# Certain Subclasses of Analytic Functions with Complex Order

Authors: A. Selvam; P. Sooriya Kala; N. Marikkannan
Journal: The Scientific World Journal
 (2013)

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## Abstract

Two new subclasses of analytic functions of complex order are introduced. Apart from establishing coefficient bounds for these classes, we establish inclusion relationships involving (n-δ) neighborhoods of analytic functions with negative coefficients belonging to these subclasses.


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## Body

## 1. Introduction

Let𝒜(n) denote the class of functions of the form
(1)f(z)=z-∑k=n+1∞‍akzk(ak≥0,n∈ℕ:={1,2,3,…}),
which are analytic and univalent in the open disc
(2)Δ={z∈ℂ:|z|<1}.A functionf(z)∈𝒜(n) is star-like of complex order b, denoted as f(z)∈S*(b) if and only if it satisfies
(3)ℜ{1+1b(zf′f-1)}>0(z∈Δ).A functionf(z)∈𝒜(n) is convex of complex order b, denoted as f(z)∈C(b) if and only if it satisfies
(4)ℜ{1+1b(zf′′f′)}>0(z∈Δ).
These classes, S*(b), and C(b) are introduced and studied by Nasr and Aouf [1] and Wiatrowski [2].For the two functionsfj(j=1,2) given by
(5)fj(z)=z+∑k=2∞‍ak,jzk(j=1,2),
the Hadamard product or convolution, denoted by (f1*f2)(z), is given by
(6)(f1*f2)(z)=z+∑k=2∞‍ak,1ak,2zk.Givenf(z) of the form (1) and δ≥0, we define n-δ neighborhood of a function f∈𝒜(n) as
(7)Nn,δ(f)={g∈𝒜(n)∣∑k=n+1∞‍k|ak-bk|≤δ}.
In particular, for the identity function e(z)=z,
(8)Nn,δ(e)={g∈𝒜(n)∣∑k=n+1∞‍k|bk|≤δ}.The concept of NeighborhoodNn,δ of a function f is introduced and studied by Ruscheweyh [3] and extended further by Silverman [4].For complex numbersα1,α2,…,αq and β1,β2,…,βs (βj∈ℂ∖𝒵0-; 𝒵0-={0,-1,-2,…} for j=1,2,…,s), we define the generalized hypergeometric function Fsq(α1,α2,…,αq;β1,β2,…,βs;z) as
(9)Fsq(α1,α2,…,αq;β1,β2,…,βs;z)=∑k=0∞‍(α1)k(α2)k⋯(αq)kzk(β1)k(β2)k⋯(βs)kk!,(q≤s+1;=∑k=0∞‍(α1)k(α2)k⋯(αq)kzk(β1)k(β2)k⋯(βs)kk!5.,q,s∈ℕ0:=ℕ∪{0};z∈U),
where ℕ denotes the set of all positive integers and (x)k is the Pochhammer symbol defined in terms of gamma functions as
(10)(x)k=Γ(x+k)Γ(x)={1ifk=0x(x+1)⋯(x+k-1)ifk∈ℕ.
Corresponding to the function gq,s(α1,β1;z) defined by
(11)gq,s(α1,β1;z)=zqFs(α1,α2,…,αq;β1,β2,…,βs;z)
recently in [5], an operator 𝒟λ,μm(α1,β1)f(z):𝒜→𝒜 is defined by
(12)𝒟λ,μ0(α1,β1)f(z)∶=f(z)*gq,s(α1,β1;z),𝒟λ,μ1(α1,β1)f(z)∶=(1-λ+μ)(f(z)*gq,s(α1,β1;z))+(λ-μ)z(f(z)*gq,s(α1,β1;z))′+λμz2(f(z)*gq,s(α1,β1;z))′′,𝒟λ,μm(α1,β1)f(z)∶=𝒟λ,μ1(𝒟λ,μm-1(α1,β1)f(z)),
where 0≤μ≤λ≤1 and m∈ℕ0. By the above definition, it is easy to note that
