# Inclusion Relationships for Certain  Subclasses of Meromorphic Functions Defined by Using the Extended Multiplier Transformations

Authors: R. M. El-Ashwah
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

Let∑ denote the class of analytic functions in the punctured unit disc U*={z:0<|z|<1}. Set φλ,ℓm(z)=1/z+∑k=0∞[ℓ+λ(k+1)/ℓ]mzk(m∈ℕ0;ℓ>0;λ≥0;z∈U*), and define φλ,ℓm,μ in terms of the Hadamard product by φλ,ℓm(z)*φλ,ℓm,μ(z)=1/z(1-z)μ(μ>0;z∈U*). In this paper, we introduce several new subclasses of analytic functions defined by means of the operator Iμm(λ,ℓ)f(z)=φλ,ℓm,μ(z)*f(z)(f∈∑;m∈ℕ0;ℓ>0;λ≥0;μ>0).Inclusion properties of these classes and some applications involving integral operator are also considered.


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## Body

## 1. Introduction

Let∑ denote the class of functions of the form: (1.1)f(z)=1z+∑k=0∞akzk,
which are analytic in the punctured open unit disk U*={z:z∈ℂ and 0<|z|<1}=U∖{0}. We denote by ∑S(η),∑K(η), and ∑C(η,β)(0≤η,β<1) the subclasses of ∑ consisting of all meromorphic functions which are, respectively, starlike of order η in U, convex of order η in U, and close-to-convex of order η and type β in U (see [1–3]).LetM be the class of all function φ which are analytic and univalent in U and for which φ(U) is convex with (1.2)φ(0)=1,Re{φ(z)}>0(z∈U).For two functionsf and g analytic in U, we say that f is subordinate to g and write f≺g in U or f(z)≺g(z), if there exists a Schwarz function w(z), which is analytic in U with w(0)=0 and |w(z)|<1(z∈U), such that f(z)=g(w(z)). It is known that (1.3)f(z)≺g(z)⟹f(0)=g(0),f(U)⊂g(U).
Furthermore, if the function g is univalent in U (see, [4, page 4]),(1.4)f(z)≺g(z)⟺f(0)=g(0),f(U)⊂g(U).Making use of the principle of subordination between analytic functions, we define the subclasses∑S(η;φ),∑K(η,φ), and ∑C(η,β;φ,ψ) of the class ∑ for 0≤η,β<1 and φ,ψ∈M, which are defined by (1.5)∑S(η,φ)={f:f∈∑,11-η(-zf′(z)f(z)-η)≺φ(z)(z∈U)},∑K(η,φ)={f:f∈∑,11-η(-[1+zf′′(z)f′(z)]-η)≺φ(z)(z∈U)},∑C(η,β;φ,ψ)={f:f∈∑,∃g∈S(η;φ)s.t.11-β(-zf′(z)g(z)-β)≺φ(z)(z∈U)},
respectively. For special choices for the parameters η and β as well as for special choices for the function φ and ψ, we will obtain various subclasses of meromorphic function of the above classes (see [5–7]).Form∈ℕ0=ℕ∪{0}(ℕ={1,2,…}),  we define the multiplier transformation Jm(λ,ℓ) for functions f∈∑ (see [8, 9] with p=1) by (1.6)Jm(λ,l)f(z)=1z+∑k=0∞(l+λ(k+1)l)makzk(l>0;λ≥0;z∈U*).
Obviously, we have (1.7)Jm1(λ,l)(Jm2(λ,l)f(z))=Jm2(λ,l)(Jm1(λ,l)f(z))=Jm1+m2(λ,l)f(z),
for all integers m1 and m2.We note that(i)
Jm(1,ℓ)f(z)=I(m,ℓ)f(z)(see [10, 11]);(ii)
Jm(λ,1)f(z)=Dλmf(z) (see [12]);(iii)
Jm(1,1)f(z)=Imf(z)(see [13]).Setting(1.8)φλ,lm(z)=1z+∑k=0∞(l+λ(k+1)l)mzk(m∈N0;l>0;λ≥0;z∈U*),
we define a new function φλ,ℓm,μ(z) in terms of the Hadamard product (or convolution) by (1.9)φλ,lm(z)*φλ,lm,μ(z)=1z(1-z)μ(μ>0;z∈U*).
Essentially Choi et al. [14] motivated the Choi-Saigo-Srivastava operator for analytic functions, which includes an integral operator considered earlier by Noor [15] and others [16–18]; we now introduce the operator Iμm(λ,ℓ):∑→∑,which is defined here by; (1.10)Iμm(λ,l)f(z)=φλ,lm,μ(z)*f(z)(f∈∑;m∈N0;l>0;λ≥0;μ>0).
We note that(i)
I20(1,1)f(z)=zf′(z)+2f(z)and I21(1,1)f(z)=f(z);(ii)
Iμm(1,ℓ)f(z)=Iℓ,μmf(z) (see [11]).It is easily verified from the definition of the operatorIμm(λ,ℓ) that (1.11)λz(Iμm+1(λ,l)f(z))′=lIμm(λ,l)f(z)-(λ+l)Iμm+1(λ,l)f(z)(λ>0),(1.12)z(Iμm(λ,l)f(z))′=μIμ+1m(λ,l)f(z)-(μ+1)Iμm(λ,l)f(z).Next, by using the operatorIμm(λ,ℓ) defined by (1.10), we introduce the following subclasses of meromorphic functions: (1.13)∑Sλ,lm,μ(η;φ)={f:f∈∑andIμm(λ,l)f(z)∈∑S(η;φ)}(φ∈M;λ,l,μ>0;m∈N0;0≤η<1),∑Kλ,lm,μ(η;φ)={f:f∈∑andIμm(λ,l)f(z)∈∑K(η;φ)}(φ∈M;λ,l,μ>0;m∈N0;0≤η<1),∑Cλ,lm,μ(η,β;φ,ψ)={f:f∈∑andIμm(λ,l)f(z)∈∑C(η,β;φ,ψ)}(φ,ψ∈M;λ,l,μ>0;m∈N0;0≤η,β<1).
We also note that (1.14)f(z)∈∑Kλ,lm,μ(η;φ)⟺-zf′(z)∈∑Sλ,lm,μ(η;φ).
In particular, we set (1.15)∑Sλ,lm,μ(η;1+Az1+Bz)=∑Sλ,lm,μ(η;A,B)(-1<B<A≤1),∑Kλ,lm,μ(η;1+Az1+Bz)=∑Kλ,lm,μ(η;A,B)(-1<B<A≤1).The main object of this paper is to investigate several inclusion properties of the classes mentioned above. Some applications involving integral operator are also considered.


