# On Certain Subclasses of Analytic Multivalent Functions Using Generalized Salagean Operator

Authors: Adnan Ghazy Alamoush; Maslina Darus
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

We introduce and study two subclasses of multivalent functions denoted byMp,nm,α,β,σ(λ1;λ2) and Np,nm,α,β,σ(μ,δ;γ). Further, by using the method of differential subordination, certain inclusion relations between the two subclasses aforementioned are given. Moreover, several consequences of the main results are also discussed.


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## Body

## 1. Introduction

LetA(p,n) denote the class of the functions f of the form(1)fz=zp+∑k=p+n∞akzk,n,p∈N=1,2,3,…,which are analytic in the open unit disc U={z∈C:|z|<1}, and let denote A≔A(1,1).A functionf∈A(p,n) is said to be multivalent starlike functions of order α in U, if it satisfies the following inequality:(2)Rzf′zfz>α,z∈U,0≤α<p,p∈N,and we denote this class by Sp,n∗(α).A functionf∈A(p,n) is said to be multivalent convex functions of order α in U, if it satisfies the following inequality:(3)R1+zf′′zf′z>α,z∈U,0≤α<p,p∈N,and we denote this class by Cp,n(α).For a functionf∈A(p,n), Goyal et al. [1] introduced the following generalized Salagean differential operator:(4)Dσ0fz=fz,(5)Dσ1fz=Dσfz=1-σfz+σpzf′z,σ≥0,(6)Dσmfz=DσDσm-1fz,m∈N.If f is given by (1), then from (5) and (6) we have(7)Dσmfz=zp+∑k=p+n∞1+kp-1σmakzk.Remark 1.
Forσ=p=1, the differential operator Dσmf(z) reduces to Salagean differential operator Dmf(z) [2].Definition 2.
LetMp,nm,α,β,σ(λ1;λ2) be the class of functions f∈A(p,n) that satisfy the condition(8)R1-λ1zDσα,β,mfz′Dσα,β,mfz+λ11+zDσα,β,mfz′′Dσα,β,mfz′>λ2,z∈U,where (0≤σ,0≤α<β≤1,λ1∈R,0≤λ2<p,m,p∈N), and let Np,nm,α,β,σ(μ,δ;γ) be the class of functions f∈A(p,n) that satisfy the conditions(9)Dσα,β,mfzDσα,β,mfz′z2p-1≠0,z∈U,RDσα,β,mfzzpμDσα,β,mfz′zp-1δ>γ,z∈U,0≤σ,δ,μ∈R,0≤α<β≤1,0≤γ<pδα+βpδ+μ,m,p∈N,where(10)Dσα,β,mfz=αDσmfz+βzDσmfz′.Remark 3.
By specifying different values, we have some well-known subclasses of the classesA(p,n) and A(n)=A(1,n) appearing from the families of the classes Mp,nm,α,β,σ(λ1;λ2) and Np,nm,α,β,σ(μ,δ;γ).(i)
Mp,n0,1,0,σ(0;λ1)=Np,n0,1,0,σ(-1,1,λ1)=Sp,n∗,(0≤λ1<p) is the class of multivalent starlike functions of order λ1.(ii)
M1,n0,1,0,σ(0;λ1)=N1,n0,1,0,σ(-1,1,λ1)=S1,n∗=Sn∗,(0≤λ1<1) is the class of starlike functions of order λ1.(iii)
Mp,n1,0,1,1(0;λ1)=Cp,n(λ1),(0≤λ1<p) is the class of multivalent convex functions of order λ1.(iv)
M1,n1,0,1,1(0;λ1)=C1,n(λ1)=Cn(λ1),(0≤λ1<1) is the class of convex functions of order γ.(v)
N1,n1,1,0,1(1,δ;λ1)=Bn(δ;λ1),(δ≥-1,0≤λ1<1) is the subclass of Bazilević functions.LetH[a,n] be denoted by the class(11)Ha,n=h∈HU:hz=a+anzn+⋯,z∈U.In this investigation, we focus on certain inequalities consisting of the following differential operator Jp,nm,α,β,σ(μ,δ):A(p,n)→H[(μ+δ),n+p]:(12)Jp,nm,α,β,σμ,δfz=μzDσα,β,mfz′Dσα,β,mfz+δ1+zDσα,β,mfz′′Dσα,β,mfz′that generalizes the expression used in the definition of class Mp,nm,α,β,σ(λ1;λ2) and we receive several properties of the expression(13)Dσα,β,mfzzpμDσα,β,mfz′zp-1δ,z∈U,including relations between classes Mp,nm,α,β,σ(λ1;λ2) and Np,nm,α,β,σ(μ,δ;γ).In order to prove our main results, we will need the following lemmas due to Miller and Mocanu [3].Lemma 4.
LetΩ⊂C and suppose that the function ψ:C2×U→C satisfies ψ(Meiθ,Keiθ;z)∉Ω for all K≥Mn,θ∈R, and z∈U. If h(z)=a+hnzn+⋯ is analytic in U and ψ(h(z),zh′(z);z)∈Ω for all z∈U, then |h(z)|<M,z∈U.Lemma 5.
LetΩ⊂C and suppose that the function ψ:C2×U→C satisfies ψ(ix,y;z)∉Ω for all x∈R,y≤-n(1+x2)/2, and z∈U. If h(z)=a+hnzn+⋯ is analytic in U and ψ(h(z),zh′(z);z)∈Ω for all z∈U, then Rh(z)>0,z∈U.


