# Subclasses of Analytic Functions Associated with Generalised Multiplier Transformations

Authors: Rashidah Omar; Suzeini Abdul Halim
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

New subclasses of analytic functions in the open unit disc are introduced which are defined using generalised multiplier transformations. Inclusion theorems are investigated for functions to be in the classes. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to be preserved for these classes.


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## Body

## 1. Introduction

LetA denote the class of functions f normalised by f(z)=z+∑n=2∞anzn in the open unit disk D:={z∈C:|z|<1}. Also let S⋆,C, and K denote, respectively, the subclasses of A consisting of functions which are starlike, convex, and close to convex in D. An analytic function f is subordinate to an analytic function g, written f(z)≺g(z)(z∈D) if there exists an analytic function w in D such that w(0)=0 and |w(z)|<1 for |z|<1 and f(z)=g(w(z)). In particular, if g is univalent in D, then f(z)≺g(z) is equivalent to f(0)=g(0) and f(D)⊂g(D). The convolution of two analytic functions φ(z)=∑n=2∞anzn and ψ(z)=∑n=0∞bnzn is defined by φ(z)*ψ(z)=∑n=0∞anbnzn=ψ(z)*φ(z).For any real numbersk and λ where k≥0, λ≥0, c≥0,  Cǎtaş [1] defined the multiplier transformations I(k,λ,c)f(z) by the following series:(1.1)I(k,λ,c)f(z)=z+∑n=2∞[1+λ(n-1)+c1+c]kanzn.
Recently, some properties of functions using the multiplier transformations have been studied in [2–6]. Using the convolution, we extend the multiplier transformation in (1.1) to be a unified operator. The approach used is similar to Noor's [7], only we generalise and extend to include powers and uses the multiplier
Cǎtaş as basis instead of the Ruscheweyh operator.Set the function(1.2)fk,c(z)=z+∑n=2∞[1+c1+λ(n-1)+c]kzn(k,λ∈R,k≥0,λ≥0,c≥0),
and note that, for λ=1, fk,c(z) is the generalised polylogarithm functions discussed in [8]. A new function fk,cμ(z) is defined in terms of the Hadamard product (or convolution) as follows:(1.3)fk,c(z)*fk,cμ(z)=z(1-z)μ(μ>0).
Motivated by [9–11] and analogous to (1.1), the following operator is introduced:(1.4)Ick(λ,μ)f(z)=fk,cμ*f(z)=z+∑n=2∞(μ)n-1(n-1)![1+λ(n-1)+c1+c]kanzn.
The operator Ick(λ,μ)f unifies other previously defined operators. For examples,(i)
Ick(λ,1)f is the I1(δ,λ,l)f given in [1],(ii)
Ick(1,1)f is the Ickf given in [12],also, for any integerk,(iii)
I0k(λ,1)f(z)≡Dλkf(z) given in [13],(iv)
I0k(1,1)f(z)≡Dkf(z) given in [14],(v)
I1k(1,1)f(z)≡Ikf(z) given in [15].The following relations are easily derived using the following definition:(1.5)(1+c)Ick+1(λ,μ)f(z)=(1-λ+c)Ick(λ,μ)f(z)+λz[Ick(λ,μ)f(z)]′,(1.6)μIck(λ,μ+1)f(z)=z[Ick(λ,μ)f(z)]′+(μ-1)Ick(λ,μ)f(z).
Let N be the class of all analytic and univalent functions ϕ in D and for which ϕ(D) is convex with ϕ(0)=1 and Re{ϕ(z)}>0 for z∈D. For ϕ,ψ∈N, Ma and Minda [16] studied the subclasses S⋆(ϕ),C(ϕ), and K(ϕ,ψ) of the class A. These classes are defined using the principle of subordination as follows:(1.7)S⋆(ϕ):={f:f∈A,zf′(z)f(z)≺ϕ(z)inD},C(ϕ):={f:f∈A,1+zf′′(z)f′(z)≺ϕ(z)inD},K(ϕ,ψ):={f:f∈A,∃g∈S⋆(ϕ)suchthatzf′(z)g(z)≺ψ(z)inD}.
Obviously, we have the following relationships for special choices ϕ and ψ:(1.8)S⋆(1+z1-z)=S⋆,C(1+z1-z)=C,K(1+z1-z,1+z1-z)=K.
Using the generalised multiplier transformations Ick(λ,μ)f, new classes Sck(λ,μ;ϕ), Cck(λ,μ;ϕ) and Kck(λ,μ;ϕ,ψ) are introduced and defined below(1.9)Sck(λ,μ;ϕ):={f∈A:Ick(λ,μ)f(z)∈S⋆(ϕ)},Cck(λ,μ;ϕ):={f∈A:Ick(λ,μ)f(z)∈C(ϕ)},Kck(λ,μ;ϕ,ψ):={f∈A:Ick(λ,μ)f(z)∈K(ϕ,ψ)}.
It can be shown easily that(1.10)f(z)∈Cck(λ,μ;ϕ)⟺zf′(z)∈Sck(λ,μ;ϕ).
Janowski [17] introduced class S⋆[A,B]=S⋆((1+Az)/(1+Bz)) and in particular for ϕ(z)=(1+Az)/(1+Bz), we set(1.11)Sck(λ,μ;1+Az1+Bz)=Sk,c⋆[μ;A,B](-1≤B<A≤1).
In [18], the authors studied the inclusion properties for classes defined using Dziok-Srivastava operator. This paper investigates the similar properties for analytic functions in the classes defined by the generalised multiplier transformations Ick(λ,μ)f. Furthermore, applications of other families of integral operators are considered involving these classes.


