Let fn(z)=z+∑k=2nakzk be the sequence of partial sums of the analytic function f(z)=z+∑k=2∞akzk. In this paper, we determine sharp lower bounds for ℜ{f(z)/fn(z)}, ℜ{fn(z)/f(z)}, ℜ{f′(z)/fn′(z)}, and ℜ{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=2∞akzk
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 [1], consisting of functions of the formf(z)=z-∑k=2∞akzk,z∈U.

For functions f∈𝒜 given by (1.1) and g∈𝒜 given by g(z)=z+∑k=2∞bkzk, we define the Hadamard product (or convolution) of f and g by(f*g)(z)=z+∑k=2∞akbkzk,z∈U.
We recall here a general Hurwitz-Lerch zeta function Φ(z,s,a) defined in [2] byΦ(z,s,a):=∑k=0∞zk(k+a)s
(a∈ℂ∖ℤ0-; s∈ℂ, when |z|<1; ℜ(s)>1, when |z|=1), where, as usual, ℤ0-:=ℤ∖ℕ, (ℤ:={0,±1,±2,±3,…}); ℕ:={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 [3], Ferreira and López [4], Garg et al. [5], Lin and Srivastava [6], Lin et al. [7], and others. Srivastava and Attiya [8] (see also Răducanu and Srivastava [9] and Prajapat and Goyal [10]) introduced and investigated the linear operator 𝒥μ,b:𝒜→𝒜 defined in terms of the Hadamard product byJμ,bf(z)=Gb,μ*f(z)(z∈U;b∈C∖Z0-;μ∈C;f∈A),
where, for convenience,Gμ,b(z):=(1+b)μ[Φ(z,μ,b)-b-μ](z∈U).
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 [8], we introduce the generalized integral operatorJμ,bm,ηf(z)=z+∑k=2∞Ckm(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 b∈ℂ∖ℤ0-; μ∈ℂ, η≥2 and m>-1. It is of interest to note that Jμ,b1,2 is the Srivastava-Attiya operator [8] 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 [13] and Bernardi [14]. Further more, we get the Jung-Kim-Srivastava integral operator [15] closely related to some multiplier transformation studied by Flett [16].

Motivated by Murugusundaramoorthy [17–19] 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 λ≥0, -1≤γ<1, and β≥0, 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 z∈U. 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 [20] 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 [20] (see [21–25]) 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 ℜ{f(z)/fn(z)}, ℜ{fn(z)/f(z)}, ℜ{f′(z)/fn′(z)}, and ℜ{fn′(z)/f′(z)}. It is seen that this study not only gives as a particular case, the results of Silverman [20], 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),0≤λ≤1, -1≤γ<1, β≥0, 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 [17], 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)(z∈U),
where
ρk=ρk(λ,γ,η)≥{1-γifk=2,3,…,n,ρn+1ifk=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+1∞akzk-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+1∞akzk-12+2∑k=2nakzk-1+(ρn+1/(1-γ))∑k=n+1∞akzk-1.
Hence we obtain
|w(z)|≤(ρn+1/(1-γ))∑k=n+1∞|ak|2-2∑k=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|≤2-2∑k=2n|ak|
or, equivalently,
∑k=2n|ak|+∑k=n+1∞ρn+11-γ|ak|≤1.

From the condition (1.14), it is sufficient to show that
∑k=2n|ak|+∑k=n+1∞ρn+11-γ|ak|≤∑k=2∞ρk1-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+1zn⟶1-1-γρn+1=ρn+1-1+γρn+1whenr⟶1-.

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(z∈U),
where ρn+1≥1-γ and
ρk≥{1-γifk=2,3,…,n,ρn+1ifk=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+1∞akzk-11+∑k=2∞akzk-1,
where
|w(z)|≤((ρn+1+1-γ)/(1-γ))∑k=n+1∞|ak|2-2∑k=2n|ak|-((ρn+1-1+γ)/(1-γ))∑k=n+1∞|ak|≤1.
This last inequality is equivalent to
∑k=2n|ak|+∑k=n+1∞ρn+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(z∈U),R{fn′(z)f′(z)}≥ρn+1ρn+1+(n+1)(1-γ)(z∈U),
where ρn+1≥(n+1)(1-γ) and
ρk≥{k(1-γ)ifk=2,3,…,n,k(ρn+1n+1)ifk=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+1∞akzk-12+2∑k=2nkakzk-1+(ρn+1/((n+1)(1-γ)))∑k=n+1∞kakzk-1.
Now |w(z)|≤1 if and only if
∑k=2nk|ak|+ρn+1(n+1)(1-γ)∑k=n+1∞k|ak|≤1.
From the condition (1.14), it is sufficient to show that
∑k=2nk|ak|+ρn+1(n+1)(1-γ)∑k=n+1∞k|ak|≤∑k=2∞ρk1-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+1∞kakzk-12+2∑k=2nkakzk-1+(1-ρn+1/((n+1)(1-γ)))∑k=n+1∞kakzk-1.
Now |w(z)|≤1 if and only if
∑k=2nk|ak|+(ρn+1(n+1)(1-γ))∑k=n+1∞k|ak|≤1.
It suffices to show that the left hand side of (2.27) is bounded previously by the condition
∑k=2∞ρk1-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 forℜ{f(z)/fn(z)}, ℜ{fn(z)/f(z)}, ℜ{f′(z)/fn′(z)}, and ℜ{fn′(z)/f′(z)} for various function classes involving the Alexander integral operator [13] and Bernardi integral operators [14], Jung-Kim-Srivastava integral operator [15] 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 ([26–29]) for further views and ideas.

Acknowledgment

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

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