# Analytic Functions Related with Starlikeness

Authors: Syed Ghoos Ali Shah; Saima Noor; Saqib Hussain; Asifa Tasleem; Akhter Rasheed; Maslina Darus
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

The aim of present investigation is to study a new class of analytic function related with the Sokol-Nunokawa class. We derived relationships of this class with strongly starlike functions and obtained many interesting results.


---

## Body

## 1. Introduction

LetA be the class of functions λ having the series representation,(1)λω=ω+∑n=2∞anωn,which are analytic in the open unit disk E=ω∈ℂ:ω<1 and normalized by λ0=0 and λ′0=1.A functionλ∈A is in class S of univalent function if and only if λω1=λω2 implies ω1=ω2, for all ω1, ω2∈E. An analytic function λ∈A subordinate to an analytic function δ denoted as λ≺δ if and only if there exists an analytic function ς in E with ς0=0 and ςω<1, ω∈E, such that λω=δςω, ω∈E. In particular, if δ is univalent in E, then λ≺δ if and only if λ0=δ0 and λE⊂δE.LetP denote the class of analytic functions of the form pz=1+p1ω+p2ω2+⋯ such that ℜpω>0 in E. A function λ∈S is in class C of normalize convex function if and only if ℜ1+zλ″ω/λ′ω>0. Also, a function λ∈S is in class S∗ if and only if ℜωλ′ω/λω>0.In 1991, Goodman [1] introduced the class of uniformly convex function UCV. A function λ∈C is in class UCV if, for every circular arc φ⊂E with center in E, the arc λφ is convex. Also, a function λ∈S is in class UST if, for every circular arc γ contained in E with center in E, the arc λγ is starlike.The classesUCV and UST were studied by Ma and Minda [2, 3] and Ronning [4, 5]. They provided the analytic characterization of class UCV and UST as follows:(2)λ∈UCVif and only ifℜ1+λ″ωλ′ω>zλ″ωλ′ω,ω∈E,and(3)λ∈USTif and only ifℜωλ′ωλω>ωλ′ωλω−1,ω∈E.In 1999, Kanas et al. [6] introduced the class k−UCV, k≥0, of k-uniformly convex functions as follows:(4)λ∈k−UCVif and only ifλ∈Aandℜ1+ωλ″ωλ′ω>kωλ″ωλ′ω,ω∈E.Furthermore, the class ofk-uniformly starlike functions is defined as follows:(5)λ∈k−USTif and only ifλ∈Aandℜωλ′ωλω>kωλ′ωλω−1,ω∈E.Many authors had investigated the interesting properties of the abovementioned classes. For details, see [6–13]. The class SS∗β [14, 15] denotes the class of strongly starlike functions of order β∈0,1, which is defined as follows:(6)SS∗β=λ∈A:argωλ′ωλω<βπ2.In 2015, Sokol et al. [16] introduced the class ℳN as follows:(7)λ∈ℳNif and only ifλ∈Aandℜωλ′ω′λ′ω>ωλ′ωλω−1,ω∈E.Fork≥0, 0≤α≤1, in a recent paper, Darus et al. [17] defined a new class k−USTα as follows:(8)k−USTα=λ∈A:ℜωλ′ω′λ′ω>k1−αωλ′ωλω+αωλ′ω′λ′ω−1,ω∈E.The classk−USTα has some interesting relationship with other classes of univalent functions. Motivated from [17–21], we introduced a new class of analytic functions and found its relation with some other existing classes of analytic functions.Definition 1.
Letλ∈A, 0≤k<1. Then, λ is in class k−CST if and only if(9)ℜωλ′ω′λ′ω>kωλ′ω′λ′ω−ωλ′ωλω−1.
For special values of parameterk, we obtained many known classes of analytic functions. For example, if we take k=0, then k−CST=C, the class of convex functions.


