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

Abstract

Let 𝒜_𝑝 denote the class of functions analytic in the open unit disc 𝕌 and given by the series 𝑓(𝑧)=𝑧^𝑝+∑^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛. For 𝑓∈𝒜_𝑝, the transformation ℐ^𝜆_𝑝,𝛿∶𝒜_𝑝→𝒜_𝑝 defined by ℐ^𝜆_𝑝,𝛿𝑓(𝑧)=𝑧^𝑝+∑^∞_𝑛=𝑝+1((𝑝+𝛿)/(𝑛+𝛿))^𝜆𝑎_𝑛𝑧^𝑛,(𝛿+𝑝∈ℂ⧵ℤ⁻₀,𝜆∈ℂ;𝑧∈𝕌), has been recently studied as fractional differintegral operator by Mishra and Gochhayat (2010). In the present paper, we observed that ℐ^𝜆_𝑝,𝛿 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.

1. Introduction and Preliminaries

Let 𝒜 be the class of functions analytic in the open unit disk 𝕌∶={𝑧∶𝑧∈ℂ,|𝑧|<1}. (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 𝑓≺𝑔in𝕌or𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝕌), (2) 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 𝑓(𝑧)=𝑔(𝜔(𝑧))(𝑧∈𝕌). (3) It follows that 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝕌)⇒𝑓(0)=𝑔(0),𝑓(𝕌)⊂𝑔(𝕌). (4) 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: 1+𝐴𝑧(),1+𝐵𝑧𝑧∈𝕌 (5) which maps conformally 𝕌 onto a disk (whenever −1≤𝐵≤1), symmetrical with respect to the real axis, which is centered at the point 1−𝐴𝐵1−𝐵²(𝐵≠±1) (6) and with its radius equal to 𝐴−𝐵1−𝐵²(𝐵≠±1). (7) 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 𝒜_𝑝(𝑝∈ℕ∶={1,2,3,…}) of 𝒜 consisting of functions of the following form: 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛𝑧^𝑛,(𝑧∈𝕌). (8)

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: Φ()=1𝑧,𝑡,𝑎𝑎^𝑡+^∞_𝑘=1𝑧^𝑘(𝑘+𝑎)^𝑡𝑎∈ℂ⧵ℤ⁻₀,ℤ⁻₀={0,−1,−2,…};𝑡∈ℂwhen𝑧∈𝕌;ℜ(𝑡)>1when𝑧∈𝜕𝕌). (9)

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 𝑙_𝑡(𝜉)=Φ(exp^{2𝜋𝑖𝜉},𝑡,1)(𝜉∈ℝ,ℜ(𝑡)>1), the polylogarithm 𝐿𝑖_𝑡(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: 𝒢_𝑡,𝑎(𝑧)=(1+𝑎)^𝑡Φ(𝑧,𝑡,𝑎)−𝑎^−𝑡=𝑧+^∞_𝑘=21+𝑎𝑘+𝑎^𝑡𝑧^𝑘(𝑧∈𝕌), (10) Srivastava-Attiya [2] introduced the linear operator ℒ_𝑡,𝑎∶𝒜₁→𝒜₁ by the following series: ℒ_𝑡,𝑎𝑓(𝑧)=𝑧+^∞_𝑘=21+𝑎𝑘+𝑎^𝑡𝑎_𝑘𝑧^𝑘(𝑧∈𝕌), (11) where the function 𝑓∈𝒜₁ is, respectively, by 𝑓(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑧^𝑘(𝑧∈𝕌). (12) The operator ℒ_𝑡,𝑎 is now popularly known in the literature as the Srivastava-Attiya operator. Various basic properties of ℒ_𝑡,𝑎 are systematically investigated in [6–11].

