AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation84925010.1155/2011/849250849250Research ArticlePartial Sums of Generalized Class of Analytic Functions Involving Hurwitz-Lerch Zeta FunctionMurugusundaramoorthyG.1UmaK.1DarusM.2GigaYoshikazu1School of Advanced SciencesVIT UniversityVellore 632014Indiavit.ac.in2School of Mathematical SciencesFaculty of Science and TechnologyUniversiti Kebangsaan MalaysiaBangi 43600Malaysiaukm.my201112062011201129012011070420112011Copyright © 2011 G. Murugusundaramoorthy et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let fn(z)=z+k=2nakzk be the sequence of partial sums of the analytic function f(z)=z+k=2akzk. In this paper, we determine sharp lower bounds for {f(z)/fn(z)}, ℜ{fn(z)/f(z)},  ℜ{f(z)/fn(z)}, and {fn(z)/f(z)}. The usefulness of the main result not only provides the unification of the results discussed in the literature but also generates certain new results.

1. Introduction and Preliminaries

Let 𝒜 denote the class of functions of the formf(z)=z+k=2akzk which are analytic and univalent in the open disc U={z:  |z|<1}. We also consider T a subclass of 𝒜 introduced and studied by Silverman , consisting of functions of the formf(z)=z-k=2akzk,zU.

For functions f𝒜 given by (1.1) and g𝒜 given by g(z)=z+k=2bkzk, we define the Hadamard product (or convolution) of f and g by(f*g)(z)=z+k=2akbkzk,zU. We recall here a general Hurwitz-Lerch zeta function Φ(z,s,a) defined in  byΦ(z,s,a):=k=0zk(k+a)s (a0-; s, when |z|<1; (s)>1, when |z|=1), where, as usual, 0-:=, (:={0,±1,±2,±3,}); :={1,2,3,}. Several interesting properties and characteristics of the Hurwitz-Lerch Zeta function Φ(z,s,a) can be found in the recent investigations by Choi and Srivastava , Ferreira and López , Garg et al. , Lin and Srivastava , Lin et al. , and others. Srivastava and Attiya  (see also Răducanu and Srivastava  and Prajapat and Goyal ) introduced and investigated the linear operator 𝒥μ,b:𝒜𝒜 defined in terms of the Hadamard product byJμ,bf(z)=Gb,μ*f(z)(zU;bCZ0-;μC;fA), where, for convenience,Gμ,b(z):=(1+b)μ[Φ(z,μ,b)-b-μ](zU). We recall here the following relationships (given earlier by [9, 10]) which follow easily by using (1.1), (1.5), and (1.6):Jbμf(z)=z+k=2(1+bk+b)μakzk. Motivated essentially by the Srivastava-Attiya operator , we introduce the generalized integral operatorJμ,bm,ηf(z)=z+k=2Ckm(b,μ)akzk=F(z), whereCkm(b,μ)=|(1+bk+b)μm!(k+η-2)!(η-2)!(k+m-1)!|, and (throughout this paper unless otherwise mentioned) the parameters μ, b are constrained as b0-; μ, η2 and m>-1. It is of interest to note that Jμ,b1,2 is the Srivastava-Attiya operator  and J0,bm,η is the well-known Choi-Saigo-Srivastava operator (see [11, 12]). Suitably specializing the parameters m, η, μ, and b in 𝒥μ,bm,ηf(z), we can get various integral operators introduced by Alexander  and Bernardi . Further more, we get the Jung-Kim-Srivastava integral operator  closely related to some multiplier transformation studied by Flett .

Motivated by Murugusundaramoorthy  and making use of the generalized Srivastava-Attiya operator 𝒥μ,bm,η, we define the following new subclass of analytic functions with negative coefficients.

