# On Certain Subclass of Meromorphic Spirallike Functions Involving the Hypergeometric Function

Authors: Lei Shi; Zhi-Gang Wang
Journal: The Scientific World Journal
 (2014)

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## Abstract

We introduce and investigate a new subclassMm1(θ,λ,η) of meromorphic spirallike functions. Such results as integral representations, convolution properties, and coefficient estimates are proved. The results presented here would provide extensions of those given in earlier works. Several other results are also obtained.


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## Body

## 1. Introduction

LetM denote the class of functions f of the form
(1)f(z)=1z+∑n=1∞‍anzn,
which are analytic in the punctured open unit disk:
(2)U*:={z:z∈C,0<|z|<1}=:U∖{0}.Letf, g∈M, where f is given by (1) and g is defined by
(3)g(z)=1z+∑n=1∞‍bnzn.
Then the Hadamard product (or convolution) f*g of f and g is defined by
(4)(f*g)(z)=1z+∑n=1∞‍anbnzn:=(g*f)(z).LetP denote the class of functions p given by
(5)p(z)=1+∑n=1∞‍pnzn(z∈U),
which are analytic in U and satisfy the condition
(6)R(p(z))>0(z∈U).Forθ which is real with |θ|<π/2, 0≤γ<1, we denote by MS*(θ,γ) and MK*(θ,γ) the subclasses of f∈M which are defined, respectively, by
(7)R(eiθzf′(z)f(z))<-γcos⁡⁡θ(z∈U*),R(eiθ(zf′(z))′f′(z))<-γcos⁡⁡θ(z∈U*).
By setting θ=0 in (7), we get the well-known subclasses of f∈M consisting of meromorphic functions which are starlike and convex of order γ(0≤γ<1), respectively. For some recent investigations on meromorphic spirallike functions and related topics, see, for example, the earlier works [1–4] and the references cited therein.Forη>1, Wang et al. [5] and Nehari and Netanyahu [6] introduced and studied the subclass M(η) of M consisting of functions f satisfying
(8)R(zf′(z)f(z))>-η(z∈U*).LetA be the class of functions of the form
(9)f(z)=z+∑n=1∞‍anzn,
which are analytic in U. A function f∈A is said to be in the class S*(θ,δ,γ) if it satisfies the condition
(10)R(eiθzf′(z)(1-δ)f(z)+δzf′(z))>γcos⁡⁡θ(z∈U;|θ|<π2;0≤δ<1;0≤γ<1).
The function class S*(θ,δ,γ) is introduced and studied recently by Orhan et al. [7]. An analogous of the class S*(θ,δ,γ) has been studied by Murugusundaramoorthy [8].For complex parametersα1,…,αl and β1,…,βm(βj≠0,-1,-2,…;j=1,2,…,m) the generalized hypergeometric function lFm(z) is defined by
(11)Flm(z)≡Flm(α1,…,αl;β1,…,βm;z):=∑n=0∞‍(α1)n⋯(αl)n(β1)n⋯(βm)nznn!(l≤m+1;l,m∈N0:=N∪{0};z∈U),
where N denotes the set of all positive integers and (a)k is the Pochhammer symbol defined by
(12)(a)n={1,n=0,a(a+1)(a+2)⋯(a+n-1),n∈N;a∈C.For a functionf∈M, we consider a linear operator (which is a meromorphically modified version of the familiar Dziok-Srivastava linear operator [9, 10]:
(13)Hml[α,β]f(z)=H(α1,…,αl;β1,…,βm)f(z)=[z-1lFm(α1,…,αl;β1,…,βm;z)]*f(z),
where αs>0, βt>0(s=1,…,l;t=1,…,m;l≤m+1;l,m∈N0).From the definition of the operatorHml[α,β]f, it is easy to observe that
(14)Hml[α,β]f(z)=1z+∑n=1∞‍ϕn(α,β;l;m)anzn,
where
(15)ϕn(α,β;l;m)=(α1)n+1⋯(αl)n+1(β1)n+1⋯(βm)n+11(n+1)!
is a positive number for all n∈N.Recently, Aouf [11], Liu and Srivastava [12], and Raina and Srivastava [13] obtained many interesting results involving the linear operator Hml[α,β]. In particular, for
(16)l=2,m=1,α1=a,β1=c,α2=1,
we obtain the following linear operator:
(17)L(a,c)f(z):=H[α1,1;β1]f(z)(z∈U*),
which was introduced and investigated earlier by Liu and Srivastava [14] and was further studied in a subsequent investigation by Srivastava et al. [15]. It should also be remarked that the linear operator Hml[α,β] is a generalization of other linear operators considered in many earlier investigations (see, e.g., [16–18]).Using the operatorHml[α,β]f, we introduce the following class of meromorphic functions.Definition 1.
For|θ|<π/2, 0≤λ<1/2, and η>1, let Mml(θ,λ,η) denote a subclass of M consisting of functions satisfying the condition that
(18)R(eiθ(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)>-ηcos⁡⁡θ,(z∈U*),
where Hml[α,β]f is given by (13).We note that, forl=2, m=1, α1=α2=1, β1=1, and θ=λ=0, the class M12(0,0,η) becomes the class M(η).In the present paper, we aim at proving some interesting properties such as integral representations, convolution properties, and coefficient estimates for the classMml(θ,λ,η).The following lemma will be required in our investigation.Lemma 2.
Suppose that the sequence{An}n=1∞ is defined by
(19)A1=(1-2λ)(η-1)cos⁡θ(1-λ)ϕ1(α,β;l;m),An+1=2(η-1)cos⁡θ(n+2)(1-λ)ϕn+1(α,β;l;m)×[1-2λ+∑k=1n‍ϕk(α,β;l;m)(1-λ+kλ)Ak].
Then
(20)An=(1-2λ)(η-1)cos⁡θ(1-λ)nϕn(α,β;l;m)×∏k=1n-1‍(k+1)(1-λ)+2(1-λ+kλ)(η-1)cos⁡θk+2,(n≥2).Proof.
From (19), we have
(21)(n+2)(1-λ)ϕn+1(α,β;l;m)An+1=2(η-1)cos⁡θ[∑k=1n‍1-2λh+∑k=1n‍ϕk(α,β;l;m)(1-λ+kλ)Ak],(n+1)(1-λ)ϕn(α,β;l;m)An=2(η-1)cos⁡θ[∑k=1n-1‍1-2λh+∑k=1n-1‍ϕk(α,β;l;m)(1-λ+kλ)Ak].
Combining (21), we find that
(22)An+1An=(n+1)(1-λ)+2(1-λ+nλ)(η-1)cos⁡θ(n+2)(1-λ)·ϕn(α,β;l;m)ϕn+1(α,β;l;m).
Thus, for n≥2, we deduce from (22) that
(23)An=AnAn-1⋯A3A2·A2A1·A1=n(1-λ)+2(1-nλ)(η-1)cos⁡θ(n+1)(1-λ)⋯×3(1-λ)+2(1+λ)(η-1)cos⁡θ4(1-λ)·2(1-λ)+2(η-1)cos⁡θ3(1-λ)·ϕn-1(α,β;l;m)ϕn(α,β;l;m)⋯ϕ2(α,β;l;m)ϕ3(α,β;l;m)·ϕ1(α,β;l;m)ϕ2(α,β;l;m)·(1-2λ)(η-1)cos⁡θ(1-λ)ϕ1(α,β;l;m)=(1-2λ)(η-1)cos⁡θ(1-λ)nϕn(α,β;l;m)×∏k=1n-1‍(k+1)(1-λ)+2(1-λ+kλ)(η-1)cos⁡θk+2.
This completes the proof of Lemma 2.