## 2. Inclusion Properties Involving the OperatorIμm(λ,ℓ)

The following lemmas will be required in our investigation.Lemma 2.1 (see [19]).
Letφ be convex univalent in U with φ(0)=1 and Re{βφ(z)+ν}>0(β,ν∈ℂ). If p is analytic in U with p(0)=1, then
(2.1)p(z)+zp′(z)βp(z)+ν≺φ(z)
implies that
(2.2)p(z)≺φ(z).Lemma 2.2 (see [20]).
Letφ be convex univalent in U and let w be analytic in U with Re{w(z)}≥0. If p(z) is analytic in U and p(0)=φ(0), then
(2.3)p(z)+w(z)zp′(z)≺p(z),
implies that
(2.4)p(z)≺φ(z).At first, with the help of Lemma2.1, we prove the following theorem.Theorem 2.3.
Letφ∈M with
(2.5)max⁡z∈U(Re{φ(z)})<min⁡(μ+1-η1-η,(l/λ)+1-η1-η)(λ,μ,l>0;0≤η<1);
then
(2.6)∑Sλ,lm,μ+1(η;φ)⊂∑Sλ,lm,μ(η;φ)⊂∑Sλ,lm+1,μ(η;φ).Proof.
We begin by showing the first inclusion relationship:(2.7)∑Sλ,lm,μ+1(η;φ)⊂∑Sλ,lm,μ(η;φ),
which is asserted by Theorem 2.3. Let f∈∑Sλ,ℓm,μ+1(η;φ)and set
(2.8)p(z)=11-η(-z(Iμm(λ,l)f(z))′Iμm(λ,l)f(z)-η),
where the function p(z) is analytic in U with p(0)=1. Then, by applying (1.11) is (2.8), we obtain
(2.9)μIμ+1m(λ,l)f(z)Iμm(λ,l)f(z)=-(1-η)p(z)+(μ+1-η).
Differentiating (2.9) logarithmically with respect to z and multiplying the resulting equation by z, we have
(2.10)11-η(-z(Iμ+1m(λ,l)f(z))′Iμ+1m(λ,l)f(z)-η)=zp′(z)-(1-η)p(z)+μ+1-η+p(z)(z∈U).
Since
(2.11)max⁡z∈U(Re{φ(z)})<μ+1-η1-η(μ>0;0≤η<1;z∈U),
we see that
(2.12)Re{μ+1-η-(1-η)φ(z)}>0(z∈U).
Applying Lemma 2.1 to (2.10), it follows that p≺φ in U, that is,
(2.13)f∈∑Sλ,lm,μ(η;φ).
For the second inclusion relationship asserted by Theorem 2.3, using arguments similar to those detailed above with (1.11), we obtain
(2.14)∑Sλ,lm,μ(η,β;φ)⊂∑Sλ,lm+1,μ(η,β;φ).
We thus complete the proof of Theorem 2.3.Theorem 2.4.
Letφ∈M with
(2.15)max⁡z∈U(Re{φ(z)})<min⁡(μ+1-η1-η,(l/λ)+1-η1-η)(λ,μ,l>0;0≤η<1);
then
(2.16)∑Kλ,lm,μ+1(η;φ)⊂∑Kλ,lm,μ(η;φ)⊂∑Kλ,lm+1,μ(η;φ).Proof.
Applying (1.14) and Theorem 2.3, we observe that
(2.17)f(z)∈∑Kλ,lm,μ+1(η;φ)⟺Iμ+1m(λ,l)f(z)∈∑K(η;φ)⟺-z(Iμ+1m(λ,l)f(z))′∈∑S(η;φ)⟺Iμ+1m(λ,l)(-zf′(z))∈∑S(η;φ)⟺-zf′(z)∈∑Sλ,lm,μ+1(η;φ)⟹-zf′(z)∈∑Sλ,lm,μ(η;φ)⟺Iμm(λ,l)(-zf′(z))∈∑S(η;φ)⟺-z(Iμm(λ,l)f(z))′∈∑S(η;φ)⟺Iμm(λ,l)f(z)∈∑K(η;φ)⟺f(z)∈∑Kλ,lm,μ(η;φ),f(z)∈∑Kλ,lm,μ(η;φ)⟺-zf′(z)∈∑Sλ,lm,μ(η;φ)⟹-zf′(z)∈Sλ,lm+1,μ(η;φ)⟺-z(Iμm+1(λ,l)f(z))′∈∑S(η;φ)⟺Iμm+1(λ,l)f(z)∈∑K(η;φ)⟺f(z)∈∑Kλ,lm+1,μ(η;φ),
which evidently prove Theorem 2.4.By setting(2.18)φ(z)=1+Az1+Bz(-1<B<A≤1;z∈U)
in Theorems 2.3 and 2.4, we deduce the following corollary.Corollary 2.5.
Suppose that(2.19)1+A1+B<min⁡(μ+1-η1-η,(l/λ)+1-η1-η)(λ,μ,l>0;0≤η<1;-1<B<A≤1).
Then, for the function classes defined by (1.15),
