# Applications ofα,β-Symmetrical Functions on a Certain Class of Spirallike Functions

Authors: Aljazi Alkhammash; Fuad Alsarari
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this note, we use the notions ofα,β-symmetrical, generalized Janowski-type and spirallike functions to define a new class Sα,βλN,M,μ defined in the open unit disk. In particular, we obtain a structural formula, a representation theorem, Marx-Strohacker inequality. Our results continue to hold the covering and distortion properties.


---

## Body

## 1. Introduction

LetHΔ denote the class of analytic functions in the open unit disk Δ=ω∈ℂ:ω<1. Also, H indicates the subclass of HΔ which has the form
(1)hω=ω+∑v=2∞avωv.The family of functionsh∈H that are univalent in Δ is represented by H~.The family ofSchwarz functions is denoted by Ω where
(2)Ω≔s∈H,s0=0,sω<1,ω∈Δ.Leth,h~ be analytic, the function h is said to be subordinate to h~ in Δ if there exists s∈Ω and hω=h~sω, and we denote this by hω≺h~ω. Whereover h~ is univalent in Δ, then the subordination is equivalent to h0=h~0 and hΔ⊂h~Δ.By using the concept of the subordination, let define the well-known Carathe˘odory class P=p∈HΔsatisfyingp0=1,Repω>0; ω∈Δ, and for any function p0=1 is said to belong to the class P has the representation pω=1+sω/1−sω, for some s∈Ω.In [1], Janowski introduced the class PN,M with −1≤M<N≤1, a function p analytic in Δ with p0=1 is said to belong to the class PN,M which has the representation pω=1+Nω/1−Mω.The classPN,M,μ of generalized Janowski functions was introduced in [2]. For arbitrary numbers N,M and μ with −1≤M<N≤1, 0≤μ<1, a function p analytic in Δ with p0=1 is said to belong to the class PN,M,μ if and only if
(3)pω≺1+1−μN+μMω1+Mω⇔pω=1+1−μN+μMsω1+Msω,s∈Ω.In order to define new classes of symmetrical functions defined inΔ, we first recall the notion of β-fold symmetric functions defined in β-fold symmetric domain, where β is any positive integer. A domain G is said to be β-fold symmetric if a rotation of G about the origin through an angle 2π/β carries G onto itself. A function h is said to be β-fold symmetric in G if for every ω in G we have
(4)hεω=εhω,ε=e2πi/β,ω∈G.The family of allβ-fold symmetric functions is denoted by S~β; we get the class of odd univalent functions for β=2. In 1995, Liczberski and Polubinski [3] constructed the theory of α,β-symmetrical functions for α=0,1,2,⋯,β−1 and β=2,3,⋯. If G is β-fold symmetric domain and α any integer, then a function h:G⟶ℂ is called α,β-symmetrical if for each ω∈G, hεω=εαhω. We note that the α,β-symmetrical functions are a generalization of the notions of even, odd, and β-symmetrical functions.In [3], we observe that the theory of α,β-symmetrical functions has many interesting applications; we now investigate some results in the classes of analytic functions.Denote the family of allα,β-symmetrical functions by S~α,β. We observe that, S~0,2, S~1,2, and S~1,β are the classes of even, odd, and β-symmetric functions, respectively.Theorem 1 (see [3], page 16).
For every mappingh:Δ↦ℂ and a β-fold symmetric set Δ, there exists exactly one sequence of α,β-symmetrical functions hα,β such that
(5)hω=∑α=0β−1hα,βω,hα,βω=1β∑m=0β−1ε−mαhεmω,ω∈Δ.Furthermore, we say thath is λ-spirallike if and only if Reeiλωh′ω/hω>0,λ is real and λ<π/2.Recently, see [4–6] obtained many interesting results for various subclasses of Janowski-type functions by using the concept of α,β-symmetrical functions.By taking motivation from the above-cited work and using the generalized Janowski functions,α,β-symmetrical functions, and λ-spirallike, we introduce a new subclass of analytic functions.Definition 2.
A functionh in H is said to belong to the class Sα,βλN,M,μ, −1≤M<N≤1,0≤μ<1 if
(6)1cosλeiλωh′ωhα,βω−isinλ∈PN,M,μ,λ<π2,ω∈Δ,
wherehα,β is defined in (5).Our defined class generalizes many classes by choosing particular values of the parameters for various choices ofα,β,λ,N,M, and μ; Definition 2 yields several known and new subclasses of H, as Sα,β0N,M,μ≔Sα,βN,M,μ introduced by the authors in [5]; S1,1λN,M,0≔SN,M,μ and S1,1λN,M,0≔SN,M,μ motivated by Polatog~lu et al. [2, 7]; S1,β0N,M,μ≔SβN,M,μ introduced by Latha and Darus [8]; S1,β01,−1,0≔Sβ defined by Sakaguchi [9]; S1,10N,M,0≔SN,M these class reduce to well-known class defined by Janowski [1].Lemma 3 (see [5]).
Ifh belongs to the class Sα,βN,M,μ, then
(7)hα,βω=ω1+Msω1−μN−M/M,ifM≠0,ωexp1−μNsω,ifM=0,
andhα,β∈Sα,βN,M,μ for some s∈Ω, where hα,β are defined by ((5)).Lemma 4 (see [2]).
Forτ is an arbitrary fixed point of Δ and Hp=pτ,p∈PN,M,μ, then the set of the values of H is in the closed disc with center at Cr and having the radius ρr, where
(8)Cr=1−M1−μN+μMr21−M2r2,0,ρr=1−μN−Mr1−M2r2,ifM≠0,Cr=1,0,ρr=1−μNr,ifM=0.Lemma 5 (see [10]).
Letp∈PN,M,μ, then
(9)1−1−μN+μMr1−Mr≤pω≤1+1−μN+μMr1+Mr,ω≤r<1.