For λ0, -1γ<1, and β0, let Pμλ(γ,β) be the subclass of 𝒜 consisting of functions of the form (1.1) and satisfying the analytic criterionR{z(Jμ,bm,ηf(z))+λz2(Jμ,bm,ηf(z))′′(1-λ)Jμ,bm,ηf(z)+λz(Jμ,bm,ηf(z))-γ}>β|z(Jμ,bm,ηf(z))+λz2(Jμ,bm,ηf(z))′′(1-λ)Jμ,bm,ηf(z)+λz(Jμ,bm,ηf(z))-1|,   where zU. Shortly we can state this condition by-R{zG(z)G(z)-γ}>β|zG(z)G(z)-1|, whereG(z)=(1-λ)F(z)+λzF(z)=z+k=2[1+λ(k-1)]|Ckm(b,μ)|akzk, and F(z)=𝒥μ,bm,ηf(z).

Recently, Silverman  determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. In the present paper and by following the earlier work by Silverman  (see ) on partial sums of analytic functions, we study the ratio of a function of the form (1.1) to its sequence of partial sums of the formfn(z)=z+k=2nakzk, when the coefficients of f(z) satisfy the condition (1.14). Also, we will determine sharp lower bounds for {f(z)/fn(z)}, {fn(z)/f(z)}, {f(z)/fn(z)}, and {fn(z)/f(z)}. It is seen that this study not only gives as a particular case, the results of Silverman , but also gives rise to several new results.

Before stating and proving our main results, we derive a sufficient condition giving the coefficient estimates for functions f(z) to belong to this generalized function class.

Lemma 1.1.

A function f(z) of the form (1.1) is in Pμλ(γ,β) if k=2(1+λ(k-1))[k(1+β)-(γ+β)]|ak|Ckm(b,μ)1-γ, where, for convenience, ρk=ρk(λ,γ,η)=(1+λ(k-1))[n(1+β  )-(γ+β)]Ckm(b,μ),0λ1, -1γ<1, β0, and Ckm(b,μ), is given by (1.9).

Proof.

The proof of Lemma 1.1 is much akin to the proof of Theorem 1 obtained by Murugusundaramoorthy , hence we omit the details.

2. Main ResultsTheorem 2.1.

If f of the form (1.1) satisfies the condition (1.14), then R{f(z)fn(z)}ρn+1(λ,γ,η)-1+γρn+1(λ,γ,η)(zU), where ρk=ρk(λ,γ,η){1-γif  k=2,3,,n,ρn+1if  k=n+1,n+2,. The result (2.1) is sharp with the function given by f(z)=z+1-γρn+1zn+1.

Proof.

Define the function w(z) by 1+w(z)1-w(z)=ρn+11-γ[f(z)fn(z)-ρn+1-1+γρn+1]=1+k=2nakzk-1+(ρn+1/(1-γ))k=n+1akzk-11+k=2nakzk-1.

It suffices to show that |w(z)|1. Now, from (2.4) we can write w(z)=(ρn+1/(1-γ))k=n+1akzk-12+2k=2nakzk-1+(ρn+1/(1-γ))k=n+1akzk-1. Hence we obtain |w(z)|(ρn+1/(1-γ))k=n+1|ak|2-2k=2n|ak|-(ρn+1/(1-γ))k=n+1|ak|. Now |w(z)|1 if and only if 2(ρn+11-γ)k=n+1|ak|2-2k=2n|ak| or, equivalently, k=2n|ak|+k=n+1ρn+11-γ|ak|1.

From the condition (1.14), it is sufficient to show that k=2n|ak|+k=n+1ρn+11-γ|ak|k=2ρk1-γ|ak| which is equivalent to k=2n(ρk-1+γ1-γ)|ak|+k=n+1(ρk-ρn+11-γ)|ak|0. To see that the function given by (2.3) gives the sharp result, we observe that for z=reiπ/n, f(z)fn(z)=1+1-γρn+1zn1-1-γρn+1=ρn+1-1+γρn+1        when    r1-.

We next determine bounds for fn(z)/f(z).

