Partial Sums of Generalized Class of Analytic Functions Involving Hurwitz-Lerch Zeta Function

Abstract

Let 𝑓_𝑛(𝑧)=𝑧+∑^𝑛𝑘=2𝑎_𝑘𝑧^𝑘 be the sequence of partial sums of the analytic function 𝑓(𝑧)=𝑧+∑^∞𝑘=2𝑎_𝑘𝑧^𝑘. In this paper, we determine sharp lower bounds for ℜ{𝑓(𝑧)/𝑓_𝑛(𝑧)},ℜ{𝑓_𝑛(𝑧)/𝑓(𝑧)},ℜ{𝑓^(𝑧)/𝑓^𝑛(𝑧)}, and ℜ{𝑓^𝑛(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=2𝑎_𝑘𝑧^𝑘(1.1) 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=2𝑎_𝑘𝑧^𝑘,𝑧∈𝑈.(1.2)

For functions 𝑓∈𝒜 given by (1.1) and 𝑔∈𝒜 given by 𝑔(𝑧)=𝑧+∑^∞𝑘=2𝑏_𝑘𝑧^𝑘, we define the Hadamard product (or convolution) of 𝑓 and 𝑔 by(𝑓∗𝑔)(𝑧)=𝑧+^∞_𝑘=2𝑎_𝑘𝑏_𝑘𝑧^𝑘,𝑧∈𝑈.(1.3) We recall here a general Hurwitz-Lerch zeta function Φ(𝑧,𝑠,𝑎) defined in [2] byΦ(𝑧,𝑠,𝑎)∶=^∞_𝑘=0𝑧^𝑘(𝑘+𝑎)^𝑠(1.4) (𝑎∈ℂ⧵ℤ⁻⁰; 𝑠∈ℂ, when |𝑧|<1; ℜ(𝑠)>1, when |𝑧|=1), where, as usual, ℤ⁻⁰∶=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 𝒥_𝜇,𝑏∶𝒜→𝒜 defined in terms of the Hadamard product by𝒥_𝜇,𝑏𝑓(𝑧)=𝒢_𝑏,𝜇∗𝑓(𝑧)𝑧∈𝑈;𝑏∈ℂ⧵ℤ⁻⁰;𝜇∈ℂ;𝑓∈𝒜,(1.5) where, for convenience,𝐺_𝜇,𝑏(𝑧)∶=(1+𝑏)^𝜇Φ(𝑧,𝜇,𝑏)−𝑏^−𝜇(𝑧∈𝑈).(1.6) We recall here the following relationships (given earlier by [9, 10]) which follow easily by using (1.1), (1.5), and (1.6):𝒥^𝜇𝑏𝑓(𝑧)=𝑧+^∞_𝑘=2𝑘1++𝑏𝑏^𝜇𝑎_𝑘𝑧^𝑘.(1.7) Motivated essentially by the Srivastava-Attiya operator [8], we introduce the generalized integral operator𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)=𝑧+^∞_𝑘=2𝐶^𝑚𝑘(𝑏,𝜇)𝑎_𝑘𝑧^𝑘=𝐹(𝑧),(1.8) where𝐶^𝑚𝑘(𝑏,𝜇)=||||1+𝑏𝑘+𝑏^𝜇𝑚!(𝑘+𝜂−2)!(𝜂−2)!(𝑘+𝑚−1)!||||,(1.9) and (throughout this paper unless otherwise mentioned) the parameters 𝜇, 𝑏 are constrained as 𝑏∈ℂ⧵ℤ⁻⁰; 𝜇∈ℂ, 𝜂≥2 and 𝑚>−1. It is of interest to note that 𝐽^1,2𝜇,𝑏 is the Srivastava-Attiya operator [8] and 𝐽^{𝑚,𝜂0,𝑏} is the well-known Choi-Saigo-Srivastava operator (see [11, 12]). Suitably specializing the parameters 𝑚, 𝜂, 𝜇, and 𝑏 in 𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(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 𝒥^{𝑚,𝜂𝜇,𝑏}, we define the following new subclass of analytic functions with negative coefficients.