## 2. Set of Lemmas

To prove our main results, we need the following lemmas.Lemma 1.
Letu=u1+iu2 and v=v1+iv2 and let Ψu,v:D⊂ℂ2⟶ℂ be a complex-valued function satisfying the conditions: (i)
Ψu,v is continuous in a domain D⊂ℂ2.(ii)
1,0∈D and Ψ1,0>0.(iii)
ℜΨiu2,v1≤0 whenever iu2,v1∈D and v1≤−1/21+u22. If hω=1+c1ω+c2ω2+⋯ is a function that is analytic in E such that hω,ωh′ω∈D and ℜΨhω,ωh′ω>0 for ω∈E, then ℜhω>0. This result is due to Miller [22].Lemma 2.
Letβ>0 and pω+βωp′ω/pω≺1+ω/1−ω. Then, pω≺1+1−2ξω/1−ω, where(10)ξ=2k−β+2k−β2+8β4.Proof.
Letpω=1−ξp1ω+ξ. Then, pω+βωp′ω/pω=1−ξp1ω+ξ+βω1−ξp1′ω/1−ξp1ω+ξ. Since pω+βωp′ω/pω≺1+ω/1−ω.
Therefore,1−ξp1ω+ξ+1−ξβωp1′ω/1−ξp1ω+ξ−k∈P,where0≤k<1.
We formulate a functionalϕu,v by taking u=p1, v=ωp1′, thus ϕu,v=1−ξu+ξ−k+β1−ξv/1−ξu+ξ. Clearly, (i)
ϕu,v is continuous in domain D⊂ℂ╲−ξ/1−ξ×ℂ(ii)
ϕ1,0=1−k>0
Considerϕiu2,v1=1−ξiu2+ξ−k+β1−ξv1/1−ξiu2+ξ. Then, ℜϕiu2,v1≤ξ−k−βξ1−δ1+u22/2ξ2+1−ξ2u22, where we have used the fact that v1≤−1+u22/2. This implies that ℜϕiu2,v1≤A+Bu22/2C, where A=ξ2ξξ−k−β1−ξ,B=1−ξ2ξ−k1−ξ−βξ,andC=1−ξ2u22+ξ2>0.
Clearly,ℜϕiu2,v1≤0, if and only if A≤0 and B≤0. From A≤0, ξ=2k−β+2k−β2+8β/4.
FromB≤0, we have 0≤ξ<1. Now, using Lemma 1, we have the required result.Lemma 3 (see [23]).
If−1≤B<A≤1,λ>0 and the complex number γ satisfies ℜγ≥−λ1−A/1−B, then the differential equation qω+ωq′ω/λqω+γ=1+Aω/1+Bω,ω∈E, has a univalent solution in E given by(11)qω=ωλ+γ1+BωλA−B/Bλ0ztλ+γ−11+BtλA−B/Bdt−γλ,forB≠0,zλ+γeλAωλ0ztλ+γ−1eλAtdt−γλ,forB=0.Ifhω=1+c1ω+c2ω2+… is analytic in E and satisfies hω+ωh′ω/λhω+γ≺1+Aω/1+Bω,ω∈E, then hω≺qω≺1+Aω/1+Bω, and qω is the best dominant.Lemma 4 (see [24]).
Leta,b and c≠0,−1,−2… be complex numbers. Then, for ℜc>ℜb>0, i2χ1a,b,c;ωΓc/Γc−bΓb01tb−11−tc−b−11−tω−adt,ii2χ1a,b,c;ω2χ1b,a,c;ω,iii2χ1a,b,c;ω1−ω2−aχ1a,c−b,c;ω/ω−1..Lemma 5.
Letp be analytic in E of the form
pω=1+∑n=m∞cnωn,cm≠0, with pω≠0 in E. If there exists a point ω0, ω0<1 such that argpω<πφ/2ω<ω0 and argpω0=πφ/2 for some φ>0, then we have ω0p′ω0/pω0=ilφ, where l≥m/2a+1/a,whenargpω0=πφ/2,l≤−m/2a+1/a,whenargpω0=−πφ/2, where pω01/φ=±iaa>0. This result is the generalization of Nunokawa’s lemma [25, 26].


