# New Criteria for Meromorphic Multivalent Alpha-Convex Functions

Authors: Muhammad Arif; Maslina Darus; Fazal Ghani; Saeed Islam
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The aim of the present paper is to obtain sufficient condition for the class of meromorphicp-valent alpha convex functions of order ξ and then to study
mapping properties of the newly defined integral operators. Many known results appeared as special consequences of our work.


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## Body

## 1. Introduction

LetΣp(n) denote the class of meromorphic functions f(z) normalized by(1)f(z)=z-p+∑k=n∞‍akzk-p+1,(p∈ℕ),
which are analytic and p-valent in the punctured unit disk 𝕌*={z:0<|z|<1}. In particular, Σ1(n)=Σ(n), Σp(1)=Σp, and Σ1(1)=Σ. For λ which is real with |λ|<π/2, α≥0, 0≤ξ<p, and n, p∈ℕ, we denote by Σ𝒮p(λ,n,ξ), Σ𝒞p(λ,n,ξ), and Σℳp(λ,n,α,ξ), the subclasses of Σp(n) consisting of all meromorphic p-valent functions of the form (1) which are defined, respectively, by
(2)-Reeiλzf′(z)f(z)>ξcosλ,(z∈𝕌*),-Reeiλ(zf′(z))′f′(z)>ξcosλ,(z∈𝕌*),-Reeiλ{(1-α)zf′(z)f(z)+α(zf′(z))′f′(z)}>ξcosλ,(z∈𝕌*).
Making λ=0, n=1 in (2), we get the well-known subclasses of Σp consisting of meromorphicp-valent functions which are starlike, convex, and alpha convex of order ξ(0≤ξ<p), respectively. For details of the classes defined by (2) and related topics, we refer the raeder to the work of Aouf and Hossen [1], Aouf and Srivastava [2], Ali and Ravichandran [3], Goyal and Prajapat [4], Joshi and Srivastava [5], Liu and Srivastava [6], Raina and Srivastava [7], Xu and Yang [8], and Owa et al. [9].Forf(z)∈Σ, Wang et al. [10] and Nehari and Netanyahu [11] introduced and studied the subclass ΣN(τ) of Σ consisting of functions f(z) satisfying
(3)-Re(zf′(z))′f′(z)<τ,(τ>1,z∈𝕌*).
We now extended this concept to define a subclass Σ𝒩p(λ,n,α,τ) of Σp(n) consisting of functions f(z) of the form (1) satisfying
(4)-Reeiλ((1-α)zf′(z)f(z)+α(zf′(z))′f′(z))<τcosλ,(τ>p,z∈𝕌*).
For α=0 and α=1 in (4), we obtain the classes Σ𝒩𝒞p(λ,n,τ) and Σ𝒩𝒮p(λ,n,τ) of Σp(n), respectively, studied by Arif  [12]; also see [13, 14].Integral operators for different classes of analytic, univalent, and multivalent functions in the open unit disk are studied by various authors; see [15–21]. We now define the following general integral operator of meromorphicp-valent functions:
(5)Gm(z)=Im,p(δ,αj;fj(z))={δzp+1∫0ztδ-1∏j=1m‍(tp(fj(t)))αjdt}1/δ.
For δ=1, we obtain the integral operator Im,p(fj(z)) studied recently in [22, 23], and, further for p=1, we obtain the integral operator introduced and studied by Mohammed and Darus [24].Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions; for some of the related work see [25–27]. The object of the present paper is to obtain sufficient conditions for the class Σℳp(λ,n,α,ξ) and then study mapping properties of the integral operator given by (5). We also consider some special cases of our results which lead to various interesting corollaries and relevance of some of these results with other known results which are also mentioned.We will assume throughout our discussion, unless otherwise stated, thatλ is real with |λ|<π/2, 0≤ξ<p, τ>p,p, n∈ℕ, αj>0 for j∈{1,…m}, δ>0, α≥0, and
(6)Jα(f)=(1-α)zf′(z)f(z)+α(zf′(z))′f′(z).To obtain our main results, we need the following lemmas.Lemma 1 (see [27]).
Ifq(z)∈Σ(n) with n≥1 and satisfies the condition
(7)|z2q′(z)+1|<nn2+1(z∈𝕌*),
then
(8)q(z)∈Σ𝒮(n).Lemma 2 (see [28]).
Letψ:ℂ2→ℂ satisfy the following condition:
(9)Re{ψ(is,t)}≤0,(s,t∈ℝ;t≤-|a+is|22,Re(a)>0,).
If the functionh(z)=a+h1z+h2z2+⋯ is analytic in 𝕌=𝕌*∖{0} and
(10)Re{ψ(h(z),zh′(z))}>0,(z∈𝕌),
then
(11)Reh(z)>0,(z∈𝕌).