## 2. Main Results

Following the same techniques and procedure given by Goswami et al. [4], we have the following results.Theorem 6.
Letf(z)∈A(p,n) with (Dσα,β,mf(z))(Dσα,β,mf(z))′/z2p-1≠0 for all z∈U, where Dσα,β,m is given by (10), and also let μ,δ∈R. If(14)RJp,nm,α,β,σμ,δfz<pδ+μ+nMM+pδα+βpδ+μ,z∈U,where pδ(α+βp)δ+μ≤M, then(15)Dσα,β,mfzzpμDσα,β,mfz′zp-1δ-pδα+βpδ+μ<M,z∈U,where the powers are the principal ones.Proof.
Let the functionh(z) be defined by(16)hz=Dσα,β,mfzzpμDσα,β,mfz′zp-1δ-pδα+βpδ+μ.From the assumptions f∈A(p,n) with (Dσα,β,mf(z))(Dσα,β,mf(z))′/z2p-1≠0 for all z∈U, we have that h∈H[0,n]. By a simple manipulation, we have(17)Jp,nm,α,β,σμ,δfz=pδ+μ+zh′zhz+pδα+βpδ+μ.Now letting(18)ψr,s,z=pδ+μ+sr+pδα+βpδ+μ,Ω=w∈C:Rw<pδ+μ+nMM+pδα+βpδ+μ,we have from (17) and (14) that(19)ψhz,zh′z;z=Jp,nm,α,β,σμ,δfz∈Ω∀z∈U.Further, for any θ∈R, K≥nM, and z∈U, since M≥pδ(α+βp)δ+μ, we also have(20)RψMeiθ,Keiθ;z=pδ+μ+KR1M+e-iθpδα+βpδ+μ≥pδ+μ+nMM+pδα+βpδ+μ,z∈U,which shows that ψ(Meiθ,Keiθ;z)∉Ω for all θ∈R, K≥nM, and z∈U. Therefore, according to Lemma 4, we obtain h(z)<M(z∈U). Hence, (15) is proven.Theorem 7.
Letf(z)∈A(p,n) with (Dσα,β,mf(z))(Dσα,β,mf(z))′/z2p-1≠0 for all z∈U, where Dσα,β,m is given by (10), and also let μ,δ∈R. If(21)RJp,nm,α,β,σμ,δfz>kμ,δ,α,β;γ,z∈U,where γ∈[0,pδ(α+βp)δ+μ) and(22)kμ,δ,α,β;γ=pδ+μ-nγ2pδα+βpδ+μ-γ,ifγ∈0,pδα+βpδ+μ2pδ+μ-npδα+βpδ+μ-γ2γ,ifγ∈pδα+βpδ+μ2,pδα+βpδ+μ,then f∈Np,nm,σ,α,β(μ,δ;γ).Proof.
Suppose that(23)hz=1pδα+βpδ+μ-γDσα,β,mfzzpμ·Dσα,β,mfz′zp-1δ-γ.Then, h(z)=1+hnzn+⋯ is analytic in U. It is easily seen from (23) that(24)Jp,nm,α,β,σμ,δfz=pδ+μ+pδα+βpδ+μ-γzh′zpδα+βpδ+μ-γhz+γ.Further, since (25)ψr,s;z=pδ+μ+pδα+βpδ+μ-γspδα+βpδ+μ-γr+γ,Ω=w∈C:Rw>kμ,δ,α,β;γ,it leads to (26)ψhz,zh′z;z=Jp,nm,α,β,σμ,δfz∈Ω∀z∈U.Also, for any x∈R, y≤-n(1+x2)/2 and z∈U, we have(27)Rψix,y;z=pδ+μ+γpδα+βpδ+μ-γypδα+βpδ+μ-γ2x2+γ2≤pδ+μ-nγpδα+βpδ+μ-γ21+x2pδα+βpδ+μ-γ2x2+γ2≡qz≤kμ,δ,α,β;γ=limx→∞⁡qz,ifγ∈0,pδα+βpδ+μ2q0,ifγ∈pδα+βpδ+μ2,pδα+βpδ+μ;that is, ψ(ix,y;z)∉Ω. Finally, by Lemma 5, we obtain that Re(h(z))>0. The proof of Theorem 7 is complete.