## 2. Inclusion Properties InvolvingIcK(λ,μ)f

In proving our results, the following lemmas are needed.Lemma 2.1 (see [19]).
Letϕ be convex univalent in D, with ϕ(0)=1 and Re[κϕ(z)+η]>0(κ,η∈C). If p is analytic in D with p(0)=1, then
(2.1)p(z)+zp′(z)κp(z)+η≺ϕ(z)⟹p(z)≺ϕ(z).Lemma 2.2 (see [20]).
Letϕ be convex univalent in D and ω be analytic in D with Re{ω(z)}≥0. If p is analytic in D and p(0)=ϕ(0), then
(2.2)p(z)+ω(z)zp′(z)≺ϕ(z)⟹p(z)≺ϕ(z).Theorem 2.3.
For any real numbersk and λ where k≥0,λ≥0 and c≥0.
Letϕ∈N and Re{ϕ(z)+(1-λ+c)/λ}>0, then Sck+1(λ,μ;ϕ)⊂Sck(λ,μ;ϕ)(μ>0).Proof.
Letf∈Sck+1(λ,μ;ϕ), and set p(z)=(z[Ick(λ,μ)f(z)]′)/(Ick(λ,μ)f(z)) where p is analytic in D with p(0)=1. Rearranging (1.5), we have
(2.3)(1+c)Ick+1(λ,μ)f(z)Ick(λ,μ)f(z)=(1-λ+c)+λz[Ick(λ,μ)f(z)]′Ick(λ,μ)f(z).
Next, differentiating (2.3) and multiplying by z gives
(2.4)z[Ick+1(λ,μ)f(z)]′Ick+1(λ,μ)f(z)=z[Ick(λ,μ)f(z)]′Ick(λ,μ)f(z)+z((z[Ick(λ,μ)f(z)]′)/(Ick(λ,μ)f(z)))′(z[Ick(λ,μ)f(z)]′)/(Ick(λ,μ)f(z))+(1-λ+c)/λ=p(z)+zp′(z)p(z)+(1-λ+c)/λ.
Since (z[Ick+1(λ,μ)f(z)]′)/(Ick+1(λ,μ)f(z))≺ϕ(z) and applying Lemma 2.1, it follows that p≺ϕ. Thus f∈Sck(λ,μ;ϕ).Theorem 2.4.
Letk,λ∈R,k≥0,λ≥0, and μ≥1. Then Sck(λ,μ+1;ϕ)⊂Sck(λ,μ;ϕ)(c≥0;ϕ∈N).Proof.
Letf∈Sck(λ,μ+1;ϕ), and from (1.6), we obtain that
(2.5)μIck(λ,μ+1)f(z)Ick(λ,μ)=z[Ick(λ,μ)f(z)]′Ick(λ,μ)+(μ-1).
Making use of the differentiation on both sides in (2.5) and setting p(z)=(z[Ick(λ,μ)f(z)]′)/(Ick(λ,μ)f(z)), we get the following:
(2.6)z[Ick(λ,μ+1)f(z)]′Ick(λ,μ+1)f(z)=p(z)+zp′(z)p(z)+(μ-1)≺ϕ(z).
Since μ≥1 and Re{ϕ(z)+(μ-1)}>0, using Lemma 2.1, we conclude that f∈Sck(λ,μ;ϕ).Corollary 2.5.
Letλ≥0,μ≥1, and -1≤B<A≤1. Then Sk+1,c⋆[μ;A,B]⊂Sk,c⋆[μ;A,B] and Sk,c⋆[μ+1;A,B]⊂Sk,c⋆[μ;A,B].Theorem 2.6.
