# Convolution Properties ofp-Valent Functions Associated with a Generalization of the Srivastava-Attiya Operator

Authors: Priyabrat Gochhayat
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

Let𝒜p denote the class of functions analytic in the open unit disc 𝕌 and given by the series f(z)=zp+∑n=p+1∞anzn. For f∈𝒜p, the transformation ℐp,δλ:𝒜p→𝒜p defined by ℐp,δλf(z)=zp+∑n=p+1∞((p+δ)/(n+δ))λanzn,(δ+p∈ℂ∖ℤ0-,λ∈ℂ;z∈𝕌), has been recently studied as fractional differintegral operator by Mishra and Gochhayat (2010). In the present paper, we observed that ℐp,δλ can also be viewed as a generalization of the Srivastava-Attiya operator. Convolution preserving properties for a class of multivalent analytic functions involving an adaptation of the popular Srivastava-Attiya transform are investigated.


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## Body

## 1. Introduction and Preliminaries

Let𝒜 be the class of functions analytic in the open unit disk
(1)𝕌:={z:z∈ℂ,|z|<1}.Suppose thatf and g are in 𝒜. We say that f is subordinate to g (or g is superordinate to f), written as
(2)f≺gin𝕌orf(z)≺g(z)(z∈𝕌),
if there exists a function ω∈𝒜, satisfying the conditions of the Schwarz lemma (i.e.,ω(0)=0 and |ω(z)|<1) such that
(3)f(z)=g(ω(z))(z∈𝕌).
It follows that
(4)f(z)≺g(z)(z∈𝕌)⇒f(0)=g(0),f(𝕌)⊂g(𝕌).
In particular, if g is univalent in 𝕌, then the reverse implication also holds (cf. [1]).For real parametersA and B such that -1≤B<A≤1, recalling the function of the form:
(5)1+Az1+Bz(z∈𝕌),
which maps conformally 𝕌 onto a disk (whenever -1≤B≤1), symmetrical with respect to the real axis, which is centered at the point
(6)1-AB1-B2(B≠±1)
and with its radius equal to
(7)A-B1-B2(B≠±1).
Furthermore, the boundary circle of the disk intersects the real axis at the point (1-A)/(1-B) and (1+A)/(1+B) provided B≠±1.In this paper we will also be dealing with the subclass𝒜p(p∈ℕ:={1,2,3,…}) of 𝒜 consisting of functions of the following form:
(8)f(z)=zp+∑n=p+1∞‍anzn,(z∈𝕌).With a view to define the Srivastava-Attiya transform we recall here a general Hurwitz-Lerch-Zeta function, which is defined in [2, 3] by the following series:
(9)Φ(z,t,a)=1at+∑k=1∞‍zk(k+a)t(a∈ℂ∖ℤ0-,ℤ0-={0,-1,-2,…};t∈ℂwhenz∈𝕌;ℜ(t)>1whenz∈∂𝕌).Important special cases of the functionΦ(z,t,a) include, for example, the Reimann zeta function ζ(t)=Φ(1,t,1), the Hurwitz zeta function ζ(t,a)=Φ(1,t,a), the Lerch zeta function lt(ξ)=Φ(exp2πiξ,t,1)(ξ∈ℝ,ℜ(t)>1), the polylogarithm Lit(z)=zΦ(z,t,1) and so on. Recent results on Φ(z,t,a), can be found in the expositions [4, 5].By making use of the following normalized function:(10)𝒢t,a(z)=(1+a)t[Φ(z,t,a)-a-t]=z+∑k=2∞‍(1+ak+a)tzk(z∈𝕌),