## 3. Corollaries and Consequences

We will discuss some interesting consequences of the main theorems that extend some previous results obtained in ([4, 5]).Puttingα=1,β=0 in Theorems 6 and 7, we get the following corollaries.Corollary 8.
Letf(z)∈A(p,n) with (Dσmf(z))(Dσmf(z))′/z2p-1≠0 for all z∈U, where Dσm is given by (7), and also let μ,δ∈R. If(28)RμzDσmfz′Dσmfz+δ1+zDσmfz′′Dσmfz′<pδ+μ+nMM+pδ,z∈U,where pδ≤M, then(29)DσmfzzpμDσmfz′zp-1δ-pδ<M,z∈U,where the powers are the principal ones.Corollary 9.
Letf(z)∈A(p,n) with (Dσmf(z))(Dσmf(z))′/z2p-1≠0 for all z∈U, where Dσm is given by (7), and also let μ,δ∈R. If(30)RμzDσmfz′Dσmfz+δ1+zDσmfz′′Dσmfz′>φμ,δ;γ,z∈U,where γ∈[0,pδ) and(31)φμ,δ;γ=kμ,δ,1,0;γ=pδ+μ-nγ2pδ-γ,ifγ∈0,pδ2pδ+μ-npδ-γ2γ,ifγ∈pδ2,pδ,then(32)RDσmfzzpμDσmfz′zp-1δ>γ,z∈U,where the powers are the principal ones.Takingμ=1-λ1 and δ=λ1 in Corollaries 8 and 9, respectively, we obtain the following special cases.Corollary 10.
Letf(z)∈A(p,n) with (Dσmf(z))(Dσmf(z))′/z2p-1≠0 for all z∈U, where Dσm is given by (7), and also let λ1∈R. If(33)R1-λ1zDσmfz′Dσmfz+λ11+zDσmfz′′Dσmfz′<p+nMM+pλ1,z∈U,where pλ1≤M, then(34)Dσmfzzp1-λ1Dσmfz′zp-1λ1-pλ1<M,z∈U,where the powers are the principal ones.Corollary 11.
Letf(z)∈A(p,n) with (Dσmf(z))(Dσmf(z))′/z2p-1≠0 for all z∈U, where Dσm is given by (7), and also let λ1∈R. If(35)R1-λ1zDσmfz′Dσmfz+λ11+zDσmfz′′Dσmfz′>χλ1;γ,z∈U,where γ∈[0,pλ1) and(36)χλ1;γ=k1-λ1,λ1,1,0;γ=p-nγ2pλ1-γ,ifγ∈0,pλ12p-npλ1-γ2γ,ifγ∈pλ12,pλ1,then(37)RDσmfzzp1-λ1Dσmfz′zp-1λ1>γ,z∈U,where the powers are the principal ones.Next, upon takingα=0, β=1 in Theorems 6 and 7, we obtain the following results.Corollary 12.
Letf(z)∈A(p,n) with (Dσmf(z))′[(Dσmf(z))′+z(Dσmf(z))′′]/z2(p-1)≠0 for all z∈U, where Dσm is given by (7), and also let μ,δ∈R. If(38)RJp,nm,0,1,σμ,δfz<pδ+μ+nMM+p2δ+μ,z∈U,where p2δ+μ≤M, then(39)Dσmfz′zp-1μDσmfz′+zDσmfz′′zp-1δ-p2δ+μ<M,z∈U,where the powers are the principal ones.Corollary 13.