## 2. Main Results

We begin by proving the following integral representation for the classMml(θ,λ,η).Theorem 3.
Letf∈Mml(θ,λ,η). Then
(24)f(z)=(z-1+∑n=1∞‍ϕn-1(α,β;l,m)zn)*[z-1·exp⁡(∫0z‍((1+e-2iθ)ω(t)]t)-1((1+e-2iθ)(1-2λ)(η-1)×(1+e-2iθ)ω(t))×((1+e-2iθ)ω(t)]t)-1((1+e-2iθ)(1-2λ)(η-1)(1+e-2iθ)ω(t)]ω(t)]t)-1[(1-λ)(1-ω(t))-λ(η-1)(1+e-2iθ)×(1+e-2iθ)ω(t)]t)-1((1+e-2iθ)(1-2λ)(η-1)(1+e-2iθ)ω(t)]ω(t)]t)-1)dt∫0z‍)],(z∈U*),
where ω is analytic in U with ω(0)=0 and |ω(z)|<1.Proof.
Suppose thatf∈Mml(θ,λ,η) and
(25)τ(z):=((+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′)×((Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1)+ηcos⁡θ+isinθ(+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′))×((η-1)cos⁡θ)-1,(z∈U).
We know that τ∈P, which implies
(26)((((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′)×((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1)+ηcos⁡θ+isinθ(((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′))×((η-1)cos⁡θ)-1((((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′)=1+ω(z)1-ω(z),
where ω is analytic in U with ω(0)=0 and |ω(z)|<1. We find from (26) that
(27)(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′=-1+[1+(η-1)(1+e-2iθ)]ω(z)1-ω(z),
which follows
(28)(Hml[α,β]f(z))′Hml[α,β]f(z)+1z=(1-2λ)(η-1)(1+e-2iθ)ω(z)z[(1-λ)(1-ω(z))-λ(η-1)(1+e-2iθ)ω(z)].
Integrating both sides of (28) yields
(29)log⁡(zHml[α,β]f(z))=∫0z‍(1-2λ)(η-1)(1+e-2iθ)ω(t)[(1-λ)(1-ω(t))-λ(η-1)(1+e-2iθ)ω(t)]tdt.
From (29), we obtain
(30)Hml[α,β]f(z)=z-1·exp⁡(∫0z(×(1+e-2iθ)ω(t)]t)-1((1-2λ)(1+e-2iθ)(η-1)ω(t))×([(1+e-2iθ)ω(t)(1-λ)(1-ω(t))-λ(η-1)∫0z(×(1+e-2iθ)ω(t)]t)-1((1-2λ)(1+e-2iθ)(η-1)ω(t))×(1+e-2iθ)ω(t)]t)-1)dt).
Thus, the assertion (24) of Theorem 3 follows directly from (30).Next, we derive a convolution property for the classMml(θ,λ,η).Theorem 4.
Letξ∈C and |ξ|=1. Then f∈Mml(θ,λ,η) if and only if
(31)f*{(1-ξ)·[1z-∑n=1∞‍nϕn(α,β;l;m)zn]+[ξη+ξ(η-1)e-2iθ-1]·[1z+∑n=1∞‍1-λ+nλ1-2λϕn(α,β;l;m)zn]}≠0,(z∈U*).Proof.
From the definition (18), we know that f∈Mml(θ,λ,η) if and only if
(32)((((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′)×((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1)+ηcos⁡θ+isinθ(((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′))×((η-1)cos⁡θ)-1((((1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′)-1(eiθ(1-2λ)z(Hml[α,β]f(z))′)≠1+ξ1-ξ,(z∈U*;|ξ|=1),
which is equivalent to
(33)-(1-ξ)z(Hml[α,β]f(z))′+[ξη+ξ(η-1)e-2iθ-1]·(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′1-2λ≠0.
On the other hand, we find from (14) that
(34)-z(Hml[α,β]f(z))′=1z-∑n=1∞‍nϕn(α,β;l;m)anzn=f(z)*[1z-∑n=1∞‍nϕn(α,β;l;m)zn],(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′1-2λ=1z+∑n=1∞‍1-λ+nλ1-2λϕn(α,β;l;m)anzn=f(z)*[1z+∑n=1∞‍1-λ+nλ1-2λϕn(α,β;l;m)zn].
Combining (33) and (34), we get assertion (31) of Theorem 4.Now, we discuss the coefficient estimates for functions in the classMml(θ,λ,η).Theorem 5.
Suppose thatf∈Mml(θ,λ,η). Then
(35)|a1|≤(1-2λ)(η-1)cos⁡θ(1-λ)ϕ1(α,β;l;m),|an|≤(1-2λ)(η-1)cos⁡θ(1-λ)nϕn(α,β;l;m)×∏k=1n-1‍(k+1)(1-λ)+2(1-λ+kλ)(η-1)cos⁡θk+2,(n∈N∖{1}).Proof.
Letf∈Mml(θ,λ,η). Then there exists τ∈P such that
(36)eiθ(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′=(η-1)cos⁡θτ(z)-ηcos⁡θ-isinθ,(z∈U*).
It follows from (36) that
(37)eiθ(1-2λ)z(Hml[α,β]f(z))′=[(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′]×[(η-1)cos⁡θτ(z)-ηcos⁡θ-isinθ].
Combining (1) and (37), we have
