# Convolution Properties for Some Subclasses of Meromorphic Functions of Complex Order

Authors: M. K. Aouf; A. O. Mostafa; H. M. Zayed
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

Making use of the operatorLυ for functions of the form fz=1/z+∑k=1∞akzk-1, which are analytic in the punctured unit disc U∗={z:z∈C and 0<|z|<1}=U∖{0}, we introduce two subclasses of meromorphic functions and investigate convolution properties, coefficient estimates, and containment properties for these subclasses.


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## Body

## 1. Introduction

LetΣ denote the class of meromorphic functions of the form(1)fz=1z+∑k=1∞akzk-1,which are analytic in the punctured unit disc U∗={z:z∈C and 0<|z|<1}=U∖{0}. Let g(z)∈Σ be given by(2)gz=1z+∑k=1∞bkzk-1;then, the Hadamard product (or convolution) of f(z) and g(z) is given by(3)f∗gz=1z+∑k=1∞akbkzk=g∗fz.We recall some definitions which will be used in our paper.Definition 1.
For two functionsf(z) and g(z), analytic in U, we say that the function f(z) is subordinate to g(z) in U and written f(z)≺g(z), if there exists a Schwarz function w(z), analytic in U with w(0)=0 and w(z)<1 such that f(z)=g(w(z))  (z∈U). Furthermore, if the function g(z) is univalent in U, then we have the following equivalence (see [1]):(4)fz≺gz⟺f0=g0,fU⊂gU.Now, consider Bessel’s function of the first kind of orderυ where υ is an unrestricted (real or complex) number, defined by (see Watson [2, page 40]) (see also Baricz [3, page 7])(5)Jυz=∑k=0∞-1kΓk+1Γk+υ+1z22k+ν,which is a particular solution of the second order linear homogenous Bessel differential equation (see, e.g., Watson [2, page 38]) (see also Baricz [3, page 7])(6)z2w′′z+zw′z+z2-υ2wz=0.Also, let us define(7)Lυz=2υΓυ+1zυ/2+1Jυz1/2=1z+∑k=1∞-1kΓυ+14kΓk+1Γk+υ+1zk-1z∈U∗.The operator Lυ is a modification of the operator introduced by Szász and Kupán [4] for analytic functions.By using the Hadamard product (or convolution), we define the operatorLυ as follows:(8)Lυfz=Lυz∗fz=1z+∑k=1∞-1kΓυ+14kΓk+1Γk+υ+1akzk-1.It is easy to verify from (8) that(9)zLυ+1fz′=υ+1Lυfz-υ+2Lυ+1fz.Definition 2.
For0≤λ<1, -1≤B<A≤1, and b∈C∗=C∖{0}, let ΣSλ∗[b;A,B] be the subclass of Σ consisting of function f(z) of the form (1) and satisfying the analytic criterion(10)1+1b-zf′z1-λfz-λzf′z-1≺1+Az1+Bz.Also, letΣKλ[b;A,B] be the subclass of Σ consisting of function f(z) of the form (1) and satisfying the analytic criterion(11)1+1b-zzf′z′1-λzf′z-λzzf′z′-1≺1+Az1+Bz.It is easy to verify from (10) and (11) that(12)fz∈ΣKλb;A,B⟺-zf′z∈ΣSλ∗b;A,B.We note that(i)
ΣS0∗b;A,B=ΣS∗b;A,B and ΣK0[b;A,B]=ΣK[b;A,B] (see Bulboacă et al. [5]);(ii)
ΣS0∗[b;1,-1]=ΣS(b) and ΣK0[b;1,-1]=ΣK(b) (see Aouf [6]);(iii)
ΣS0∗[(1-α)e-iμcosμ;1,-1]=ΣSμ(α) and ΣK0[(1-α)e-iμcosμ;1,-1]=ΣKμ(α)  (μ∈R,μ≤π/2,0≤α<1) (see Ravichandran et al. [7, with p=1]).Definition 3.
For0≤λ<1,-1≤B<A≤1,b∈C∗ and υ is an unrestricted (real or complex) number, let(13)ΣSλ,υ∗b;A,B=fz∈Σ:Lυfz∈ΣSλ∗b;A,B,ΣKλ,υb;A,B=fz∈Σ:Lυfz∈ΣKλb;A,B.It is easy to show that(14)fz∈ΣKλ,υb;A,B⟺-zf′z∈ΣSλ,υ∗b;A,B.The object of the present paper is to investigate some convolution properties, coefficient estimates, and containment properties for the subclassesΣSλ,υ∗[b;A,B] and ΣKλ,υ[b;A,B].


