1. Introduction and Preliminaries
Let 𝒜 be the class of functions analytic in the open unit disk
(1)𝕌:={z:z∈ℂ, |z|<1}.
Suppose that f and g are in 𝒜. We say that f is subordinate to g (or g is superordinate to f), written as
(2)f≺g in 𝕌 or f(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 parameters A 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 subclass 𝒜p (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∈ℂ when z∈𝕌; ℜ(t)>1 when z∈∂𝕌).
Important special cases of the function Φ(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 function f∈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, transformation ℐp,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)︸(m times),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 function φ(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 of ℐp,δλ
We state and prove the following convolution preserving properties of ℐp,δλ.
Theorem 2.
Let τ>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 functions fj(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.
Since φj(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 ×∫0zt(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) as z→-1.
This completes the proof of Theorem 2.
Taking μ=δ we get the following consequence.
Corollary 3.
Let τ>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.
Taking δ=1 in Corollary 3, we get the result due to Özkan (cf. [16, Theorem 1]).