# Mapping Properties of Some Classes of Analytic Functions under Certain Integral Operators

Authors: Serap Bulut
Journal: Journal of Mathematics
 (2013)

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## Abstract

We consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under certain integral operators.


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## Body

## 1. Introduction

Let𝒜 be the class of all functions of the following form:
(1)f(z)=z+∑k=2∞akzk
which are analytic in the open unit disc
(2)𝕌={z∈ℂ:|z|<1}.A functionf∈𝒜 is said to be spiral-like if there exists a real number λ(|λ|<π/2) such that
(3)Re{eiλzf′(z)f(z)}>0(z∈𝕌).
The class of all spiral-like functions was introduced by Spacek [1] in 1933 and we denote it by 𝒮λ*. Later in 1969, Robertson [2] considered the class 𝒞λ of analytic functions in 𝕌 for which zf′(z)∈𝒮λ*.Let𝒫kλ(ρ) be the class of functions p(z) analytic in 𝕌 with p(0)=1 and
(4)∫02π‍|Reeiλp(z)-ρcosλ1-ρ|dθ≤kπcosλ,z=reiθ,
where k≥2,0≤ρ<1,λ is real with |λ|<π/2.Forλ=0, this class was introduced in [3] and for ρ=0, see [4]. For k=2,λ=0 and ρ=0, the class 𝒫kλ(ρ) reduces to the class 𝒫 of functions p(z) analytic in 𝕌 with p(0)=1 and whose real part is positive.The following definition of fractional derivative by Owa [5] (also by Srivastava and Owa [6]) will be required in our investigation.The fractional derivative of orderγ is defined, for a function f, by
(5)Dzγf(z)=1Γ(1-γ)ddz∫0zf(ξ)(z-ξ)γdξ(0≤γ<1),
where the function f is analytic in a simply connected region of the complex z-plane containing the origin, and the multiplicity of (z-ξ)-γ is removed by requiring log(z-ξ) to be real when z-ξ>0.It readily follows from (5) that
(6)Dzγzk=Γ(k+1)Γ(k+1-γ)zk-γ(0≤γ<1,k∈ℕ={1,2,…}).UsingDzγf, Owa and Srivastava [7] introduced the operator Ωγ:𝒜→𝒜, which is known as an extension of fractional derivative and fractional integral, as follows:
(7)Ωγf(z)=Γ(2-γ)zγDzγf(z)=z+∑k=2∞Γ(k+1)Γ(2-γ)Γ(k+1-γ)akzk.
Note that
(8)Ω0f(z)=f(z).In [8], Al-Oboudi and Al-Amoudi defined the linear multiplier fractional differential operator (namely, generalized Al-Oboudi differential operator) Dδn,γ as follows:
(9)D0f(z)=f(z),Dδ1,γf(z)=(1-δ)Ωγf(z)+δz(Ωγf(z))′Dδ1,γf(z)=Dδγ(f(z)),δ≥0,0≤γ<1,Dδ2,γf(z)=Dδγ(Dδ1,γf(z)),⋮Dδn,γf(z)=Dδγ(Dδn-1,γf(z)),n∈ℕ.Iff is given by (1), then by (7) and (9), we see that
(10)Dδn,γf(z)=z+∑k=2∞Ψk,n(γ,δ)akzk,n∈ℕ0=ℕ∪{0},
where
(11)Ψk,n(γ,δ)=[Γ(k+1)Γ(2-γ)Γ(k+1-γ)(1+(k-1)δ)]n.Remark 1.
(i) Whenγ=0, we get Al-Oboudi differential operator [9].
(ii) Whenγ=0 and δ=1, we get Sălăgean differential operator [10].
(iii) Whenn=1 and δ=0, we get Owa-Srivastava fractional differential operator [7].Definition 2.
A functionf∈𝒜 is said to belong to the class ℛkλ(ρ,b;n,γ,δ) if and only if
(12)1+1b(z(Dδn,γf(z))′Dδn,γf(z)-1)∈𝒫kλ(ρ),
where k≥2,0≤ρ<1,λ is real with |λ|<π/2,b∈ℂ-{0} and D is the generalized Al-Oboudi differential operator.Definition 3.