Srivastava-Attiya [2] introduced the linear operator ℒt,a:𝒜1→𝒜1 by the following series:
(11)ℒt,af(z)=z+∑k=2∞‍(1+ak+a)takzk(z∈𝕌),
where the function f∈𝒜1 is, respectively, by
(12)f(z)=z+∑k=2∞‍akzk(z∈𝕌).
The operator ℒt,a is now popularly known in the literature as the Srivastava-Attiya operator. Various basic properties of ℒt,a are systematically investigated in [6–11].For a functionf∈𝒜p and represented by the series (8), the transformation
(13)ℐp,δλ:𝒜p→𝒜p
defined by
(14)ℐp,δλf(z)=zp+∑n=p+1∞‍(p+δn+δ)λanzn,(δ+p∈ℂ∖ℤ0-,λ∈ℂ;z∈𝕌),
has been recently studied as fractional differintegral operator by the authors [12]. We observed that ℐp,δλ can also be viewed as a generalization of the Srivastava-Attiya operator (take p=1,λ=t,δ=a in (14)), suitable for the study of multivalent functions. (Also see [13] for a variant.)Furthermore, transformationℐp,δλ generalizes several previously studied familiar operators. For example taking λ=0 we get the identity transformation; the choices λ=-1,δ=0 yield the Alexander transformation and p=1,λ a negative integer, δ=0 give the Sălăgean operator. Some more interesting particular cases are also pointed out by the authors in [12] (also see [14]).Using (14) it can be verified that
(15)z(ℐp,δλℐp,μ1f(z))′=(p+μ)ℐp,δλf(z)-μ(ℐp,δλℐp,μ1f(z)).
For the functions fj(z)∈𝒜p given by
(16)fj(z)=zp+∑n=p+1∞‍an,jzn(j=1,2;z∈𝕌),
their Hadamard product (or convolution) is defined by
(17)f1(z)*f2(z)=zp+∑n=p+1∞‍an,1an,2zn=f2(z)*f1(z)(z∈𝕌).
Observe that when λ=m∈ℕ, the operator ℐp,δλ given by (14) can be represented in terms of convolution as follows:
(18)ℐp,δmf(z)=ψp,δm(z)*f(z),
where
(19)ψp,δm(z)=ψp,δ(z)*ψp,δ(z)*⋯*ψp,δ(z)︸(mtimes),ψp,δ(z)=zp+∑n=p+1∞‍(p+δn+δ)zn(z∈𝕌).
In the sequel to earlier investigations, in the present paper we find a convolution result involving ℐp,δλ is also presented.With a view to state a well-known result, we denote by℘(ρ) the class of functions as follows:
(20)φ(z)=1+c1z+c2z2+⋯(z∈𝕌),
which are analytic in 𝕌 and satisfy ℜ(φ(z))>ρ(0≤ρ<1). A function φ∈℘(ρ) if and only if (φ(z)-ρ)/(1-ρ)∈℘(0). The following result is a consequence of the principle of subordination.Lemma 1.
Let the functionφ(z), given by (20), be in the class ℘(ρ). Then
(21)ℜ(φ(z))≥2ρ-1+2(1-ρ)1+|z|(0≤ρ<1;z∈𝕌).