For a function 𝑓∈𝒜_𝑝 and represented by the series (8), the transformation ℐ^𝜆_𝑝,𝛿∶𝒜_𝑝→𝒜_𝑝 (13) defined by ℐ^𝜆_𝑝,𝛿𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑝+𝛿𝑛+𝛿^𝜆𝑎_𝑛𝑧^𝑛,𝛿+𝑝∈ℂ⧵ℤ⁻₀,𝜆∈ℂ;𝑧∈𝕌, (14) has been recently studied as fractional differintegral operator by the authors [12]. We observed that ℐ^𝜆_𝑝,𝛿 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 ℐ^𝜆_𝑝,𝛿 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 𝑧ℐ^𝜆_𝑝,𝛿ℐ¹_𝑝,𝜇𝑓(𝑧)^′=𝑝+𝜇ℐ^𝜆_𝑝,𝛿𝑓(𝑧)−𝜇ℐ^𝜆_𝑝,𝛿ℐ¹_𝑝,𝜇𝑓(𝑧). (15) For the functions 𝑓_𝑗(𝑧)∈𝒜_𝑝 given by 𝑓_𝑗(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛,𝑗𝑧^𝑛𝑗=1,2;𝑧∈𝕌, (16) their Hadamard product (or convolution) is defined by 𝑓₁(𝑧)∗𝑓₂(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑎_𝑛,1𝑎_𝑛,2𝑧^𝑛=𝑓₂(𝑧)∗𝑓₁(𝑧)(𝑧∈𝕌). (17) Observe that when 𝜆=𝑚∈ℕ, the operator ℐ^𝜆_𝑝,𝛿 given by (14) can be represented in terms of convolution as follows: ℐ^𝑚_𝑝,𝛿𝑓(𝑧)=𝜓^𝑚_𝑝,𝛿(𝑧)∗𝑓(𝑧), (18) where 𝜓^𝑚_𝑝,𝛿(𝑧)=𝜓_𝑝,𝛿(𝑧)∗𝜓_𝑝,𝛿(𝑧)∗⋯∗𝜓_𝑝,𝛿(𝑧)_(𝑚times),𝜓_𝑝,𝛿(𝑧)=𝑧^𝑝+^∞_𝑛=𝑝+1𝑝+𝛿𝑧𝑛+𝛿^𝑛(𝑧∈𝕌). (19) In the sequel to earlier investigations, in the present paper we find a convolution result involving ℐ^𝜆_𝑝,𝛿 is also presented.

With a view to state a well-known result, we denote by ℘(𝜌) the class of functions as follows: 𝜑(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯(𝑧∈𝕌), (20) 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 ℜ𝜑(𝑧)2≥2𝜌−1+1−𝜌|𝑧|.1+0≤𝜌<1;𝑧∈𝕌 (21)

2. Convolution Properties of ℐ^𝜆_𝑝,𝛿

We state and prove the following convolution preserving properties of ℐ^𝜆_𝑝,𝛿.

Theorem 2. Let 𝜏>0 and −1≤𝐵_𝑗<𝐴_𝑗≤1(𝑗=1,2). If each of the functions 𝑓_𝑗(𝑧)∈𝒜_𝑝(𝑗=1,2) satisfies the following subordination condition: 𝜏ℐ^𝜆_𝑝,𝛿𝑓_𝑗(𝑧)𝑧^𝑝+()ℐ1−𝜏^{𝜆𝑝,𝛿}ℐ¹_𝑝,𝜇𝑓_𝑗(𝑧)𝑧^𝑝≺1+𝐴_𝑗𝑧1+𝐵_𝑗𝑧𝑗=1,2;𝛿,𝜇>−𝑝;𝑧∈𝕌 (22) then 𝜏ℐ^𝜆_𝑝,𝛿𝐻(𝑧)𝑧^𝑝+()ℐ1−𝜏^{𝜆𝑝,𝛿}ℐ¹_𝑝,𝜇𝐻(𝑧)𝑧^𝑝≺1+1−2𝜒𝑧,1−𝑧𝛿,𝜇>−𝑝;𝑧∈𝕌 (23) where 𝐻(𝑧)=ℐ^𝜆_𝑝,𝛿ℐ¹_𝑝,𝜇𝑓₁∗𝑓₂(𝑧)𝛿,𝜇>−𝑝;𝑧∈𝕌,4𝐴𝜒=1−¹−𝐵₁𝐴₂−𝐵₂1−𝐵₁1−𝐵₂×1−𝑝+𝜇𝜏¹₀𝑢^{(𝑝+𝜇)/𝜏−1}.1+𝑢𝑑𝑢 (24)(25) The result is the best possible for 𝐵₁=𝐵₂=−1.

