functionshmz,m≥Mof Theorem 12.Definition 9.
The functionhz, in the classSis said to be(i)
Starlike of orderτ0≤τ<1, ifRezh′z/hz>τ;(ii)
Convex of orderτ0≤τ<1, ifRe1+zh′z/h′z>τforz∈U.
In the following theorems, the radii results for the functions inα−UCSPTβm,Mq,ϕ,δ to be starlike or convex of order τ are derived.Theorem 13.
hz, inα−UCSPTβm,Mq,ϕ,δ, will be starlike of orderτ0≤τ<1inz<R1, whereR1is the largest value for which(42)∑m=2Mβmcosϕ−δ1−τ/mRm−1α+1m,q−cosϕ−δ+1−∑m=2Mβmcosϕ−δm−τRm−1m,qm,qα+1−cosϕ−δ≤1−τ.
form≥M+1. The result is sharp.Proof.
To get the required result, we need to prove thatzh′z/hz−1≤1−τ, for z<R1.
Using appropriate substitutions,(43)zh′zhz−1≤∑m=2Mcosϕ−δβmRm−11−1/m/m,qα+1−cosϕ−δ+∑m=M+1∞m−1dmRm−11−∑m=2Mcosϕ−δβmRm−11−1/m/m,qα+1−cosϕ−δ−∑m=M+1∞dmRm−1≤1−τ,for z≤R, if and only if(44)∑m=2Mcosϕ−δβm1−τ/mRm−1m,qα+1−cosϕ−δ−∑m=M+1∞m−τdmRm−1≤1−τ.
Sincef∈α−UCSPTβm,Mq,ϕ,δ, by the coefficient estimate, we can take(45)dm=1−∑m=2Mβmcosϕ−δμmm,qα+1−cosϕ−δ,where μm≥0, ∑m=M+1∞μm≤1 and m≥M+1.
Let us now choose a positive integerm0=m0R, for each fixed R so that m−τRm−1/m is a maximum.
Hence, it follows that(46)∑m=M+1∞m−τdmRm−1≤1−∑m=2Mβmcosϕ−δm0−τRm0−1m0,qm0,qα+1−cosϕ−δ.
∴hz is starlike of order τ in z<R1 if(47)∑m=2Mβmcosϕ−δ1−τ/mRm−1m,qα+1−cosϕ−δ+1−∑m=2Mβmcosϕ−δm0−τRm0−1m0,qm0,qα+1−cosϕ−δ≤1−τ.
We get the valueR0 and m0R0 so that(48)∑m=2Mβmcosϕ−δ1−τ/mR0m−1m,qα+1−cosϕ−δ+1−∑m=2Mβmcosϕ−δm0−τR0m0−1m0,qm0,qα+1−cosϕ−δ=1−τ.
R0 is the radius of starlikeness of order τ for functions hz∈α−UCSPTβm,Mq,ϕ,δ. The result is sharp for hz given by (34).
For functions in the classα−UCSPTβm,Mq,ϕ,δ, we state the theorem regarding the radius of convexity of order τ.Theorem 14.
Lethz∈α−UCSPTβm,Mq,ϕ,δ. Then,hzis convex of orderτ0≤τ<1in the discz<R2, whereR2is the greatest value for which(49)∑m=2Mcosϕ−δβmm−τRm−1m,qα+1−cosϕ−δ+1−∑m=2Mβmcosϕ−δm−τRm−1m,qα+1−cosϕ−δ≤1−τ,for m≥M+1. The result is sharp.