Letf(z)∈A(p,n) with (Dσmf(z))′[(Dσmf(z))′+z(Dσmfz)′′]/z2(p-1)≠0 for all z∈U, where Dσm is given by (7), and also let μ,δ∈R. If(40)RJp,nm,0,1,σμ,δfz>ϕμ,δ;γ,z∈U,where γ∈[0,p2δ+μ) and(41)ϕμ,δ;γ=kμ,δ,0,1;γ=pδ+μ-nγ2p2δ+μ-γ,ifγ∈0,p2δ+μ2pδ+μ-np2δ+μ-γ2γ,ifγ∈p2δ+μ2,p2δ+μ,then(42)RDσmfz′zp-1μ·Dσmfz′+zDσmfz′′zp-1δ>γ,z∈U,where the powers are the principal ones.Takingμ=1-λ1 and δ=λ1 in Corollaries 12 and 13, respectively, we obtain the following special cases.Corollary 14.
Letf(z)∈A(p,n) with (Dσmf(z))′[(Dσmf(z))′+z(Dσmf(z))′′]/z2(p-1)≠0 for all z∈U, where Dσm is given by (7), and also let λ1∈R. If(43)RJp,nm,0,1,σ1-λ1,λ1fz<p+nMM+pλ1+1,z∈U,where pλ1+1≤M, then(44)Dσmfz′zp-11-λ1·Dσmfz′+zDσmfz′′zp-1λ1-pλ1+1<M,z∈U,where the powers are the principal ones.Corollary 15.
Letf(z)∈A(p,n) with (Dσmf(z))′[(Dσmf(z))′+z(Dσmfz)′′]/z2(p-1)≠0 for all z∈U, where Dσm is given by (7), and also let λ1∈R. If(45)RJp,nm,0,1,σ1-λ1,λ1fz>ψλ1;γ,z∈U,where γ∈[0,pλ1+1) and(46)ψλ1;γ=k1-λ1,λ1,0,1;γ=pδ+μ-nγ2pλ1+1-γ,ifγ∈0,pλ1+12pδ+μ-np2δ+μ-γ2γ,ifγ∈pλ1+12,pλ1+1,then(47)RDσmfz′zp-11-λ1·Dσmfz′+zDσmfz′′zp-1λ1>γ,z∈U,where the powers are the principal ones.In the next result, we will find the relation betweenMp,nm,α,β,σ(λ1;γ) and Mp,nm,α,β,σ(1-λ1,λ1;γ). For this purpose, taking μ=1-λ1 and δ=λ1 in Theorem 7, we obtain the following result.Corollary 16.
Letf(z)∈A(p,n) with (Dσα,β,mf(z))(Dσα,β,mf(z))′/z2p-1≠0 for all z∈U, where Dσα,β,m is given by (10), and also let λ1∈R. If(48)fz∈Mp,nm,α,β,σλ1;ϱλ1,α,β;γ,where γ∈[0,pλ1(α+βp)) and(49)ϱλ1,α,β;γ=k1-λ1,λ1,α,β;γ=p-nγ2pλ1α+βp-γ,ifγ∈0,pλ1α+βp2p-npλ1α+βp-γ2γ,ifγ∈pλ1α+βp2,pλ1α+βp,then f(z)∈Np,nm,α,β,σ(1-λ1,λ1;γ).Takingλ1=0 and n=1 in the above corollary, we get the next special result.Corollary 17.
Letf(z)∈A(p) with (Dσα,β,mf(z))(Dσα,β,mf(z))′/z2p-1≠0 for all z∈U, where Dσα,β,m is given by (10), and also let λ1∈R. If(50)fz∈Mpm,α,β,σϱα,β;γ,where γ∈[0,α+βp) and(51)ϱα,β;γ=k1,0,α,β;γ=p-γ2α+βp-γ,ifγ∈0,α+βp2p-α+βp-γ2γ,ifγ∈α+βp2,α+βp,then f(z)∈Npm,α,β,σ(1,0;γ).Again, for the special cases ofμ and δ, Theorems 6 and 7 reduce at once to some results obtained by [4, 5].Remark 18.
Takingp=1 and m=0 in (7) and α=1 and β=0 in (10), we get a known result obtained by Irmak et al. [5].Remark 19.
Takingm=0 in (7) and α=1-β in (10), we get a known result obtained by Goswami et al. [4].



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