## 2. Main Results

Unless otherwise mentioned, we assume throughout this paper that0≤λ<1,-1≤B<A≤1,b∈C∗ and υ is an unrestricted (real or complex) number.Theorem 4.
Iff(z)∈Σ, then f(z)∈ΣSλ∗[b;A,B] if and only if(15)zfz∗1-λ-1M+λ+1zz1-z2≠0forz∈U,where M=Mθ=e-iθ+B/(A-B)b, θ∈[0,2π), and also M=0.Proof.
It is easy to verify that(16)fz∗1z1-z=fz,fz∗1z1-z2-21-z2=-zf′z∀z∈U∗;f∈Σ.(i) In view of (10), f(z)∈ΣSλ∗[b;A,B] if and only if (10) holds. Since the function 1+[B+(A-B)b]z/1+Bz is analytic in U, it follows that (1-λ)f(z)-λzf′z≠0 for z∈U∗ or z(1-λ)f(z)-λzf′(z)≠0 for z∈U; this is equivalent to (15) holdding for M=0. To prove (15) for all M≠0, we write (10) by using the principle of subordination as(17)-zf′z1-λfz-λzf′z=1+B+A-Bbwz1+Bwz,where w(z) is Schwarz function, analytic in U with w(0)=0 and w(z)<1; hence,(18)z-zf′z1+Beiθ-1-λfz-λzf′z·1+B+A-Bbeiθ≠0forz∈U,θ∈0,2π.Using (16), (18) may be written as(19)zfz∗1-λ-1e-iθ+B/A-Bb+λ+1zz1-z2≠0forz∈U.Thus, the first part of Theorem 4 was proved.
(ii) Reversely, because assumption (15) holds for M=0, it follows that z(1-λ)f(z)-λzf′(z)≠0 for z∈U. This implies that φ(z)=-zf′(z)/(1-λ)f(z)-λzf′(z) is analytic in U (i.e., it is regular in z=0, with φ(0)=1).
Since it was shown in the first part of the proof that assumption (18) is equivalent to (15), we obtain that(20)-zf′z1-λfz-λzf′z≠1+B+A-Bbeiθ1+Beiθforz∈U,θ∈0,2π.Assume that(21)ψz=1+B+A-Bbeiθ1+Beiθ.Relation (20) means that φ(U)∩ψ(∂U)=∅. Thus, the simply connected domain is included in a connected component of C∖ψ(∂U). From this, using the fact that φ(0)=ψ(0) and the univalence of the function ψ, it follows that φ(z)≺ψ(z); this implies that f(z)∈ΣSλ,υ∗[b;A,B]. Thus, the proof of Theorem 4 is completed.Remark 5.
(i) Puttingλ=0 in Theorem 4, we obtain the result obtained by Bulboacă et al. [5, Theorem 1].
(ii) Puttingλ=0, b=1, and eiθ=x in Theorem 4, we obtain the result obtained by Ponnusamy [8, Theorem 2.1].
(iii) Puttingλ=0, b=(1-α)e-iμcosμ(μ∈R,μ≤π/2,0≤α<1), A=1, B=-1, and eiθ=x in Theorem 4, we obtain the result obtained by Ravichandran et al. [7, Theorem 1.2 with p=1].Theorem 6.
Iff(z)∈Σ, then f(z)∈ΣKλ[b;A,B] if and only if(22)zfz∗1-3z+2λ-1M+λ+1z2z1-z3≠0forz∈U,where M=Mθ=e-iθ+B/(A-B)b, θ∈[0,2π), and also M=0.Proof.
Putting(23)gz=1-λ-1M+λ+1zz1-z2,then(24)-zg′z=1-3z+2λ-1M+λ+1z2z1-z3.From (12) and using the identity(25)-zf′z∗gz=fz∗-zg′z,we obtain the required result from Theorem 4.Remark 7.
(i) Puttingλ=0 in Theorem 6, we obtain the result obtained by Bulboacă et al. [5, Theorem 2].
(ii) Puttingλ=0, b=1, and eiθ=x in Theorem 4, we obtain the result obtained by Ponnusamy [8, Theorem 2.2].Theorem 8.
Iff(z)∈Σ, then f(z)∈ΣSλ,υ∗[b;A,B] if and only if(26)1+∑k=1∞-1k1-kλΓυ+14kΓk+1Γk+υ+1akzk≠0,(27)1+∑k=1∞-1kΓυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1--kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+1akzk≠0,for all θ∈[0,2π).Proof.
Iff(z)∈Σ, from Theorem 4, we have f(z)∈ΣSλ,υ∗[b;A,B] if and only if(28)zLυfz∗1-λ-1M+λ+1zz1-z2≠0forz∈U,where M=Mθ=e-iθ+B/(A-B)b, θ∈[0,2π), and also M=0. Since(29)1-λ+1zz1-z2=1z+∑k=1∞1-kλzk-1,z∈U∗,it is easy to show that (28) holds for M=0 if and only if (26) holds. Also, (30)1-λ-1M+λ+1zz1-z2=1z+∑k=1∞1-kλ-kλ-1Mzk-1,z∈U∗;we may easily check that (28) is equivalent to (27). This completes the proof of Theorem 8.Theorem 9.