## 4.1. Closure Theorems

Theorem 11.
α−UCSPTβm,Mq,ϕ,δis closed under convex linear combination.Proof.
Lethz and fz belong to α−UCSPTβm,Mq,ϕ,δ and be defined as(36)hz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞dmzm,fz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞bmzm,with dm,bm≥0, 0≤βi≤1, and 0≤∑m=2Mβm≤1.
TakingGz=1−μhz+μfz, 0≤μ≤1, we prove.
Gz∈α−UCSPTβm,Mq,ϕ,δ. Considering(37)Gz=z−∑m=2Mβmcosϕ−δzmm,qm,qα+1−cosϕ−δ−∑m=M+1∞1−μdm+μbmzm,
and as(38)∑m=M+1∞m,qm,qα+1−cosϕ−δ1−μdm+μbm=1−μ∑m=M+1∞m,qα+1−cosϕ−δm,qdm+μ∑m=M+1∞α+1m,q−cosϕ−δm,qbm≤1−μcosϕ−δ1−∑m=2Mβm+μcosϕ−δ1−∑m=2Mβm=cosϕ−δ1−∑m=2Mβm.which shows that Gz is in α−UCSPTβm,Mq,ϕ,δ, by Theorem 10.
Hence, the classα−UCSPTβm,Mq,ϕ,δ is convex.Theorem 12.
LethMz=z−∑j=2Mβjcosϕ−δzj/j,qj,qα+1−cosϕ−δand(39)hmz=z−∑j=2Mβjcosϕ−δzjj,qj,qα+1−cosϕ−δ−1−∑j=2Mβjcosϕ−δzmm,qm,qα+1−cosϕ−δ,with m≥M+1.
Then, a functionhwill belong to the classα−UCSPTβm,Mq,ϕ,δif and only ifhz=∑m=M∞μmhmz, whereμm≥0and∑m=M∞μm=1.Proof.
Lethz=∑m=M∞μmhmz. Then,(40)hz=z−∑j=2Mβjcosϕ−δzjj,qj,qα+1−cosϕ−δ−∑m=M+1∞1−∑j=2Mβjcosϕ−δμmzmm,qm,qα+1−cosϕ−δ.
From the above,(41)∑m=M+1∞1−∑j=2Mβjμmcosϕ−δm,qm,qα+1−cosϕ−δm,qm,qα+1−cosϕ−δ=∑m=M+1∞μmcosϕ−δ1−∑j=2Mβj=cosϕ−δ1−μM1−∑j=2Mβj≤1−∑j=2Mβjcosϕ−δ,which implies that f∈α−UCSPTβm,Mq,ϕ,δ by using Theorem 10.
To prove the converse part let us takeμm=m,qm,qα+1−cosϕ−δdm/cosϕ−δ1−∑j=2Mβj, for m≥M+1 and μM=1−∑m=M+1∞μm, then, we obtain hz=∑m=M∞μmhmz.Corollary 1.
For the classα−UCSPTβm,Mq,ϕ,δ, the extreme points are the functionshmz,m≥Mof Theorem 12.Definition 9.
The functionhz, in the classSis said to be(i)
Starlike of orderτ0≤τ<1, ifRezh′z/hz>τ;(ii)
Convex of orderτ0≤τ<1, ifRe1+zh′z/h′z>τforz∈U.
In the following theorems, the radii results for the functions inα−UCSPTβm,Mq,ϕ,δ to be starlike or convex of order τ are derived.Theorem 13.
hz, inα−UCSPTβm,Mq,ϕ,δ, will be starlike of orderτ0≤τ<1inz<R1, whereR1is the largest value for which(42)∑m=2Mβmcosϕ−δ1−τ/mRm−1α+1m,q−cosϕ−δ+1−∑m=2Mβmcosϕ−δm−τRm−1m,qm,qα+1−cosϕ−δ≤1−τ.
form≥M+1. The result is sharp.Proof.
To get the required result, we need to prove thatzh′z/hz−1≤1−τ, for z<R1.
Using appropriate substitutions,(43)zh′zhz−1≤∑m=2Mcosϕ−δβmRm−11−1/m/m,qα+1−cosϕ−δ+∑m=M+1∞m−1dmRm−11−∑m=2Mcosϕ−δβmRm−11−1/m/m,qα+1−cosϕ−δ−∑m=M+1∞dmRm−1≤1−τ,for z≤R, if and only if(44)∑m=2Mcosϕ−δβm1−τ/mRm−1m,qα+1−cosϕ−δ−∑m=M+1∞m−τdmRm−1≤1−τ.
Sincef∈α−UCSPTβm,Mq,ϕ,δ, by the coefficient estimate, we can take(45)dm=1−∑m=2Mβmcosϕ−δμmm,qα+1−cosϕ−δ,where μm≥0, ∑m=M+1∞μm≤1 and m≥M+1.
Let us now choose a positive integerm0=m0R, for each fixed R so that m−τRm−1/m is a maximum.
Hence, it follows that(46)∑m=M+1∞m−τdmRm−1≤1−∑m=2Mβmcosϕ−δm0−τRm0−1m0,qm0,qα+1−cosϕ−δ.
∴hz is starlike of order τ in z<R1 if(47)∑m=2Mβmcosϕ−δ1−τ/mRm−1m,qα+1−cosϕ−δ+1−∑m=2Mβmcosϕ−δm0−τRm0−1m0,qm0,qα+1−cosϕ−δ≤1−τ.
We get the valueR0 and m0R0 so that(48)∑m=2Mβmcosϕ−δ1−τ/mR0m−1m,qα+1−cosϕ−δ+1−∑m=2Mβmcosϕ−δm0−τR0m0−1m0,qm0,qα+1−cosϕ−δ=1−τ.
R0 is the radius of starlikeness of order τ for functions hz∈α−UCSPTβm,Mq,ϕ,δ. The result is sharp for hz given by (34).
For functions in the classα−UCSPTβm,Mq,ϕ,δ, we state the theorem regarding the radius of convexity of order τ.Theorem 14.
Lethz∈α−UCSPTβm,Mq,ϕ,δ. Then,hzis convex of orderτ0≤τ<1in the discz<R2, whereR2is the greatest value for which(49)∑m=2Mcosϕ−δβmm−τRm−1m,qα+1−cosϕ−δ+1−∑m=2Mβmcosϕ−δm−τRm−1m,qα+1−cosϕ−δ≤1−τ,for m≥M+1. The result is sharp.


## 5. Conclusion

The concept of q-calculus is used widely in quantum physics and fractional calculus of mathematics. The classesα−UCSPTq,ϕ,δ and α−SPpTq,ϕ,δ were introduced and analyzed in this paper. Obtaining the coefficient inequalities, containment theorem, and rotational invariance are of paramount importance as they throw more light on these classes of functions. The results of this study have ample capacity to facilitate more research in this area and beyond.



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*Source: 5361581-2022-08-05.xml*