Letλ≥0 and μ≥1. Then Cck+1(λ,μ;ϕ)⊂Cck(λ,μ;ϕ) and Cck(λ,μ+1;ϕ)⊂Cck(λ,μ;ϕ).Proof.
Using (1.10) and Theorem 2.3, we observe that
(2.7)f(z)∈Cck+1(λ,μ;ϕ)⟺zf′(z)∈Sck+1(λ,μ;ϕ)⟹zf′(z)∈Sck(λ,μ;ϕ)⟺Ick(λ,μ)zf′(z)∈S⋆(ϕ)⟺z[Ick(λ,μ)f(z)]′∈S⋆(ϕ)⟺Ick(λ,μ)f(z)∈C(ϕ)⟺f∈Cck(λ,μ;ϕ).
To prove the second part of theorem, using the similar manner and applying Theorem 2.4, the result is obtained.Theorem 2.7.
Letλ≥0,c≥0 and Re{(1-λ+c)/λ}>0.
ThenKck+1(λ,μ;ϕ,ψ)⊂Kck(λ,μ;ϕ,ψ) and  Kck(λ,μ+1;ϕ,ψ)⊂Kck(λ,μ;ϕ,ψ)(ϕ,ψ∈N).Proof.
Letf∈Kck+1(λ,μ;ϕ,ψ). In view of the definition of the class Kck+1(λ,μ;ϕ,ψ), there is a function g∈Sck+1(λ,μ;ϕ) such that
(2.8)z[Ick+1(λ,μ)f(z)]′Ick+1(λ,μ)g(z)≺ψ(z).
Applying Theorem 2.3, then g∈Sck(λ,μ;ϕ) and let q(z)=(z[Ick(λ,μ)g(z)]′)/(Ick(λ,μ)g(z))≺ϕ(z).
Let the analytic functionp with p(0)=1 as
(2.9)p(z)=z[Ick(λ,μ)f(z)]′Ick(λ,μ)g(z).
Thus, rearranging and differentiating (2.9), we have
(2.10)[Ick(λ,μ)zf′(z)]′Ick(λ,μ)g(z)=p(z)[Ick(λ,μ)g(z)]′Ick(λ,μ)g(z)+p′(z).
Making use (1.5), (2.9), (2.10), and q(z), we obtain that
(2.11)z[Ick+1(λ,μ)f(z)]′Ick+1(λ,μ)g(z)=[Ick+1(λ,μ)zf′(z)]Ick+1(λ,μ)g(z)=(1-λ+c)Ick(λ,μ)zf′(z)+λz[Ick(λ,μ)zf′(z)]′(1-λ+c)Ick(λ,μ)g(z)+λz[Ick(λ,μ)g(z)]′=((1-λ+c)Ick(λ,μ)zf′(z))/(Ick(λ,μ)g(z))+(λz[Ick(λ,μ)zf′(z)]′)/(Ick(λ,μ)g(z))(1-λ+c)+(λz[Ick(λ,μ)g(z)]′)/(Ick(λ,μ)g(z))=(1-λ+c)p(z)+λ[p(z)q(z)+p′(z)](1-λ+c)+λq(z)=p(z)+zp′(z)q(z)+(1-λ+c)/λ≺ψ(z).
Since q(z)≺ϕ(z) and Re{(1-λ+c)/λ}>0, then Re{q(z)+(1-λ+c)/λ}>0. Using Lemma 2.2, we conclude that p(z)≺ψ(z) and thus f∈Kck(λ,μ;ϕ,ψ). By using similar manner and (1.6), we obtain the second result.In summary, using subordination technique inclusion properties has been established for certain analytic functions defined via the generalised multiplier transformation.