Iff(z)∈Σ, then f(z)∈ΣKλ,υ[b;A,B] if and only if(31)1+∑k=1∞-1kkλ-1k-1Γυ+14kΓk+1Γk+υ+1akzk≠0,(32)1+∑k=1∞-1kk-1Γυ+1·kλ-1A-Bb4kA-BbΓk+1Γk+υ+1+kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+1akzk≠0,for all θ∈[0,2π).Proof.
Iff(z)∈Σ, from Theorem 6, we have f(z)∈ΣKλ,υ[b;A,B] if and only if(33)zLυfz∗1-3z+2λ-1M+λ+1z2z1-z3≠0forz∈U,where M=Mθ=e-iθ+B/(A-B)b, θ∈[0,2π), and also M=0. Since(34)1-3z+2λ+1z2z1-z3=1z+∑k=1∞kλ-1k-1zk-1,z∈U∗,it is easy to show that (33) holds for M=0 if and only if (31) holds. Also, (35)1-3z+2λ-1M+λ+1z2z1-z3=1z+∑k=1∞k-1kλ-1M+kλ-1zk-1;for z∈U∗, we may easily check that (33) is equivalent to (32). This completes the proof of Theorem 9.Unless otherwise mentioned, we assume throughout the remainder of this section thatυ is a real number (υ>-1).Theorem 10.
Iff(z)∈Σ satisfies inequalities(36)∑k=1∞kλ-1Γυ+14kΓk+1Γk+υ+1ak<1,(37)∑k=1∞1-kλA-Bb+k1-λ1+BΓυ+14kΓk+1Γk+υ+1ak<A-Bb,then f(z)∈ΣSλ,υ∗[b;A,B].Proof.
We have(38)1-∑k=1∞-1kkλ-1Γυ+14kΓk+1Γk+υ+1akzk≥1-∑k=1∞-1kkλ-1Γυ+14kΓk+1Γk+υ+1akzk≥1-∑k=1∞kλ-1Γυ+14kΓk+1Γk+υ+1akzk≥1-∑k=1∞kλ-1Γυ+14kΓk+1Γk+υ+1ak>0,forz∈U,which implies inequality (36). Also,(39)1+∑k=1∞-1kΓυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1-kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+1akzk≥1-∑k=1∞-1kΓυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1-kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+1akzk≥1-∑k=1∞Γυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1+k1-λ1+B4kA-BbΓk+1Γk+υ+1akzk≥1-∑k=1∞Γυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1+k1-λ1+B4kA-BbΓk+1Γk+υ+1ak>0,forz∈U,which implies inequality (37). Thus, the proof of Theorem 10 is completed.Using similar arguments to those in the proof of Theorem10, we obtain the following theorem.Theorem 11.
Iff(z)∈Σ satisfies inequalities(40)∑k=1∞kλ-1k-1Γυ+14kΓk+1Γk+υ+1ak<1,∑k=1∞k-1Γυ+1kλ-1A-Bb4kΓk+1Γk+υ+1+k1-λ1+B4kΓk+1Γk+υ+1ak<A-Bb,then f(z)∈ΣKλ,υ[b;A,B].Now, using the method due to Ahuja [9], we will prove the containment relations for the subclasses ΣSλ,υ∗[b;A,B] and ΣKλ,υ[b;A,B].Theorem 12.
Forυ>-1, we have ΣSλ,υ+1∗[b;A,B]⊂ΣSλ,υ∗[b;A,B].Proof.
Sincef(z)∈ΣSλ,υ+1∗[b;A,B], we see from Theorem 8 that(41)1+∑k=1∞-1k1-kλΓυ+24kΓk+1Γk+υ+2akzk≠0,1+∑k=1∞-1kΓυ+2·1-kλA-Bb4kA-BbΓk+1Γk+υ+2-kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+2akzk≠0.We can write (41) as(42)1+∑k=1∞υ+1k+υ+1zk∗1+∑k=1∞-1k1-kλΓυ+14kΓk+1Γk+υ+1akzk≠0,1+∑k=1∞υ+1k+υ+1zk∗1+∑k=1∞-1kΓυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1-kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+1akzk≠0,since(43)1+∑k=1∞υ+1k+υ+1zk∗1+∑k=1∞k+υ+1υ+1zk=1+∑k=1∞zk.By using the property, if f≠0 and g∗h≠0, then f∗(g∗h)≠0; (42) can be written as(44)1+∑k=1∞-1k1-kλΓυ+14kΓk+1Γk+υ+1akzk≠0,1+∑k=1∞-1kΓυ+1·1-kλA-Bb4kA-BbΓk+1Γk+υ+1-kλ-1e-iθ+B4kA-BbΓk+1Γk+υ+1akzk≠0,which means that f(z)∈ΣSλ,υ∗[b;A,B]. This completes the proof of Theorem 12.Using the same arguments as in the proof of Theorem12, we obtain the following theorem.Theorem 13.
Forυ>-1, we have ΣKλ,υ+1[b;A,B]⊂ΣKλ,υ[b;A,B].



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