## 2. Sufficiency Criteria for the ClassΣℳp(λ,n,α,ξ)

In this section we establish a new sufficiency criteria for the subclassΣℳp(λ,n,α,ξ) of Σp(n).Theorem 3.
Iff(z)∈Σp(n) satisfies
(12)|{(zpf(z)(-zf′(z)pf(z))α)eiλ/(p-ξ)cosλeiλJα(f)(zpf(z)(-zf′(z)pf(z))α)eiλ/(p-ξ)cosλ+ξcosλ+ipsinλ}+(p-ξ)cosλ(zpf(z)(-zf′(z)pf(z))α)eiλ/(p-ξ)cosλ|<nn2+1(p-ξ)cosλ(z∈𝕌*),
then f(z)∈Σℳp(λ,n,α,ξ), where Jα(f) is given by (6).Proof.
Let us set a functionq(z) by
(13)q(z)=1z(zpf(z)(-zf′(z)pf(z))α)eiλ/(p-ξ)cosλ=1z+αeiλanbn(p-ξ)cosλzn+⋯
for f(z)∈Σp(n). Then clearly (13) shows that q(z)∈Σ(n).
Logarithmic differentiating of (13) gives
(14)q′(z)q(z)=eiλ(p-ξ)cosλ×[(1-α)f′(z)f(z)+α(zf′(z))′zf′(z)+pz]-1z
which further implies
(15)|z2q′(z)+1|=|(zpf(z)(-zf′(z)pf(z))α)eiλ/(p-ξ)cosλeiλ(p-ξ)cosλ×[Jα(f)+ξcosλ+ipsinλ]+1(zpf(z)(-zf′(z)pf(z))α)eiλ/(p-ξ)cosλ|.
Thus using (12), we get
(16)|z2q′(z)+1|≤nn2+1,(z∈𝕌*).
Therefore by Lemma 1, we have q(z)∈Σ𝒮(n).
From (14), we can write
(17)zq′(z)q(z)=1(p-ξ)cosλ[eiλJα(f)+ξcosλ+ipsinλ].
Since q(z)∈Σ𝒮(n), it implies that Re(-zq′(z)/q(z))>0. Therefore, we get
(18)1(p-ξ)cosλ[-ReeiλJα(f)-ξcosλ]=Re(-zq′(z)q(z))>0
or
(19)-ReeiλJα(f)>ξcosλ.
And therefore f(z)∈Σℳp(λ,n,α,ξ).
By takingα=0 and α=1 in Theorem 3, we obtain Corollaries 4 and 5, respectively, proved by Arif  [12].Corollary 4.
Iff(z)∈Σp(n) satisfies
(20)|(zpf(z))eiλ/(p-ξ)cosλ{eiλzf′(z)f(z)+ξcosλ+ipsinλ}+(p-ξ)cosλ{eiλzf′(z)f(z)+ξcosλ+ipsinλ}|<n(p-ξ)cosλn2+1(z∈𝕌*),
then f(z)∈Σ𝒮p(λ,n,ξ).Corollary 5.
Iff(z)∈Σp(n) satisfies
(21)|(zp+1f′(z)-p)eiλ/(p-ξ)cosλ×{eiλ(zf′′(z)f′(z)+1)+ξcosλ+ipsinλ}+(p-ξ)cosλ(zp+1f′(z)-p)eiλ/(p-ξ)cosλ|<nn2+1(p-ξ)cosλ,(z∈𝕌*),
then f(z)∈Σ𝒞p(λ,n,ξ).Remarks. We note that by simple computation (13) gives
(22)f(z)=[-pα∫0z‍(tq(t))(p-ξ)cosλ/αeiλtp/α+1]α.
By taking suitable meromorphic starlike function for q(z) in (22) such as q(z)=(1-zn)2/z, which satisfies the inequality of Lemma 1, we can conclude that the functionfof (22) is the subclass of a meromorphic function.