For 𝜆≥0, −1≤𝛾<1, and 𝛽≥0, let 𝑃^𝜆𝜇(𝛾,𝛽) be the subclass of 𝒜 consisting of functions of the form (1.1) and satisfying the analytic criterionℜ⎧⎪⎨⎪⎩𝑧𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)^+𝜆𝑧²𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)^(1−𝜆)𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)+𝜆𝑧𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)^−𝛾⎫⎪⎬⎪⎭>𝛽||||||𝑧𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)^+𝜆𝑧²𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)^(1−𝜆)𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)+𝜆𝑧𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(𝑧)^−1||||||,(1.10) where 𝑧∈𝑈. Shortly we can state this condition by−ℜ𝑧𝐺^(𝑧)𝐺(𝑧)−𝛾>𝛽||||𝑧𝐺^(𝑧)𝐺(𝑧)−1||||,(1.11) where𝐺(𝑧)=(1−𝜆)𝐹(𝑧)+𝜆𝑧𝐹^(𝑧)=𝑧+^∞_𝑘=2[1+𝜆(𝑘−1)]||𝐶^𝑚𝑘(𝑏,𝜇)||𝑎_𝑘𝑧^𝑘,(1.12) and 𝐹(𝑧)=𝒥^{𝑚,𝜂𝜇,𝑏}𝑓(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 form𝑓_𝑛(𝑧)=𝑧+^𝑛_𝑘=2𝑎_𝑘𝑧^𝑘,(1.13) when the coefficients of 𝑓(𝑧) satisfy the condition (1.14). Also, we will determine sharp lower bounds for ℜ{𝑓(𝑧)/𝑓_𝑛(𝑧)}, ℜ{𝑓_𝑛(𝑧)/𝑓(𝑧)}, ℜ{𝑓^(𝑧)/𝑓^𝑛(𝑧)}, and ℜ{𝑓^𝑛(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 𝑃^𝜆𝜇(𝛾,𝛽) if ^∞_𝑘=2(1+𝜆(𝑘−1))[𝑘(1+𝛽)−(𝛾+𝛽)]||𝑎_𝑘||𝐶^𝑚𝑘(𝑏,𝜇)≤1−𝛾,(1.14) where, for convenience, 𝜌_𝑘=𝜌_𝑘(𝜆,𝛾,𝜂)=(1+𝜆(𝑘−1))[𝑛(1+𝛽)−(𝛾+𝛽)]𝐶^𝑚𝑘(𝑏,𝜇),(1.15)  0≤𝜆≤1, −1≤𝛾<1, 𝛽≥0, and 𝐶^𝑚𝑘(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 Results

Theorem 2.1. If 𝑓 of the form (1.1) satisfies the condition (1.14), then ℜ𝑓(𝑧)𝑓_𝑛(𝑧)≥𝜌^𝑛+1(𝜆,𝛾,𝜂)−1+𝛾𝜌_𝑛+1(𝜆,𝛾,𝜂)(𝑧∈𝑈),(2.1) where 𝜌_𝑘=𝜌_𝑘(𝜆,𝛾,𝜂)≥𝜌1−𝛾if𝑘=2,3,…,𝑛,_𝑛+1if𝑘=𝑛+1,𝑛+2,….(2.2) The result (2.1) is sharp with the function given by 𝑓(𝑧)=𝑧+1−𝛾𝜌_𝑛+1𝑧^𝑛+1.(2.3)