## 2. Main Results

Theorem 6.
A functionh belongs to the class Sα,βλN,M,μ if and only if
(10)hω=∫0ωBtqtdt,
where(11)qt=exp∫0t1βu∑m=0β−1Bεmu−βdu,(12)Bω=cosλpω+isinλe−iλ,p∈PN,M,μ.Proof.
Suppose thath∈Sα,βλN,M,μ, we have
(13)ωh′ωhα,βω=cosλpω+isinλe−iλ,p∈PN,M,μ.
Replacingω by εmω in (13), we obtain
(14)εm1−αωh′εmωhα,βω=Bεmω.
From (13) and (14), we get
(15)h′εmω=Bεmωεmα−1h′ωBω.
By differentiation (13), we have
(16)hα,β′ω=ωh′′ω+h′ωBω−ωh′ωB′ωB2ω.
From (5) and (15), we get
(17)hα,β′ω=1βh′ωBω∑m=0β−1Bεmω.
From (5) and (17), we get
(18)h′′ωh′ω=B′ωBω+1βω∑m=0β−1Bεmω−β.
Integrating repeatedly, we get the required structural formula(19)hω=∫0ωBtqtdt,
which proves the necessity. To prove the sufficiency of (10), suppose that (10) holds with p∈PN,M,μ. The function h defined by (10) is obviously in H with h0=0 and h′0=1. The following identity can be verified by differentiation
(20)ωqω=∫0εmω1β∑m=0β−1ε−αmBt.qtdt,
whereq and B are given by (10) and (11), respectively. Also, using (10), we have
(21)h′ω=Bω.qω,
which shows thath′≠0 in Δ.
From (10), since ε is the root of unity, we conclude that
(22)hα,βω=∫0εmω1β∑m=0β−1ε−αmBt.qtdt.
Using (20), (21), and (22), we arrive at the result
(23)hα,βω=ωh′ωBω,
thus proving the sufficiency of (10).Lemma 7.
Ifh∈Sα,βλN,M,μ, then
(24)hα,βω=ω1+Msωe−iλ1−μN−Mcosλ/M,M≠0,ωe1−μNcosλe−iλsω,M=0,
for somes∈Ω, where hα,β are defined by (5).Proof.
By using Lemma3 and similar technique proof in Theorem 6 in [5].Corollary 8.
Marx-Strohacker inequality for the classSα,βλN,M,μ is
(25)loghα,βωωeiλ/1−μNcosλ<1,ifM=0,hα,βωωMeiλ/1−μN−Mcosλ−1<1,ifM≠0.Proof.
The proof of this corollary is a simple consequence of Lemma7. Indeed,
(26)hα,βωω=1+Msωe−iλ1−μN−Mcosλ/M⇒hα,βωωMeiλ/1−μN−Mcosλ−1<1,hα,βωω=e1−μNcosλe−iλsω⇒loghα,βωωeiλ/1−μNcosλ<1.The next covering, starlikeness and distortion theorems for the classSα,βλN,M,μ hold.Theorem 9.
Leth∈Sα,βλN,M,μ, with −1≤M<N≤1 and 0≤μ<1. Then,
(27)hω=∫0ω1+γs~ζ1+Ms~ζ1+Msζe−iλ1−μN−Mcosλ/M,dζ,ifM≠0,∫0ω1+e−iλ1−μNcosλs~ζee−iλ1−μNcosλsζdζ,ifM=0,
for somes,s~∈Ω and
(28)γ=1−μN+μMcosλ+iMsinλe−iλ.Proof.
Supposing thath∈Sα,βλN,M,μ, it follows that there exists a function s~∈Ω such that
(29)eiλωh′ω/hα,βω−isinλcosλ=1+1−μN+μMs~ω1+Ms~ω,ω∈Δ.
Combining the above relation with Lemma7, we have
(30)h′ω=1+1−μN+μMs~ω1+s~ω1+Msωe−iλ1−μN−Mcosλ/M,ifM≠0,1+e−iλ1−μNcosλs~ωee−iλ1−μNcosλsω,ifM=0.
Integrating the above equation along the line connecting the origin withω∈Δ, we obtain our result.Theorem 10.
The radius of starlikeness of the classh∈Sα,βλN,M,μ is