(38)eiθ(1-2λ)[-1z+∑n=1∞‍nϕn(α,β;l;m)anzn]=[(1-2λ)1z+∑n=1∞‍(1-λ+nλ)ϕn(α,β;l;m)anzn]·[-eiθ+(η-1)cos⁡θτ1z+(η-1)cos⁡θτ2z2+⋯].
Evaluating the coefficient of zn in both sides of (38) yields
(39)2eiθ(1-λ)ϕ1(α,β;l;m)a1=(1-2λ)(η-1)cos⁡θτ2,eiθ(n+1)(1-λ)ϕn(α,β;l;m)an=(η-1)cos⁡θ[∑k=1n-1‍(1-2λ)τn+1hhhhhhhhhh+∑k=1n-1‍ϕk(α,β;l;m)(1-λ+kλ)akτn-k],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh(n≥2).
By observing the fact that |τn|≤2 for n∈N, we find from (39) that
(40)|a1|≤(1-2λ)(η-1)cos⁡θ(1-λ)ϕ1(α,β;l;m),(41)|an|≤2(η-1)cos⁡θ(n+1)(1-λ)ϕn(α,β;l;m)×[1-2λ+∑k=1n-1‍ϕk(α,β;l;m)(1-λ+kλ)|ak|],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh(n≥2).
Now we define the sequence {An}n=1∞ as follows:
(42)A1=(1-2λ)(η-1)cos⁡θ(1-λ)ϕ1(α,β;l;m),An+1=2(η-1)cos⁡θ(n+2)(1-λ)ϕn+1(α,β;l;m)×[1-2λ+∑k=1n‍ϕk(α,β;l;m)(1-λ+kλ)Ak].
In order to prove that
(43)|an|≤An(n∈N),
we use the principle of mathematical induction. Note that
(44)|a1|≤A1=(1-2λ)(η-1)cos⁡θ(1-λ)ϕ1(α,β;l;m).
Therefore, assume that
(45)|ak|≤Ak(k=1,2,…,n;n∈N).
Combining (41) and (42), we get
(46)|an+1|≤2(η-1)cos⁡θ(n+2)(1-λ)ϕn+1(α,β;l;m)×[1-2λ+∑k=1n‍ϕk(α,β;l;m)(1-λ+kλ)|ak|]≤2(η-1)cos⁡θ(n+2)(1-λ)ϕn+1(α,β;l;m)×[1-2λ+∑k=1n‍ϕk(α,β;l;m)(1-λ+kλ)Ak]=An+1.
Hence, by the principle of mathematical induction, we have
(47)|an|≤An(n∈N),
as desired. By means of Lemma 2 and (42), we know that (20) holds. Combining (47) and (20), we readily get the coefficient estimates asserted by Theorem 5.Remark 6.
By settingθ=0, l=2, m=1, α1=α2=β1=1, and λ=0 in Theorem 5, we get the corresponding result due to Wang et al. [5].In what follows, we present some sufficient conditions for functions belonging to the classMml(θ,λ,η).Theorem 7.
Letζ be a real number with 0≤ζ<1. If f∈M satisfies the condition
(48)|(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hmlf(z)+λz(Hml[α,β]f(z))′+1|≤1-ζ,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh(z∈U*),
then f∈Mml(θ,λ,η) provided that
(49)cos⁡θ≥1-ζη-1.Proof.
From (48), it follows that
(50)(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′=-1+(1-ζ)ω(z),
where ω is analytic in U with ω(0)=0 and |ω(z)|<1. Thus, we have
(51)R(eiθ(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)-λz(Hml[α,β]f(z))′)=R(eiθ(-1+(1-ζ)ω(z)))=-cos⁡θ+(1-ζ)R(eiθω(z))≥-cos⁡θ-(1-ζ)|ω(z)|>-cos⁡θ-(1-ζ)≥-ηcos⁡θ,
provided that cos⁡θ≥(1-ζ)/(η-1). This completes the proof of Theorem 7.If we takeζ=1-(η-1)cos⁡θ in Theorem 7, we obtain the following result.Corollary 8.
Iff∈M satisfies the inequality
(52)|(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′+1|≤(η-1)cos⁡θ,
then f∈Mml(θ,λ,η).Theorem 9.
If a functionf∈M given by (1) satisfies the inequality
(53)∑n=1∞‍[(1-λ)(1+n)secθ+(η-1)(1-λ+nλ)]×ϕn(α,β;l,m)|an|≤(1-2λ)(η-1),
then it belongs to the class Mml(θ,λ,η).Proof.
In virtue of Corollary8, it suffices to show that condition (52) holds. We observe that
(54)|(1-2λ)z(Hml[α,β]f(z))′(1-λ)Hml[α,β]f(z)+λz(Hml[α,β]f(z))′+1|=(1-λ)×|∑n=1∞‍(1+n)ϕn(α,β;l,m)anzn+11-2λ+∑n=1∞‍(1-λ+nλ)ϕn(α,β;l,m)anzn+1|<(1-λ)∑n=1∞‍(1+n)ϕn(α,β;l,m)|an|1-2λ-∑n=1∞‍(1-λ+nλ)ϕn(α,β;l,m)|an|.
The last expression is bounded by (η-1)cos⁡θ, if
(55)(1-λ)∑n=1∞‍(1+n)ϕn(α,β;l,m)|an|≤(η-1)cos⁡θ[∑n=1∞‍1-2λH-∑n=1∞‍(1-λ+nλ)ϕn(α,β;l,m)|an|],
which is equivalent to
(56)∑n=1∞‍[(1-λ)(1+n)secθ+(η-1)(1-λ+nλ)]×ϕn(α,β;l,m)|an|≤(1-2λ)(η-1).
This completes the proof of Theorem 9.



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