(13)𝒟λ,μm(α1,β1)f(z)=z+∑k=2∞‍[1+(k-1)(λ-μ+kμλ)]m×(α1)k-1(α2)k-1…(αq)k-1(β1)k-1(β2)k-1…(βs)k-1(k-1)!akzk.
Let us take for convenience that
(14)Bk=(α1)k-1(α2)k-1…(αq)k-1(β1)k-1(β2)k-1…(βs)k-1(k-1)!,Ck=1+(k-1)(λ-μ+kμλ).
Hence, we have
(15)𝒟λ,μm(α1,β1)f(z)=z+∑k=2∞‍CkmBkakzk.
For suitable values of αi′s, βj′s, q, s, m, λ, and μ, we can deduce several operators such as Sălăgean derivative operator [6], Ruscheweyh derivative operator [7], fractional calculus operator [8], Carlson-Shaffer operator [9], Dziok-Srivatsava operator [10], and also the operator introduced by Abubaker and Darus [11].Definition 1.
For0≤α≤1, we let A be the subclass of 𝒜(n) consisting of functions of the form (1) that satisfy
(16)|1b(11111+αz[𝒟λ,μm(α1,β1)f(z)]′)-1z[𝒟λ,μm(α1,β1)f(z)]′+αz[𝒟λ,μm(α1,β1)f(z)]′)-111111×([𝒟λ,μm(α1,β1)f(z)]′(1-α)𝒟λ,μm(α1,β1)f(z)11111+αz[𝒟λ,μm(α1,β1)f(z)]′)-1)|<γ,
where z∈Δ, b∈ℂ∖{0}, 0<γ≤1, and 𝒟λ,μm(α1,β1)f(z) are as given in (15).Definition 2.
For0≤α≤1 we let B be the subclass of 𝒜(n), consisting of functions of the form (1) that satisfy
(17)|1b((1-α)𝒟λ,μm(α1,β1)f(z)z𝒟λ,μm(α1,β1)f(z)z+α[𝒟λ,μm(α1,β1)f(z)]′-1)|<γ,
where z∈Δ, b∈ℂ∖{0}, 0<γ≤1, and 𝒟λ,μm(α1,β1)f(z) are as given in (15).By specializing the parameters involved in the above definitions, we could arrive at several known as well as new classes. For example, by takingλ=1, μ=0, q=2, s=1, α1=β1, and α2=1 and the above classes reduced to
(18)A1={f∈𝒜(n)∣|1b(1111111111111111+αz[𝔻mf(z)]′)-1z[𝔻mf(z)]′111111111111111×(111111111111+αz[𝔻mf(z)]′)-1(1-α)𝔻mf(z)11111111111111111111+αz[𝔻mf(z)]′)-1+αz[𝔻mf(z)]′)-1)|<γ},
where 𝔻mf(z) denote the Sălăgean derivative of order m given by
(19)𝔻mf(z)=z-∑k=n+1∞‍kmakzk,B1={f∈𝒜(n)∣|1b((1-α)𝔻mf(z)z111111111111111+α[𝒟mf(z)]′-1𝔻mf(z)z)|<γ}|1b((1-α)𝔻mf(z)z.Similarly, on takingq=2, s=1, α1=η-1, (η>-1), α2=1, β1=1, one gets
(20)A2={f∈𝒜(n)∣|1b(+αz[Dmf(z)]′)-1z[Dmf(z)]′×((1-α)Dmf(z)+αz[Dmf(z)]′)-1+αz[Dmf(z)]′)-1)|<γ},
where Dmf(z) is the operator introduced and studied by Abubaker and Darus [11] given by
(21)Dmf(z)=z-∑k=n+1∞‍Ckm(η+k-1k-1)zk,B2={f∈𝒜(n)∣|1b((1-α)Dmf(z)z111111111111111+α[Dmf(z)]′-1Dmf(z)z)|<γ}.Further, by takingm=0 in the definition of the classes A and B, we could arrive at Sn(q,s,α,b,γ) and Rn(q,s,α,b,γ) which were introduced and studied by Murugusundaramoorthy et al. [12].In this paper, we establish the coefficient inequalities for the classesA and B and several inclusion relationships involving n-δ neighborhoods of analytic univalent functions with negative and missing coefficients belonging to these classes.