## 2. Convolution Properties ofℐp,δλ

We state and prove the following convolution preserving properties ofℐp,δλ.Theorem 2.
Letτ>0 and -1≤Bj<Aj≤1(j=1,2). If each of the functions fj(z)∈𝒜p(j=1,2) satisfies the following subordination condition:
(22)τℐp,δλfj(z)zp+(1-τ)ℐp,δλℐp,μ1fj(z)zp≺1+Ajz1+Bjz(j=1,2;δ,μ>-p;z∈𝕌)
then
(23)τℐp,δλH(z)zp+(1-τ)ℐp,δλℐp,μ1H(z)zp≺1+(1-2χ)z1-z(δ,μ>-p;z∈𝕌),
where
(24)H(z)=ℐp,δλℐp,μ1(f1*f2)(z)(δ,μ>-p;z∈𝕌),(25)χ=1-4(A1-B1)(A2-B2)(1-B1)(1-B2)×[1-p+μτ∫01‍u(p+μ)/τ-11+udu].
The result is the best possible for B1=B2=-1.Proof.
Suppose that each of the functionsfj(z)∈𝒜p(j=1,2) satisfies the condition (22). Set
(26)φj(z):=τℐp,δλfj(z)zp+(1-τ)ℐp,δλℐp,μ1fj(z)zp(j=1,2;δ,μ>-p;z∈𝕌).
Then, by making use of the identity (15) in (26) we get
(27)ℐp,δλℐp,μ1fj(z)=(p+μ)τzp-(p+μ)/τ∫0z‍t(p+μ)/τ-1φj(t)dt(j=1,2).
Therefore, a simple computation, by using (24) and (27), shows that
(28)ℐp,δλℐp,μ1H(z)=(p+μ)τzp-(p+μ)/τ∫0z‍t((p+μ)/τ)-1φ0(t)dt,
where
(29)φ0(z)=τℐp,δλH(z)zp+(1-τ)ℐp,δλℐp,μ1H(z)zp=(p+μ)τzp-(p+μ)/τ∫0z‍t(p+μ)/τ-1(φ1*φ2)(t)dt.
The proof will be completed by finding the best possible lower bound for ℜ(φ0(z)). A change of variable also gives
(30)φ0(z)=(p+μ)τ∫01‍u(p+μ)/τ-1(φ1*φ2)(uz)du.
Sinceφj(z)∈℘(ρj)(j=1,2), where ρj=((1-Aj)/(1-Bj))(j=1,2), it follows from a result in [15] that
(31)(φ1*φ2)(z)∈℘(ρ3)(ρ3=1-2(1-ρ1)(1-ρ2))
and the bound ρ3 is the best possible. An application of Lemma 1, in (30), yields
(32)ℜ{φ0(z)}≥(p+μ)τ∫01‍u(p+μ)/τ-1[2ρ3-1+2(1-ρ3)1+u|z|]du>(p+μ)τ∫01‍u(p+μ)/τ-1[2ρ3-1+2(1-ρ3)1+u]du=1-4(A1-B1)(A2-B2)(1-B1)(1-B2)×[1-p+μτ∫01‍u((p+μ)/τ)-11+udu]=χ.
In order to show that χ is the best possible in the assertion (23) when B1=B2=-1, we consider the function fj(z)∈𝒜p given by
(33)ℐp,δλℐp,μ1fj(z)=(p+μ)τzp-(p+μ)/τ×∫0z‍t(p+μ)/τ-1(1+Ajt1-t)dt(j=1,2).
It is readily checked that fj satisfies (22) with Bj=-1. Since
(34)(1+A1t1-t)*(1+A2t1-t)=1-(1+A1)(1+A2)+(1+A1)(1+A2)1-z,
it follows from (30) and Lemma 1 that
(35)φ0(z)=(p+μ)τz(p+μ)/τ×∫0z‍t(p+μ)/τ-1(1-(1+A1)(1+A2)+(1+A1)(1+A2)1-t)dt=(p+μ)τ∫01‍u(p+μ)/τ-1((1+A1)(1+A2)1-uz1-(1+A1)(1+A2)(p+μ)τ∫01‍u(p+μ)/τ-1l+(1+A1)(1+A2)1-uz)du→1-(1+A1)(1+A2)×(1-(p+μ)τ∫01‍u(p+μ)/τ-11+udu)asz→-1.
This completes the proof of Theorem 2.Takingμ=δ we get the following consequence.Corollary 3.
Letτ>0 and -1≤Bj<Aj≤1(j=1,2). If each of the functions fj(z)∈𝒜p(j=1,2) satisfies the following subordination condition:
(36)τℐp,δλfj(z)zp+(1-τ)ℐp,δλ+1fj(z)zp≺1+Ajz1+Bjz(j=1,2;δ>-p;z∈𝕌)
then
(37)τℐp,δλH(z)zp+(1-τ)ℐp,δλ+1H(z)zp≺1+(1-2χ)z1-z(δ>-p;z∈𝕌),
where
(38)H(z)=ℐp,δλ+1(f1*f2)(z)(δ>-p;z∈𝕌),χ=1-4(A1-B1)(A2-B2)(1-B1)(1-B2)[1-p+δτ∫01‍u(p+δ)/τ-11+udu].
The result is best possible for B1=B2=-1.Remark 4.
Takingδ=1 in Corollary 3, we get the result due to Özkan (cf. [16, Theorem 1]).



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