(2.20)∑Sλ,lm,μ+1(η;A,B)⊂∑Sλ,lm,μ(η;A,B)⊂∑Sλ,lm+1,μ(η;A,B),∑Kλ,lm,μ+1(η;A,B)⊂∑Kλ,lm,μ(η;A,B)⊂∑Kλ,lm+1,μ(η;A,B).Next by using Lemma2.2, one obtains the following inclusion relationships for the class ∑Cλ,ℓm,μ(η,β;φ,ψ).Theorem 2.6.
Letφ,ψ∈M with
(2.21)max⁡z∈U(Re{φ(z)})<min⁡(μ+1-η1-η,(l/λ)+1-η1-η)(λ,μ,l>0;0≤η<1).
Then
(2.22)∑Cλ,lm,μ+1(η,β;φ,ψ)⊂∑Cλ,lm,μ(η,β;φ,ψ)⊂∑Cλ,lm+1,μ(η,β;φ,ψ).Proof.
We begin by proving that(2.23)∑Cλ,lm,μ+1(η,β;φ,ψ)⊂∑Cλ,lm,μ(η,β;φ,ψ),
which is the first inclusion relationship asserted by Theorem 2.6. Let
(2.24)f∈∑Cλ,lm,μ+1(η,β;φ,ψ).
Then, in view of the definition of the function class ∑Cλ,ℓm,μ+1(η,β;φ,ψ), there exists a function k(z)∈∑S(η;φ) such that
(2.25)11-β(-z(Iμ+1m(λ,l)f(z))′k(z)-β)≺ψ(z).
Choose the function g(z) such that Iμ+1m(λ,ℓ)g(z)=k(z). Then g(z)∈∑Sλ,ℓm,μ+1(η;φ)and
(2.26)11-β(-z(Iμ+1m(λ,l)f(z))′Iμ+1m(λ,l)g(z)-β)≺ψ(z).
Now let
(2.27)p(z)=11-β(-z(Iμ+1m(λ,l)f(z))′Iμ+1m(λ,l)g(z)-β),
where the function p(z) is analytic in U with p(0)=1. Using (1.12), we find that
(2.28)11-β(-z(Iμ+1m(λ,l)f(z))′Iμ+1m(λ,l)g(z)-β)=11-β(Iμ+1m(λ,l)(-zf′(z))Iμ+1m(λ,l)g(z)-β)=11-β(z(Iμm(λ,l)(-zf′(z))′+(μ+1)Iμm(λ,l)(-zf′(z))z(Iμm(λ,l)g(z))′+(μ+1)Iμm(λ,l)g(z)-β)=11-β×((z(Iμm(λ,l)(-zf′(z))′/Iμm(λ,l)g(z))+(μ+1)(Iμm(λ,l)(-zf′(z))/Iμm(λ,l)g(z))(z(Iμ+1m(λ,l)g(z))′/Iμm(λ,l)g(z))+(μ+1)-β).
Since
(2.29)g∈∑Sλ,lm,μ+1(η;φ)⊂∑Sλ,lm,μ(η;φ),
by Theorem 2.3, then we set
(2.30)q(z)=11-η(-z(Iμm(λ,l)g(z))′Iμm(λ,l)g(z)-η),
where q≺φ in U with the assumption that φ∈M. Then by (2.27) and (2.28), we observe that
(2.31)Iμm(λ,l)(-zf′(z))=(1-β)p(z)Iμm(λ,l)g(z)+βIμm(λ,l)g(z),(2.32)11-β(-z(Iμ+1m(λ,l)f(z))′Iμ+1m(λ,l)g(z)-β)=11-β((z(Iμm(λ,l)(-zf′(z))′/Iμm(λ,l)g(z))+(μ+1)((1-β)p(z)+β)μ+1-η-(1-η)q(z)-β).
Differentiating both sides of (2.31) with respect to z, multiplying by z and dividing by Iμm(λ,ℓ)g(z), we obtain
(2.33)z(Iμm(λ,l)(-zf′(z))′Iμm(λ,l)g(z)=(1-β)zp′(z)-[(1-β)p(z)+β][(1-η)q(z)+η].
Now making use of (2.26), (2.32), and (2.33), we get
(2.34)11-β(-z(Iμm(λ,l)f(z))′Iμm(λ,l)g(z)-β)=p(z)+zp′(z)(μ+1-η)-(1-η)q(z)≺ψ(z).
Since μ>0 and q≺φ in U with
(2.35)max⁡z∈U(Re{φ(z)})<μ+1-η1-η,
we have
(2.36)Re{μ+1-η-(1-η)q(z)}>0(z∈U).
Hence, by taking
(2.37)w(z)=1(μ+1-η)-(1-η)q(z)
in (2.34), and then applying Lemma 2.2, we can show that p≺ψ in U, so that
(2.38)f∈∑Cλ,lm,μ(η,β;φ,ψ).
For the second inclusion relationship asserted by Theorem 2.6, using arguments similar to those detailed above with (1.11), we obtain
(2.39)∑Cλ,lm,μ(η,β;φ,ψ)⊂∑Cλ,lm+1,μ(η,β;φ,ψ).
We thus complete the proof of Theorem 2.6.