## 3.  Some Properties of the Integral OperatorGm(z)

In this section, we discuss some mapping properties of the integral operatorGm(z).Theorem 6.
Forj∈{1,…m}, let fj(z)∈Σp(n) and satisfy (20). If
(23)∑j=1m‍αj<(p+1)(2-δ)(p-ξ),0<δ<2,
then Gm(z)∈Σ𝒩p(λ,n,δ,η) with η>p, and Gm(z) is given by (5).Proof.
From (5), we obtain
(24)δzp+1Gmδ-1(z)Gm′(z)+(p+1)zpGmδ(z)=δzδ-1∏j=1m‍(zpfj(z))αj.
Dividing both sides by zpGmδ-1(z), we have
(25)δzGm′(z)+(p+1)Gm(z)=δzδ-p-1Gm1-δ(z)∏j=1m‍(zpfj(z))αj.
Differentiating again logarithmically, we have
(26)δzGm′′(z)+(p+δ+1)Gm′(z)δzGm′(z)+(p+1)Gm(z)=(δ-p-1)1z+(1-δ)Gm′(z)Gm(z)+∑j=1m‍αj(fj′(z)fj(z)+pz).
Now by simple computation, we get
(27)(1-1δ)zGm′(z)Gm(z)+1δ(zGm′(z))′Gm′(z)=1δ∑j=1m‍αj(zfj′(z)fj(z)+p)-1δ(2+2p-δ)+Gm(z)zGm′(z)×[(p+1)∑j=1m‍αj(zfj′(z)fj(z)+p)+(p+1)(δ-p-1)],
or equivalently we have
(28)-eiλ{(1-1δ)zGm′(z)Gm(z)+1δ(zGm′(z))′Gm′(z)}=1δ∑j=1m‍αj(-eiλzfj′(z)fj(z)-peiλ)+1δ(2+2p-δ)eiλ+Gm(z)zGm′(z)×[(p+1-δ)-∑j=1m‍αj(zfj′(z)fj(z)+p)](p+1)eiλ.
By taking real part on both sides, we obtain
(29)-Reeiλ{(1-1δ)zGm′(z)Gm(z)+1δ(zGm′(z))′Gm′(z)}=1δ∑j=1m‍αj(-Reeiλzfj′(z)fj(z)-pcosλ)+1δ(2+2p-δ)cosλ+ReGm(z)zGm′(z)×[(p+1-δ)-∑j=1m‍αj(zfj′(z)fj(z)+p)](p+1)eiλ,
which further implies that
(30)-Reeiλ{(1-1δ)zGm′(z)Gm(z)+1δ(zGm′(z))′Gm′(z)}≤1δ∑j=1m‍αj(-Reeiλzfj′(z)fj(z)-pcosλ)+1δ(2+2p-δ)cosλ+|Gm(z)zGm′(z)[∑j=1m‍(p+1-δ)-∑j=1m‍αj(zfj′(z)fj(z)+p)](p+1)eiλ|.
Let
(31)η=|Gm(z)zGm′(z)[∑j=1m‍(p+1-δ)-∑j=1m‍αj(zfj′(z)fj(z)+p)](p+1)eiλ|+1δ(2+2p-δ)+1δ∑j=1m‍αj(-1cosλReeiλzfj′(z)fj(z)-p).
Clearly we have
(32)η>1δ(2+2p-δ)+1δcosλ∑j=1m‍αj(-Reeiλzfj′(z)fj(z)-pcosλ).
Then by using (23)  and Theorem 3 with α=0, we obtain
(33)η>1δ[∑j=1m‍αj(ξ-p)+(2+2p-δ)]>p.
Therefore Gm(z)∈Σ𝒩p(λ,n,δ,η) with η>p.
Makingδ=1in Theorem 6, we have the following.Corollary 7.
Forj∈{1,…m}, let fj(z)∈Σp(n) and satisfy (20). If
(34)∑j=1m‍αj<(p+1)(p-ξ),
then Im,p(fj(z))∈Σ𝒩𝒞p(λ,n,η)with η>p.Theorem 8.
Forj∈{1,…m}, let fj(z)∈Σ𝒮p(0,n,ξ). If
(35)∑j=1m‍αj<p+1-δp-ξ,
then Gm(z)∈Σ𝒮p(0,n,ρ), where 0<ρ≤p with ρ=p+1-δ+(ξ-p)∑j=1mαj.Proof.
From (26), we obtain
(36)δ(zGm′′(z)/Gm′(z)+1)+(p+1)δ(zGm′(z)/Gm(z))+(p+1)zGm′(z)Gm(z)=(δ-p-1)+(1-δ)zGm′(z)Gm(z)+∑j=1m‍αj(zfj′(z)fj(z)+p).
Let
(37)-eiλzGm′(z)Gm(z)=q(z)=((p-ρ)p(z)+ρ)cosλ+isinλ,
such that q(z) is analytic in 𝕌* with q(0)=p. Then (36)  can be written as
(38)q(z)+zq′(z)eiλ(p+1)eiλ-δq(z)=(p+1-δ)δeiλ+1δ∑j=1m‍αj(-eiλzfj′(z)fj(z)-peiλ).
Taking real part on both sides, we have
(39)Re(q(z)+zq′(z)eiλ(p+1)eiλ-δq(z))=p+1-δδcosλ+1δ∑j=1m‍αj(-Reeiλzfj′(z)fj(z)-pcosλ)>1δ(p+1-δ-(p-ξ)∑j=1m‍αj)cosλ>0,
where we have used (35)  and the assumption that fj(z)∈Σ𝒮p(λ,n,ξ). Let us put
(40)ψ(u,v)=u+veiλ(p+1)eiλ-δu.
Then, for s,t∈ℝ such that t≤-(p2+s2)/2, we have
(41)Re(ψ(is,t))=Re{is+teiλ(p+1)eiλ-iδs}=-[(p+1)-δssinλ](p2+s2)2[(p+1)2cos2λ+δ2s2(1-(p+1)sinλ)2]<0.
Thus using Lemma 2, we conclude that Req(z)>0, which is equivalent to
(42)-ReeiλzGm′(z)Gm(z)>ρcosλ.
That is, Gm(z)∈Σ𝒮p(λ,n,ρ).



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