Proof. Define the function 𝑤(𝑧) by 1+𝑤(𝑧)1−𝑤(𝑧)=1𝜌−^𝑛+1𝛾𝑓𝑓_𝑛((𝑧𝑧))−𝜌^𝑛+1−1+𝛾=1+∑_𝑛𝑘=2𝜌_𝑛+1𝑎_𝑘𝑧^𝑘−1+𝜌_𝑛+1/(1−𝛾)∑^{∞𝑘=𝑛+1}𝑎_𝑘𝑧^𝑘−11+∑^𝑛𝑘=2𝑎_𝑘𝑧^𝑘−1.(2.4)
It suffices to show that |𝑤(𝑧)|≤1. Now, from (2.4) we can write 𝑤(𝑧)=𝜌^𝑛+1/(1−𝛾)∑^{∞𝑘=𝑛+1}𝑎_𝑘𝑧^𝑘−12+2∑^𝑛𝑘=2𝑎_𝑘𝑧^𝑘−1+𝜌_𝑛+1/(1−𝛾)∑^{∞𝑘=𝑛+1}𝑎_𝑘𝑧^𝑘−1.(2.5) Hence we obtain ||𝑤(𝑧)||≤𝜌^𝑛+1/(1−𝛾)∑^{∞𝑘=𝑛+1}||𝑎_𝑘||2−2∑^𝑛𝑘=2||𝑎_𝑘||−𝜌_𝑛+1/(1−𝛾)∑^{∞𝑘=𝑛+1}||𝑎_𝑘||.(2.6) Now |𝑤(𝑧)|≤1 if and only if 2𝜌^𝑛+11−𝛾^∞_𝑘=𝑛+1||𝑎_𝑘||≤2−2^𝑛_𝑘=2||𝑎_𝑘||(2.7) or, equivalently, ^𝑛_𝑘=2||𝑎_𝑘||+^∞_𝑘_=𝑛+1𝜌_𝑛+11−𝛾||𝑎^𝑘||≤1.(2.8)
From the condition (1.14), it is sufficient to show that ^𝑛_𝑘=2||𝑎_𝑘||+^∞_𝑘_=𝑛+1𝜌_𝑛+11−𝛾||𝑎^𝑘||≤^∞_𝑘=2𝜌_𝑘1−𝛾||𝑎^𝑘||(2.9) which is equivalent to ^𝑛_𝑘=2𝜌^𝑘−1+𝛾1−𝛾||𝑎^𝑘||+^∞_𝑘_=𝑛+1𝜌^𝑘−𝜌_𝑛+11−𝛾||𝑎^𝑘||≥0.(2.10) To see that the function given by (2.3) gives the sharp result, we observe that for 𝑧=𝑟𝑒^{𝑖𝜋/𝑛}, 𝑓(𝑧)𝑓_𝑛(𝑧)=1+1𝜌−_𝑛+1𝛾𝑧^𝑛⟶1−1−𝛾𝜌_𝑛+1=𝜌^𝑛+1−1+𝛾𝜌_𝑛+1when𝑟⟶1⁻.(2.11)

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

Theorem 2.2. If 𝑓 of the form (1.1) satisfies the condition (1.14), then ℜ𝑓^𝑛(𝑧)𝑓(𝑧)≥𝜌_𝑛+1𝜌+1^𝑛+1−𝛾(𝑧∈𝑈),(2.12) where 𝜌_𝑛+1≥1−𝛾 and 𝜌_𝑘≥𝜌1−𝛾if𝑘=2,3,…,𝑛,_𝑛+1if𝑘=𝑛+1,𝑛+2,….(2.13) The result (2.12) is sharp with the function given by (2.3).