A functionf∈𝒜 is said to belong to the class 𝒱kλ(ρ,b;n,γ,δ) if and only if
(13)1+1bz(Dδn,γf(z))′′(Dδn,γf(z))′∈𝒫kλ(ρ),
where k≥2,0≤ρ<1,λ is real with |λ|<π/2,b∈ℂ-{0} and D is the generalized Al-Oboudi differential operator.Remark 4.
(i) Lettingγ=0 and b=1 in Definition 2, we have the class 𝒱kλ(ρ,δ,n) introduced by Dileep and Latha [11].
(ii) Forn=0 and b=1, we obtain the classes ℛkλ(ρ) and 𝒱kλ(ρ), respectively, introduced and studied by Noor et al. [12] and Moulis [13].
(iii) Forλ=0 and n=0, we have the classes ℛk(ρ,b) and 𝒱k(ρ,b), respectively, introduced and studied by Noor et al. [14].
(iv) Fork=2, λ=0 and n=0, we have the classes 𝒮ρ*(b) and 𝒞ρ(b), respectively, introduced by Frasin [15].Definition 5.
Letn∈ℕ0, l=(l1,…,lm)∈ℕ0m, and kj>0(1≤j≤m). One defines the integral operator In,m,l,k:𝒜m→𝒜 as
(14)In,m,l,k(f1,…,fm)=F,(15)Dδn,γF(z)=∫0z‍(Dδl1,γf1(t)t)k1⋯(Dδlm,γfm(t)t)kmdt(z∈𝕌),
where f1,…,fm∈𝒜 and D is the generalized Al-Oboudi differential operator.Remark 6.
The integral operatorDδn,γF generalizes many operators which were introduced and studied recently.(i) Forγ=0, we have the integral operator
(16)DδnF(z)=∫0z‍(Dδl1f1(t)t)k1⋯(Dδlmfm(t)t)kmdt
introduced by Bulut [16]. Here D is the Al-Oboudi differential operator.(ii) Forn=0, γ=0 and l1=⋯=lm=l∈ℕ0, we have the integral operator
(17)F(z)=∫0z‍(Dδlf1(t)t)k1⋯(Dδlfm(t)t)kmdt
introduced by Bulut [17]. Here D is the Al-Oboudi differential operator.(iii) Forγ=0 and λ=1, we have the integral operator
(18)DnF(z)=∫0z‍(Dl1f1(t)t)k1⋯(Dlmfm(t)t)kmdt
introduced by Breaz et al. [18]. Here D is the Sălăgean differential operator.(iv) Forn=0 and Dδ0,γfj=D01,0fj=fj∈𝒜(1≤j≤m), we have the integral operator
(19)Fm(z)=∫0z‍(f1(t)t)k1⋯(fm(t)t)kmdt
introduced by D. Breaz and N. Breaz [19].(v) Forn=0, m=1, k1=k∈[0,1], k2=⋯=km=0 and Dδ0,γf1=D01,0f1=f∈𝒮*(consists of functions that are analytic, univalent and starlike), we have the integral operator
(20)Fk(z)=∫0z‍(f(t)t)kdt
studied by Miller et al. [20].(vi) Forn=0, m=1, k1=1, k2=⋯=km=0 and Dδ0,γf1=D01,0f1=f∈𝒜, we have the integral operator of Alexander
(21)F(z)=∫0z‍f(t)tdt
introduced by Alexander [21].Definition 7.
Letn∈ℕ0, l=(l1,…,lm)∈ℕ0m, and kj>0(1≤j≤m). One defines the integral operator Jn,m,l,k:𝒜m→𝒜 as
(22)Jn,m,l,k(f1,…,fm)=G,(23)Dδn,γG(z)=∫0z‍[(Dδl1,γf1(t))′]k1⋯[(Dδlm,γf1(t))′]kmdt(z∈𝕌),
where f1,…,fm∈𝒜 and D is the generalized Al-Oboudi differential operator.Remark 8.
The integral operatorDδn,γG generalizes many operators which were introduced and studied recently.(i)
Forn=0 and Dδ0,γfj=D01,0fj=fj∈𝒜(1≤j≤m), we have the integral operator
(24)G(z)=∫0z‍(f1′(t))k1⋯(fm′(t))kmdt
introduced by Breaz et al. [22].(ii)
Forn=0, m=1, k1=k∈ℂ, k2=⋯=km=0 and Dδ0,γf1=D01,0f1=f∈𝒜, we have the integral operator
(25)Gk(z)=∫0z‍(f′(t))kdt
introduced by Pfaltzgraff [23] (see also Pascu and Pescar [24]).In this paper, we investigate some propeties of the above integral operatorsDδn,γF and Dδn,γG for the classes
(26)ℛkλ(ρ,b;n,γ,δ),𝒱kλ(ρ,b;n,γ,δ).