## 2. Coefficient Inequalities

Theorem 3.
Let the functionf∈𝒜(n) as given in (1). Then, f∈A if and only if
(22)∑k=n+1∞{[1+α(k-1)](γ|b|-1)+k}CkmBkak≤γ|b|.‍Proof.
Let the functions of form (1) belong to the class A. Then, in view of (15) and (16), we get
(23)|∑k=n+1∞([1+α(k-1)]-k)CkmBkakzkz-∑k=n+1∞[1+α(k-1)]CkmBkakzk|<γ|b|,∑k=n+1∞([1+α(k-1)]-k)CkmBkakrk-11-∑k=n+1∞[1+α(k-1)]CkmBkakrk-1<γ|b|.
Letting r→1- through real values, we get the required assertion (22). Conversely, suppose f(z) satisfies (22), then, in view of (16), consider
(24)|z[𝒟λ,μm(α1,β1)f(z)]′1-(1-α)𝒟λ,μm(α1,β1)f(z)1-αz[𝒟λ,μm(α1,β1)f(z)]′|-γ|b||[𝒟λ,μm(α1,β1)f(z)]′(1-α)𝒟λ,μm(α1,β1)f(z)1111111+αz[𝒟λ,μm(α1,β1)f(z)]′|=|∑k=n+1∞‍([1+α(k-1)]-k)CkmBkakzk|-γ|b||z-∑k=n+1∞‍[1+α(k-1)]CkmBkakzk|<∑k=n+1∞‍{([1+α(k-1)](γ|b|-1)+k)}×CkmBkak-γ|b|≤0.
Hence, the result follows.Similarly, we prove the following.Theorem 4.
Let the functionf∈𝒜(n) be as defined in (1). Then, f∈B if and only if
(25)∑k=n+1∞‍[1+α(k-1)]CkmBkak≤γ|b|.Corollary 5.
Let the functionf∈𝒜(n) as given in (1). Then, f(z)∈A1 if and only if
(26)∑k=n+1∞‍{[1+α(k-1)](γ|b|-1)+k}kmak≤γ|b|.Corollary 6.
Let the functionf∈𝒜(n) be as defined in (1). Then, f∈B1 if and only if
(27)∑k=n+1∞‍[1+α(k-1)]kmak≤γ|b|.Corollary 7.
Let the functionf∈𝒜(n) as given in (1). Then, f∈A2 if and only if
(28)∑k=n+1∞‍{[1+α(k-1)](γ|b|-1)+k}Ckm×(η+k-1k-1)ak≤γ|b|.Corollary 8.
Let the functionf(z)∈𝒜(n) be as defined in (1). Then, f∈B2 if and only if
(29)∑k=n+1∞‍[1+α(k-1)]Ckm(η+k-1k-1)ak≤γ|b|.


## 3. Inclusion Relationships

Theorem 9.
If(30)δ=γ|b|(1+n)[(1+nα)(γ|b|-1)+n+1]Bn+1Cn+1m(γ|b|>1),
then A⊂Nn,δ(e).Proof.
Letf∈A. Then, in view of (22), we have
(31)Cn+1mBn+1[(1+nα)(γ|b|-1)n+1]∑k=n+1∞‍ak≤γ|b|,(32)∑k=n+1∞‍ak≤γ|b|Cn+1mBn+1[(1+nα)(γ|b|-1)n+1].
Consider
(33)Cn+1mBn+1∑k=n+1∞‍kak≤γ|b|+(1+nα)(1-γ|b|)Bn+1Cn+1m∑k=n+1∞‍ak≤γ|b|+(1+nα)(1-γ|b|)γ|b|(1+nα)(γ|b|-1)+n+1≤γ|b|(n+1)(1+nα)(γ|b|-1)+n+1.
Hence,
(34)∑k=n+1∞‍kak≤γ|b|(n+1)[(1+nα)(γ|b|-1)+n+1]Bn+1Cn+1m=δ.
Hence, the result follows.In similar manner, we establish the following result.Theorem 10.
If(35)δ=γ|b|(1+n)(1+nα)Bn+1Cn+1m,
then B⊂Nn,δ(e).Corollary 11.
If(36)δ=γ|b|(1+n)[(1+nα)(γ|b|-1)+n+1](n+1)m,
then A1⊂Nn,δ(e).Corollary 12.
If(37)δ=γ|b|(1+n)(1+nα)(n+1)m,
then B1⊂Nn,δ(e).Corollary 13.
If(38)δ=γ|b|(1+n)[(1+nα)(γ|b|-1)+n+1]Cn+1m(η+nn),
then A2⊂Nn,δ(e).Corollary 14.
If(39)δ=γ|b|(1+n)(1+nα)Cn+1m(η+nn),
then B2⊂Nn,δ(e).