Proof. Suppose that each of the functions 𝑓_𝑗(𝑧)∈𝒜_𝑝(𝑗=1,2) satisfies the condition (22). Set 𝜑_𝑗(𝑧)ℐ∶=𝜏^{𝜆𝑝,𝛿}𝑓_𝑗(𝑧)𝑧^𝑝+()ℐ1−𝜏^{𝜆𝑝,𝛿}ℐ¹_𝑝,𝜇𝑓_𝑗(𝑧)𝑧^𝑝𝑗=1,2;𝛿,𝜇>−𝑝;𝑧∈𝕌. (26)
Then, by making use of the identity (15) in (26) we get ℐ^𝜆_𝑝,𝛿ℐ¹_𝑝,𝜇𝑓_𝑗(𝑧)=𝑝+𝜇𝜏𝑧^{𝑝−(𝑝+𝜇)/𝜏𝑧}₀𝑡^{(𝑝+𝜇)/𝜏−1}𝜑_𝑗(𝑡)𝑑𝑡𝑗=1,2. (27) Therefore, a simple computation, by using (24) and (27), shows that ℐ^𝜆_𝑝,𝛿ℐ¹_𝑝,𝜇𝐻(𝑧)=𝑝+𝜇𝜏𝑧^{𝑝−(𝑝+𝜇)/𝜏𝑧}₀𝑡^{((𝑝+𝜇)/𝜏)−1}𝜑₀(𝑡)𝑑𝑡, (28) where 𝜑₀(𝑧)ℐ=𝜏^{𝜆𝑝,𝛿}𝐻(𝑧)𝑧^𝑝+()ℐ1−𝜏^{𝜆𝑝,𝛿}ℐ¹_𝑝,𝜇𝐻(𝑧)𝑧^𝑝=𝑝+𝜇𝜏𝑧^{𝑝−(𝑝+𝜇)/𝜏𝑧}₀𝑡^{(𝑝+𝜇)/𝜏−1}𝜑₁∗𝜑₂(𝑡)𝑑𝑡. (29) The proof will be completed by finding the best possible lower bound for ℜ(𝜑₀(𝑧)). A change of variable also gives 𝜑₀(𝑧)=𝑝+𝜇𝜏¹₀𝑢^{(𝑝+𝜇)/𝜏−1}𝜑₁∗𝜑₂(𝑢𝑧)𝑑𝑢. (30)
Since 𝜑_𝑗(𝑧)∈℘(𝜌_𝑗)(𝑗=1,2), where 𝜌_𝑗=((1−𝐴_𝑗)/(1−𝐵_𝑗))(𝑗=1,2), it follows from a result in [15] that 𝜑₁∗𝜑₂(𝑧)∈℘𝜌₃𝜌₃=1−21−𝜌₁1−𝜌₂ (31) and the bound 𝜌₃ is the best possible. An application of Lemma 1, in (30), yields ℜ𝜑₀(𝑧)≥𝑝+𝜇𝜏¹₀𝑢^{(𝑝+𝜇)/𝜏−1}2𝜌₃2−1+1−𝜌₃|𝑧|>1+𝑢𝑑𝑢𝑝+𝜇𝜏¹₀𝑢^{(𝑝+𝜇)/𝜏−1}2𝜌₃2−1+1−𝜌₃4𝐴1+𝑢𝑑𝑢=1−¹−𝐵₁𝐴₂−𝐵₂1−𝐵₁1−𝐵₂×1−𝑝+𝜇𝜏¹₀𝑢^{((𝑝+𝜇)/𝜏)−1}1+𝑢𝑑𝑢=𝜒. (32) In order to show that 𝜒 is the best possible in the assertion (23) when 𝐵₁=𝐵₂=−1, we consider the function 𝑓_𝑗(𝑧)∈𝒜_𝑝 given by ℐ^𝜆_𝑝,𝛿ℐ¹_𝑝,𝜇𝑓_𝑗(𝑧)=𝑝+𝜇𝜏𝑧^{𝑝−(𝑝+𝜇)/𝜏}×^𝑧₀𝑡^{(𝑝+𝜇)/𝜏−1}1+𝐴_𝑗𝑡.1−𝑡𝑑𝑡𝑗=1,2 (33) It is readily checked that 𝑓_𝑗 satisfies (22) with 𝐵_𝑗=−1. Since 1+𝐴₁𝑡∗1−𝑡1+𝐴₂𝑡1−𝑡=1−1+𝐴₁1+𝐴₂+1+𝐴₁1+𝐴₂,1−𝑧 (34) it follows from (30) and Lemma 1 that 𝜑₀(𝑧)=𝑝+𝜇𝜏𝑧^{(𝑝+𝜇)/𝜏}×^𝑧₀𝑡^{(𝑝+𝜇)/𝜏−1}1−1+𝐴₁1+𝐴₂+1+𝐴₁1+𝐴₂=1−𝑡𝑑𝑡𝑝+𝜇𝜏¹₀𝑢^{(𝑝+𝜇)/𝜏−1}1−1+𝐴₁1+𝐴₂+1+𝐴₁1+𝐴₂1−𝑢𝑧𝑑𝑢→1−1+𝐴₁1+𝐴₂×1−𝑝+𝜇𝜏¹₀𝑢^{(𝑝+𝜇)/𝜏−1}1+𝑢𝑑𝑢as𝑧→−1. (35) This completes the proof of Theorem 2.