## 3. Inclusion Properties Involving the Integral OperatorFc

In this section, we consider the integral operatorFc (see, [4, page 11]) defined by (3.1)Fc(f)=Fc(f)(z)=czc+1∫0ztcf(t)dt(f∈∑;c>0).
From the definition (3.1), it is easily verified that (3.2)z(Iμm(λ,l)Fc(f)(z))′=cIμm(λ,l)f(z)-(c+1)Iμm(λ,l)Fc(f)(z).
By using (3.2) we can prove the following theorems (see Cho et al. [11]).Theorem 3.1.
Letφ∈M with
(3.3)max⁡z∈U(Re{φ(z)})<c+1-η1-η(c>0;0≤η<1).
If  f∈∑Sλ,ℓm,μ(η;φ), then
(3.4)Fc(f)∈∑Sλ,lm,μ(η;φ).Theorem 3.2.
Letφ∈M with
(3.5)max⁡z∈U(Re{φ(z)})<c+1-η1-η(c>0;0≤η<1).
If f∈∑Kλ,ℓm,μ(η;φ), then
(3.6)Fc(f)∈∑Kλ,lm,μ(η;φ).From Theorems3.1 and 3.2, we can easily deduce the following.Corollary 3.3.
Suppose that(3.7)1+A1+B<c+1-η1-η(c>0;-1<B<A≤1;0≤η<1).
Then for the function classes defined by (1.15), the following inclusion relationships hold true:
(3.8)f∈∑Sλ,lm,μ(η;A,B)⟹Fc(f)∈∑Sλ,lm,μ(η;A,B),f∈∑Kλ,lm,μ(η;A,B)⟹Fc(f)∈∑Kλ,lm,μ(η;A,B).Theorem 3.4.
Letφ,ψ∈M with
(3.9)max⁡z∈U(Re{φ(z)})<c+1-η1-η(c>0;0≤η<1).
If f∈∑Cλ,ℓm,μ(η,β;φ,ψ), then
(3.10)Fc(f)∈∑Cλ,lm,μ(η,β;φ,ψ).Remark 3.5.
Puttingλ=1 in all the above results, we will obtain the results obtained by Cho et al. [11].



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