(31)r=21−μN−Mcosλ+1−μ2N−M2cos2λ+41−μN+μMMcos2λ+M2sin2λ,ifM≠0,11−μNcosλ,ifM=0.This radius is sharp because the extremal function is(32)hα,βω=ω1+Mωe−iλ1−μN−Mcosλ/M,M≠0,ωe1−μNcosλe−iλω,M=0.Proof.
Since(33)eiλωh′ω/hα,βω−isinλcosλ=pω,p∈PN,M,μ,
using Lemma4, that is
(34)pω−1−M1−μN+μMr21−M2r2≤1−μN−Mr1−M2r2.
Using (33) in (34) and after straightforward calculations, we get
(35)1−1−μN−Mcosλr−1−μN+μMMcos2λ+M2sin2λr21−M2r2,ifM≠0,1−1−μNcosλr,ifM=0,≤Rωh′ωhα,βω≤1+1−μN−Mcosλr−1−μN+μMMcos2λ+M2sin2λr21−M2r2,ifM≠0,1+1−μNcosλr,ifM=0,
whereω≤r<1. The above inequalities shows that this theorem is true.Remark 11.
(i)
ForN=−M=1,λ=μ=0, we obtain r=1(ii)
ForN=−M=1,μ=0, we obtain r=1/cosλ+sinλWe also note that if we give another special values toN,M,α,β, and μ, we obtain radius of starlikeness of the subclass of λ-spirallike functions.Corollary 12.
Ifh∈Sα,βλN,M,μ, then
(36)1−1−μN+μMrcosλ−1−Mrsinλ1−Mr,ifM≠0,1−1−μNrcosλ−sinλ,ifM=0,≤ωh′ωhα,βω≤1+1−μN+μMrcosλ+1+Mrsinλ1+Mr,ifM≠0,1+1−μNrcosλ+sinλ,ifM=0,
whereω≤r<1.Proof.
For an arbitrary functionh∈Sα,βλN,M,μ, we have
(37)1cosλeiλωh′ωhα,βω−isinλ=pω,p∈PN,M,μ.
Using Lemma5 and after the straightforward calculations, we get the result.Theorem 13.
ForM≥0, if h∈Sα,βλN,M,μ, then
(38)cosλ1−1−μN+μMr1−Mr−sinλ1−Msωcosλe−iλ1−μN−M/M,ifM≠0,cosλ1−1−μNr−sinλexp−cosλ1−μNr,ifM=0,≤h′ω≤cosλ1+1−μN+μMr1+Mr+sinλ1+Msωcosλe−iλ1−μN−M/M,ifM≠0,cosλ1+1−μNr+sinλexpcosλ1−μNr,ifM=0,
whereω≤r<1.Proof.
For functionh∈Sα,βλN,M,μ, according to Lemma 7, we have to distinguish the next two cases.
(i)
ForM≠0, then there exists a function s∈Ω such that hα,βω=ω1+Msωcosλe−iλ1−μN−M/M, and by Lemma 5, for, ω≤r<1, we get(39)cosλ1−1−μN+μMr1−Mr−sinλ1+Msωcosλe−iλ1−μN−M/M≤h′ω≤cosλ1+1−μN+μMr1+Mr+sinλ1+Msωcosλe−iλ1−μN−M/M.
Sinces∈Ω, we have
(40)1−Mr≤1+Msω≤1+Mr,ω≤r<1.
ForM>0, we can easily use the fact that −1≤M<N≤1,0≤μ<1 and ω≤r<1; we have
(41)1−Mrcosλe−iλ1−μN−M/M≤1+Msωcosλe−iλ1−μN−M/M≤1+Mrcosλe−iλ1−μN−M/M,
and from (39), we obtain
(42)cosλ1−1−μN+μMr1−Mr−sinλ1−Msωcosλe−iλ1−μN−M/M≤h′ω≤cosλ1+1−μN+μMr1+Mr+sinλ1+Msωcosλe−iλ1−μN−M/M.(ii)
IfM=0, there exists a function s∈Ω such that hα,βω=ωexpcosλe−iλ1−μNsω, ,ω≤r<1 and therefore(43)cosλ1−1−μNsω−sinλexpcosλe−iλ1−νNsω≤h′ω≤cosλ1+1−μNsω+sinλexpcosλe−iλ1−μNsω.
Since(44)expcosλe−iλ1−μNsω=expcosλ1−μNRee−iλsω,ω∈Δ,
using a similar computation as in the previous case, we deduce(45)exp−cosλ1−μNr≤expcosλe−iλ1−μNsω≤expcosλ1−μNr,ω≤r<1.
Thus, (43) yields to
(46)cosλ1−1−μNr−sinλexp−cosλ1−μNr≤h′ω≤cosλ1+1−μNr+sinλexpcosλ1−μNr,
forω≤r<1, which completes the proof of our theorem.



---

*Source: 9004239-2022-06-30.xml*