Taking 𝜇=𝛿 we get the following consequence.

Corollary 3. Let 𝜏>0 and −1≤𝐵_𝑗<𝐴_𝑗≤1(𝑗=1,2). If each of the functions 𝑓_𝑗(𝑧)∈𝒜_𝑝(𝑗=1,2) satisfies the following subordination condition: 𝜏ℐ^𝜆_𝑝,𝛿𝑓_𝑗(𝑧)𝑧^𝑝+()ℐ1−𝜏^{𝜆+1𝑝,𝛿}𝑓_𝑗(𝑧)𝑧^𝑝≺1+𝐴_𝑗𝑧1+𝐵_𝑗𝑧𝑗=1,2;𝛿>−𝑝;𝑧∈𝕌 (36) then 𝜏ℐ^𝜆_𝑝,𝛿𝐻(𝑧)𝑧^𝑝+()ℐ1−𝜏^{𝜆+1𝑝,𝛿}𝐻(𝑧)𝑧^𝑝≺1+1−2𝜒𝑧,1−𝑧𝛿>−𝑝;𝑧∈𝕌 (37) where 𝐻(𝑧)=ℐ^𝜆+1_𝑝,𝛿𝑓₁∗𝑓₂(𝑧),4𝐴𝛿>−𝑝;𝑧∈𝕌𝜒=1−¹−𝐵₁𝐴₂−𝐵₂1−𝐵₁1−𝐵₂1−𝑝+𝛿𝜏¹₀𝑢^{(𝑝+𝛿)/𝜏−1}.1+𝑢𝑑𝑢 (38) 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]).

Acknowledgment

The present investigation is partially supported under the University of Pretoria, Post Doctoral Fellowship Grant and the Fast Track Research Project for Young Scientiest, Department of Science and Technology, Government of India, sanction Letter no. 100/IFD/12100/2010-11.