## 4. Neighborhoods forAσ and Bσ

In this section, we determine the neighborhood properties ofAσ and Bσ. Here, the classes Aσ consist of functions f∈𝒜(n) for which there exists a function g(z)∈A such that
(40)|f(z)g(z)-1|<1-σ(z∈Δ,0≤σ<1).
In the same way, we define Bσ, consisting of functions f(z)∈𝒜(n) for which there exists another function g(z)∈B such that
(41)|f(z)g(z)-1|<1-σ(z∈Δ,0≤σ<1).Theorem 15.
Ifg∈A and
(42)σ=1-δn+1×{×Bn+1Cn+1m-γ|b|)-1[(1+nα)(γ|b|-1)+n+1]111×Bn+1Cn+1m111×([(1+nα)(γ|b|-1)+n+1]1111111×Bn+1Cn+1m-γ|b|)-1}11111111111111111(γ|b|>1),
then Nn,δ(g)⊂Aσ.Proof.
Supposef∈Nn,δ(g), then
(43)∑k=n+1∞‍k|ak-bk|≤δ,∑k=n+1∞‍|ak-bk|≤δn+1.
Sinceg∈A, we have
(44)∑k=n+1∞‍bk≤b|k|[(1+nα)(γ|b|-1)+n+1]Bn+1Cn+1m.
Consider
(45)|f(z)g(z)-1|11<∑k=n+1∞|ak-bk|1-∑k=n+1∞bk11≤δn+1{[(1+nα)(γ|b|-1)+n+1]Bn+1Cn+1m[(1+nα)(γ|b|-1)+n+1]Bn+1Cn+1m-γ|b|}11=1-σ.
Therefore, f∈Aσ for σ given by (42).Theorem 16.
Ifg∈B and
(46)σ=1-δn+1{(1+nα)Bn+1Cn+1m(1+nα)Bn+1Cn+1m-γ|b|},
then Nn,δ(g)⊂Bσ.Corollary 17.
Ifg∈A1 and
(47)σ=1-δn+1{(n+1)m-γ|b|)-1[(1+nα)(γ|b|-1)+n+1](n+1)m11111111×([(1+nα)(γ|b|-1)+n+1]11111111111×(n+1)m-γ|b|)-1}11111111111111111111111(γ|b|>1),
then Nn,δ(g)⊂A1σ.Corollary 18.
Ifg∈B1 and
(48)σ=1-δn+1{(1+nα)(n+1)m(1+nα)(n+1)m-γ|b|},
then Nn,δ(g)⊂B1σ.Corollary 19.
Ifg∈A2 and
(49)σ=1-δn+1{[(1+nα)(γ|b|-1)+n+1](η+nn)Cn+1m×(×(η+nn)Cn+1m-γ|b|)-1[(1+nα)(γ|b|-1)+n+1]11×(η+nn)Cn+1m-γ|b|)-1}1111111111111111111(γ|b|>1),
then Nn,δ(g)⊂A2σ.Corollary 20.
Ifg∈B2 and
(50)σ=1-δn+1{(1+nα)(η+nn)Cn+1m(1+nα)(η+nn)Cn+1m-γ|b|},
then Nn,δ(g)⊂B2σ.



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*Source: 958796-2013-10-29.xml*