## 2. Main Results

Theorem 9.
Letfj∈ℛkλ(ρj,b;n,γ,δ) for 1≤j≤m with 0≤ρj<1,b∈ℂ-{0}. Also let λ be real with |λ|<π/2,kj>0(1≤j≤m). If
(27)0≤1+∑j=1m‍kj(ρj-1)<1,
then the integral operator F defined by (15) is in the class 𝒱kλ(η,b;n,γ,δ) with
(28)η=1+∑j=1mkj(ρj-1).Proof.
Sincefj∈𝒜(1≤j≤m), by (10), we have
(29)Dδlj,γfj(z)z=1+∑k=2∞Ψk,lj(γ,δ)ak,jzk-1,Dδlj,γfj(z)z≠0
for all z∈𝕌. By (15), we get
(30)(Dδn,γF(z))′=(Dδl1,γf1(z)z)k1⋯(Dδlm,γfm(z)z)km.
This equality implies that
(31)ln(Dδn,γF(z))′=k1lnDδl1,γf1(z)z+⋯+kmlnDδlm,γfm(z)z
or equivalently
(32)ln(Dδn,γF(z))′=k1[lnDδl1,γf1(z)-lnz]+⋯+km[lnDδlm,γfm(z)-lnz].
By differentiating the above equality, we get
(33)(Dδn,γF(z))′′(Dδn,γF(z))′=∑j=1mkj[(Dδlj,γfj(z))′Dδlj,γfj(z)-1z].
Hence, we obtain from this equality that
(34)z(Dδn,γF(z))′′(Dδn,γF(z))′=∑j=1mkj(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1).
Then by multiplying the above relation with 1/b, we have
(35)1bz(Dδn,γF(z))′′(Dδn,γF(z))′=∑j=1mkj1b(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)=∑j=1mkj[1+1b(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)]-∑j=1mkj
or equivalently
(36)eiλ(1+1bz(Dδn,γF(z))′′(Dδn,γF(z))′)=(1-∑j=1mkj)eiλ+∑j=1mkjeiλ[1+1b(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)].
Subtracting and adding cosλ∑j=1m‍kjρj on the left hand side and then taking real part, we have
(37)Re{eiλ(1+1bz(Dδn,γF(z))′′(Dδn,γF(z))′)-ηcosλ}=∑j=1mkjRe{eiλ[1+1b(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)]a-ρjcosλz(Dδlj,γfj(z))′Dδlj,γfj(z)},
where η is given by (28). Integrating (37) and then using (28), we have
(38)∫02π‍|Re{eiλ(1+1bz(Dδn,γF(z))′′(Dδn,γF(z))′)-ηcosλ}|dθ≤∑j=1mkj∫02π‍|Re{eiλ[1+1b(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)]aaaaaaaaaiaaaaaaa-ρjcosλ(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)}|dθ.
Since fj∈ℛkλ(ρj,b;n,γ,δ)(1≤j≤m), we get
(39)∫02π‍|Re{eiλ[1+1b(z(Dδlj,γfj(z))′Dδlj,γfj(z)-1)]asdddd-ρjcosλz(Dδlj,γfj(z))′Dδlj,γfj(z)}|dθ≤(1-ρj)kπcosλ
for 1≤j≤m. Using (39) in (38), we obtain
(40)∫02π‍|Re{eiλ(1+1bz(Dδn,γF(z))′′(Dδn,γF(z))′)-ηcosλ}|dθ≤kπcosλ∑j=1mkj(1-ρj).
Hence, we obtain F∈𝒱kλ(η,b;n,γ,δ) with η is given by (28).By settingn=0, γ=0, b=1, l1=⋯=lm=l in Theorem 9, we obtain the following.Corollary 10 (see [11, Theorem 1]).
Letfj∈𝒱kλ(ρj,δ,n) for 1≤j≤m with 0≤ρj<1. Also let λ be real with |λ|<π/2,kj>0(1≤j≤m). If
(41)0≤1+∑j=1mkj(ρj-1)<1,
then the integral operator F(z) defined by (17) is in the class 𝒱kλ(η) with
(42)η=1+∑j=1mkj(ρj-1).Remark 11.
Lettingρ1=ρ2=⋯=ρm=ρ in Corollary 10, then we have [12, Theorem 3.1].By settingn=0, λ=0 and Dδ0,γ=D01,0 in Theorem 9, we obtain the following.Corollary 12.