Theorem 2.2.

If f of the form (1.1) satisfies the condition (1.14), then R{fn(z)f(z)}ρn+1ρn+1+1-γ(zU), where ρn+11-γ and ρk{1-γif  k=2,3,,n,ρn+1if  k=n+1,n+2,. The result (2.12) is sharp with the function given by (2.3).

Proof.

We write 1+w(z)1-w(z)=ρn+1+1-γ1-γ[fn(z)f(z)-ρn+1ρn+1+1-γ]=1+k=2nakzk-1-(ρn+1/(1-γ))k=n+1akzk-11+k=2akzk-1, where |w(z)|((ρn+1+1-γ)/(1-γ))k=n+1|ak|2-2k=2n|ak|-((ρn+1-1+γ)/(1-γ))k=n+1|ak|1. This last inequality is equivalent to k=2n|ak|+k=n+1ρn+11-γ|ak|1. We are making use of (1.14) to get (2.10). Finally, equality holds in (2.12) for the extremal function f(z) given by (2.3).

We next turns to ratios involving derivatives.

Theorem 2.3.

If f of the form (1.1) satisfies the condition (1.14), then R{f(z)fn(z)}ρn+1-(n+1)(1-γ)ρn+1(zU),R{fn(z)f(z)}ρn+1ρn+1+(n+1)(1-γ)(zU), where ρn+1(n+1)(1-γ) and ρk{k(1-γ)if  k=2,3,,n,k(ρn+1n+1)if  k=n+1,n+2,. The results are sharp with the function given by (2.3).

Proof.

We write 1+w(z)1-w(z)=ρn+1(n+1)(1-γ)[f(z)fn(z)-(ρn+1-(n+1)(1-γ)ρn+1)], where w(z)=(ρn+1/((n+1)(1-γ)))k=n+1akzk-12+2k=2nkakzk-1+(ρn+1/((n+1)(1-γ)))k=n+1kakzk-1. Now |w(z)|1 if and only if k=2nk|ak|+ρn+1(n+1)(1-γ)k=n+1k|ak|1. From the condition (1.14), it is sufficient to show that k=2nk|ak|+ρn+1(n+1)(1-γ)k=n+1k|ak|k=2ρk1-γ|ak| which is equivalent to k=2n(ρk-(1-γ)k1-γ)|ak|+k=n+1(n+1)ρk-kρn+1(n+1)(1-γ)|ak|0.

To prove the result (2.18), define the function w(z) by 1+w(z)1-w(z)=(n+1)(1-γ)+ρn+1(1-γ)(n+1)[fn(z)f(z)-ρn+1(n+1)(1-γ)+ρn+1], where w(z)=-(1+ρn+1/((n+1)(1-γ)))k=n+1kakzk-12+2k=2nkakzk-1+(1-ρn+1/((n+1)(1-γ)))k=n+1kakzk-1. Now |w(z)|1 if and only if k=2nk|ak|+(ρn+1(n+1)(1-γ))k=n+1k|ak|1. It suffices to show that the left hand side of (2.27) is bounded previously by the condition k=2ρk1-γ|ak|, which is equivalent to k=2n(ρk1-γ-k)|ak|+k=n+1(ρk1-γ-ρn+1(n+1)(1-γ))k|ak|0.

Remark 2.4.

As a special case of the previous theorems, we can determine new sharp lower bounds for  {f(z)/fn(z)}, {fn(z)/f(z)}, {f(z)/fn(z)}, and {fn(z)/f(z)} for various function classes involving the Alexander integral operator  and Bernardi integral operators , Jung-Kim-Srivastava integral operator  and Choi-Saigo-Srivastava operator (see [11, 12]) on specializing the values of η, m, μ, and b.

Some other work related to partial sums and also related to zeta function can be seen in () for further views and ideas.

Acknowledgment

The third author is presently supported by MOHE: UKM-ST-06-FRGS0244-2010.

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