## 3. Inclusion Properties InvolvingFcf

In this section, we determine properties of generalised Bernardi-Libera-Livington integral operator defined by [21–24](3.1)Fc[f(z)]=c+1zc∫0ztc-1f(t)dt(c>-1,Rec≥0)=z+∑n=2∞c+1n+canzn
and satisfies the following:(3.2)cIck(λ,μ)Fc[f(z)]+z[Ick(λ,μ)Fc[f(z)]]′=(c+1)Ick(λ,μ)f(z).Theorem 3.1.
Iff∈Sck(λ,μ;ϕ), then Fcf∈Sck(λ,μ;ϕ).Proof.
Letf∈Sck(λ,μ;ϕ), then (z[Ick(λ,μ)f(z)]′)/(Ick(λ,μ)f(z))≺ϕ(z). Taking the differentiation on both sides of (3.2) and multiplying by z, we obtain
(3.3)z[Ick(λ,μ)f(z)]′Ick(λ,μ)f(z)=z[Ick(λ,μ)Fc[f(z)]]′Ick(λ,μ)Fc[f(z)]+z((z[Ick(λ,μ)Fc[f(z)]]′)/(Ick(λ,μ)Fc[f(z)]))′(z[Ick(λ,μ)Fc[f(z)]]′)/(Ick(λ,μ)Fc[f(z)])+c.
Setting p(z)=(z[Ick(λ,μ)Fc[f(z)]]′)/(Ick(λ,μ)Fc[f(z)]), we have
(3.4)z[Ick(λ,μ)f(z)]′Ick(λ,μ)f(z)=p(z)+zp′(z)p(z)+c.
Lemma 2.1 implies (z[Ick(λ,μ)Fc[f(z)]]′)/(Ick(λ,μ)Fc[f(z)])≺ϕ(z). Hence Fcf∈Sck(λ,μ;ϕ).Theorem 3.2.
Letf∈Cck(λ,μ;ϕ), then Fcf∈Cck(λ,μ;ϕ).Proof.
By using (1.10) and Theorem 3.1, it follows that
(3.5)f∈Cck(λ,μ;ϕ)⟺zf′(z)∈Sck(λ,μ;ϕ)⟹Fc[zf′(z)]∈Sck(λ,μ;ϕ)⟺z[Fc[f(z)]]′∈Sck(λ,μ;ϕ)⟺Fc[f(z)]∈Cck(λ,μ;ϕ).Theorem 3.3.
Letϕ,ψ∈N, and f∈Kck(λ,μ;ϕ,ψ), then Fcf∈Kck(λ,μ;ϕ,ψ).Proof.
Letf∈Kck(λ,μ;ϕ,ψ), then there exists function g∈Sck(λ,μ;ϕ) such that (z[Ick(λ,μ)f(z)]′)/(Ick(λ,μ)g(z))≺ψ(z). Since g∈Sck(λ,μ;ϕ) therefore from Theorem 3.1, Fc[f(z)]∈Sck(λ,μ;ϕ). Then let
(3.6)q(z)=z[Ick(λ,μ)Fc[g(z)]]′Ick(λ,μ)Fc[g(z)]≺ϕ(z).
Set
(3.7)p(z)=z[Ick(λ,μ)Fc[f(z)]]′Ick(λ,μ)Fc[g(z)].
By rearranging and differentiating (3.7), we obtain that
(3.8)[Ick(λ,μ)Fc[zf′(z)]]′Ick(λ,μ)Fc[g(z)]=p(z)[Ick(λ,μ)Fc[g(z)]]′Ick(λ,μ)Fc[g(z)]+[Ick(λ,μ)Fc[g(z)]]p′(z)Ick(λ,μ)Fc[g(z)].
Making use (3.2), (3.7), and (3.6), it can be derived that
(3.9)z[Ick(λ,μ)f(z)]′Ick(λ,μ)g(z)=p(z)+zp′(z)c+q(z).
Hence, applying Lemma 2.2, we conclude that p(z)≺ψ(z), and it follows that Fc[f(z)]∈Kck(λ,μ;ϕ,ψ).For analytic functions in the classes defined by generalised multiplier transformations, the generalised Bernardi-Libera-Livington integral operator has been shown to be preserved in these classes.


## 4. Conclusion

Results involving functions defined using the generalised multiplier transformation, namely, inclusion properties and the Bernardi-Libera-Livington integral operator were obtained using subordination principles. In [18], similar results were discussed for functions defined using the Dziok-Srivastava operator.



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