Letfj∈ℛk(ρj,b) for 1≤j≤m with 0≤ρj<1,b∈ℂ-{0}. Also let kj>0(1≤j≤m). If
(43)0≤1+∑j=1mkj(ρj-1)<1,
then the integral operator Fm(z) defined by (19) is in the class 𝒱k(η,b) with
(44)η=1+∑j=1mkj(ρj-1).Remark 13.
In Corollary12, letting(i)
ρ1=ρ2=⋯=ρm=ρ, we have [14, Theorem 2.1],(ii)
k=2, we have [25, Theorem 1].Theorem 14.
Letfj∈𝒱kλ(ρj,b;n,γ,δ) for 1≤j≤m with 0≤ρj<1,b∈ℂ-{0}. Also let λ be real with |λ|<π/2,kj>0(1≤j≤m). If
(45)0≤1+∑j=1mkj(ρj-1)<1,
then the integral operator G defined by (23) is in the class 𝒱kλ(μ,b;n,γ,δ) with
(46)μ=1+∑j=1mkj(ρj-1).Proof.
By (23), we get
(47)(Dδn,γG(z))′=[(Dδl1,γf1(z))′]k1⋯[(Dδlm,γf1(z))′]km.
This equality implies that
(48)(Dδn,γG(z))′′=∑j=1mkj[(Dδlj,γfj(z))′]kj(Dδlj,γfj(z))′′(Dδlj,γfj(z))′×∏r=1(r≠j)m[(Dδlr,γfr(z))′]kr.
Thus by using (47) and (48), we obtain
(49)z(Dδn,γG(z))′′(Dδn,γG(z))′=∑j=1mkjz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′.
Then by multiplying the above relation with 1/b, we have
(50)1bz(Dδn,γG(z))′′(Dδn,γG(z))′=∑j=1mkj1bz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′=∑j=1mkj(1+1bz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′)-∑j=1nkj
or equivalently
(51)eiλ(1+1bz(Dδn,γG(z))′′(Dδn,γG(z))′)=(1-∑j=1mkj)eiλ+∑j=1mkjeiλ(1+1bz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′).
Subtracting and adding cosλ∑j=1m‍kjρj on the left hand side and then taking real part, we have
(52)Re{eiλ(1+1bz(Dδn,γG(z))′′(Dδn,γG(z))′)-μcosλ}=∑j=1mkjRe{eiλ(1+1bz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′)-ρjcosλ},
where μ is given by (46). Integrating (52) and then using (46), we have
(53)∫02π‍|Re{eiλ(1+1bz(Dδn,γG(z))′′(Dδn,γG(z))′)-μcosλ}|dθ≤∑j=1mkj∫02π|Re{eiλ(1+1bz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′)‍asai-ρjcosλz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′}|dθ.
Since fj∈𝒱kλ(ρj,b;n,γ,δ)(1≤j≤m), we get
(54)∫02π‍|Re{eiλ(1+1bz(Dδlj,γfj(z))′′(Dδlj,γfj(z))′)-ρjcosλ}|dθ≤(1-ρj)kπcosλ
for 1≤j≤m. Using (54) in (53), we obtain
(55)∫02π‍|Re{eiλ(1+1bz(Dδn,γG(z))′′(Dδn,γG(z))′)-μcosλ}|dθ≤kπcosλ∑j=1mkj(1-ρj).
Hence, we obtain G∈𝒱kλ(μ,b;n,γ,δ) with μ given by (46).By settingn=0, λ=0 and Dδ0,γ=D01,0 in Theorem 14, we obtain the following.Corollary 15.
Letfj∈𝒱k(ρj,b) for 1≤j≤m with 0≤ρj<1,b∈ℂ-{0}. Also let kj>0(1≤j≤m). If
(56)0≤1+∑j=1mkj(ρj-1)<1,
then the integral operator G(z) defined by (24) is in the class 𝒱k(μ,b) with
(57)μ=1+∑j=1mkj(ρj-1).Remark 16.
In Corollary15, letting(i)
ρ1=ρ2=⋯=ρm=ρ, we have [14, Theorem 2.5],(ii)
k=2, we have [25, Theorem 3].



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