## 3. Main Results

In this section, we derived our main results.Theorem 1.
The functionλω=ω/1−Aω is in the class k−CST if and only if A≤1−k/1+2k.Proof.
Letλω=ω/1−Aω, then some simple calculation gives us(12)ℜωλ′ω′λ′ω=ℜ1+Aω1−Aω.
By using (9) and (12), we obtain(13)kωλ′ω′λ′ω−zλ′ωλω−1=k2Aω−11−Aω.
Using (12) and (13), it is clear that λ∈k−CST if and only if(14)ℜ1+Aω1−Aω>k2Aω−11−Aω.
It is suffices to studyω=1, A=r, and Aω=reiθ in (14).(15)ℜ1+reiθ1−reiθ=1−r21−reiθ2,=1−r21−reiθ2≥k2reiθ−11−reiθ.
Then, equation (15) leads to a simple computation which shows that
1−r2/1−reiθ≥k4r2−4rcosθ+1 or(16)k≤1−r21−reiθ4r2−4rcosθ+1.
The right-hand side of the above inequality (16), is seen to have minimum for θ=π and minimum value is 1−r/2r+1. Hence, a necessary and sufficient conditions for (16) is k≤1−r/2r+1 or A=r≤1−k/2k+1. This completes the proof.Corollary 1.
The functionλω=ω/1−ω belongs to a class k−CST if and only if k=0.Remark 1.
It is important to note that ifk1≥k2, then k1−CST⊆k2−CST.Theorem 2.
Letλ∈k−CST. Then, λ∈S∗ξ, where S∗ is the class of starlike functions of order ξ, where ξ is defined in (10) and 0≤k<1 and β=1+k.Proof.
Letωλ′ω/λω=pω, where p is analytic in unit disk and p0=1. Now, using (9), we obtain ℜpω+ωp′ω/pω>kωp′ω/pω−1=k1−ωp′ω/pω. This implies that ℜpω+βωp′ω/pω>kℜ1−ωp′ω/pω>k, where β=1+k,0≤k<1. The above relation can be written in the following Briot Bouquet differential subordination pω+ωp′ω/1/βpω≺1+ω/1−ω. Now, using Lemma 3, we have p∈Pξ this complete the proof.
Special case: fork=0 and β=1, we have ξ=1/2, that is, λω∈0−CST implies λ∈P∗1/2.Theorem 3.
Letλ∈k−CST of the form λω=ω+∑n=m+1∞anωn,am+1≠0; then, λ is strongly starlike of order β0, where β0=mink2−aβcosβπ/22+k2β2ma2+1/2a2.Proof.
Letpω=ωλ′ω/λω; then, p is of the form pω=1+∑n=m∞cnωn,cm≠0. Now using the definition of class k−CST, we have ℜpω+ωp′ω/pω>kωp′ω/pω−1.
If there exists a pointω0∈E, we have argpω<βπ/2,ω<ω0,andargpω0=βπ/2. Then, from Lemma 5, we have ω0p′ω0/pω0=ilβ, where pω01/β=±iaa>0 and(17)l:l≥m2a+1a,whenargpω0=πφ2,l≤−m2a+1a,whenargpω0=−πφ2.
For the caseargpω0=βπ/2, we obtain(18)ℜpω0+ω0p′ω0pω0=ℜiaβ+ilβ=aβcosβπ2.
Also, we havekω0p′ω0/pω0−1=kilβ−1=k1+l2β2.
Then, froml≥ma2+1/2a for β≥β0, we have(19)0≤k2−acosβπ22+k2ma2+122aβ2.
Therefore,0≤k2−acosβπ/22+k2ma2+12/2aβ2.
By using (18) and (19), we have ℜpω0+ω0p′ω0/pω0<kω0p′ω0/pω0−1, which is a contradiction, therefore, argpω<βπ/2 for ω<1. Similarly, we can prove the case argpω0=−βπ/2, by using the same method as the above and will get a contradiction. This proves that λ is strongly starlike of order β0.Theorem 4.
Ifλ∈k−CST, then ωλ′/λ≺qω=1/δω, where qω=2χ12/β;1;1/β+1;ω/ω−1, with β=1+k and 0≤k<1.Proof.
Letωλ′/λ=pω, where pω is analytic in E and p0=1. ℜpω+ωp′ω/pω>kωp′ω/pω−1=k1−ωp′ω/pω>ℜk1−ωp′ω/pω.
This implies thatℜpω+βωp′ω/pω>k, where 0≤k<1, β=1+k. The above relation can be written in the following Briot–Bouquet differential subordination, pω+βωp′ω/pω≺1+ω/1−ω. Using Lemma 4, for λ=1/β and γ=0, we have pω≺qω=1/δω=1/1/β∫01t1/β−11−tω/1−ω−2/βdt=2χ12/β,1;1/β+1;ω/ω−1−1, which is required.Remark 2.
Ifλ∈k−CST, then a2≤21−ξ and a3≤21−ξ, where ξ is defined in (10).Theorem 5.
Let0≤k<1 and λω∈k−CST; then, λE contains an open disk of radius 1/ϖ0≤2+21−ξ.Proof.
Letϖ0≠0 be a complex number such that λω≠0, for ω∈E, then λ1ω=ϖ0λω/ϖ0−λω=ω+a2+1/ϖ0ω2+….
Sinceλ1ω is univalent in E so that(20)a2+1ϖ0≤2=1ϖ0−a2≤2,=1ϖ0≤2+a2.
Sinceλω∈k−CST, then(21)a2≤21−ξ.
Using (21) in (20), 1/ϖ0≤2+21−ξ, where ξ is defined in (10).Theorem 6.
Letλ∈k−CST such that χϖ=λ−1ϖ=ϖ+d2ϖ2+d3ϖ3+…. Then,(22)d2≤21−ξ,(23)d3≤1−ξ5−8ξ,where ξ is defined in (10).Proof.
Asχϖ=λ−1ϖ, ϖ=λω and χλω=ω. Now, χϖ=ϖ+d2ϖ2+d3ϖ3+…, and λω=ω+a2ω2+a3ω3+…. Then, χλω=ω+a2+d2ω2+a3+2a2d2+d3ω3+…, this implies χλω=ω+a2+d2ω2+a3+2a2d2+d3ω3+…, where d2=−a2,d3=2a23−a3, and d3≤1−ξ5−8ξ, in which ξ is defined in (10).


## 4. Conclusions

In this article, we studied a new class of analytic functions. We derived certain necessary and sufficient condition, order of strongly starlike. We also pointed out some existing results in literature.



---

*Source: 9924434-2021-06-28.xml*