Proof. We write 1+𝑤(𝑧)1−𝑤(𝑧)=𝜌^𝑛+1+1−𝛾1−𝛾𝑓^𝑛(𝑧)=1+∑_𝑛𝑘=2𝑓(𝑧)−𝜌_𝑛+1𝜌+1^𝑛+1−𝛾𝑎_𝑘𝑧^𝑘−1−𝜌_𝑛+1/(1−𝛾)∑^{∞𝑘=𝑛+1}𝑎_𝑘𝑧^𝑘−11+∑^∞𝑘=2𝑎_𝑘𝑧^𝑘−1,(2.14) where ||𝑤(𝑧)||≤𝜌^𝑛+1+1−𝛾/(1−𝛾)∑^{∞𝑘=𝑛+1}||𝑎_𝑘||2−2∑^𝑛𝑘=2||𝑎_𝑘||−𝜌_𝑛+1−1+𝛾/(1−𝛾)∑^{∞𝑘=𝑛+1}||𝑎_𝑘||≤1.(2.15) This last inequality is equivalent to ^𝑛_𝑘=2||𝑎_𝑘||+^∞_𝑘_=𝑛+1𝜌_𝑛+11−𝛾||𝑎^𝑘||≤1.(2.16) 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)𝑓^𝑛(𝑧)≥𝜌^𝑛+1−(𝑛+1)(1−𝛾)𝜌_𝑛+1ℜ𝑓(𝑧∈𝑈),(2.17)_𝑛(𝑧)𝑓^(𝑧)≥𝜌_𝑛+1+(𝑛𝜌+^𝑛+11)(1−𝛾)(𝑧∈𝑈),(2.18) where 𝜌_𝑛+1≥(𝑛+1)(1−𝛾) and 𝜌_𝑘≥⎧⎪⎨⎪⎩𝑘𝑘(1𝜌−𝛾)if𝑘=2,3,…,𝑛,_𝑛+1𝑛+1if𝑘=𝑛+1,𝑛+2,….(2.19) The results are sharp with the function given by (2.3).

Proof. We write 1+𝑤(𝑧)1−𝑤(𝑧)=(𝑛+1𝜌)(1^𝑛+1−𝛾)𝑓^(𝑧)𝑓^𝑛(𝑧)−𝜌^𝑛+1−(𝑛+1)(1−𝛾)𝜌_𝑛+1,(2.20) where 𝑤(𝑧)=𝜌^𝑛+1/((𝑛+1)(1−𝛾))∑^{∞𝑘=𝑛+1}𝑎_𝑘𝑧^𝑘−12+2∑^𝑛𝑘=2𝑘𝑎_𝑘𝑧^𝑘−1+𝜌_𝑛+1/((𝑛+1)(1−𝛾))∑^{∞𝑘=𝑛+1}𝑘𝑎_𝑘𝑧^𝑘−1.(2.21) Now |𝑤(𝑧)|≤1 if and only if ^𝑛_𝑘=2𝑘||𝑎_𝑘||+𝜌^𝑛+1(𝑛+1)(1−𝛾)^∞_𝑘=𝑛+1𝑘||𝑎_𝑘||≤1.(2.22) From the condition (1.14), it is sufficient to show that ^𝑛_𝑘=2𝑘||𝑎_𝑘||+𝜌^𝑛+1(𝑛+1)(1−𝛾)^∞_𝑘=𝑛+1𝑘||𝑎_𝑘||≤^∞_𝑘=2𝜌_𝑘1−𝛾||𝑎^𝑘||(2.23) which is equivalent to ^𝑛_𝑘=2𝜌^𝑘−(1−𝛾)𝑘1−𝛾||𝑎^𝑘||+^∞_𝑘_=𝑛+1(𝑛+1)𝜌_𝑘−𝑘𝜌_𝑛+1(𝑛+1)(1−𝛾)||𝑎^𝑘||≥0.(2.24)
To prove the result (2.18), define the function 𝑤(𝑧) by 1+𝑤(𝑧)1−𝑤(𝑧)=(𝑛+(11)−(1𝛾)−(𝑛𝛾)++1)𝜌^𝑛+1𝑓^𝑛(𝑧)𝑓^(𝑧)−(𝑛+1)(1𝜌−^𝑛+1𝛾)+𝜌_𝑛+1,(2.25) where 𝑤(𝑧)=−1+𝜌^𝑛+1/((𝑛+1)(1−𝛾))∑^{∞𝑘=𝑛+1}𝑘𝑎_𝑘𝑧^𝑘−12+2∑^𝑛𝑘=2𝑘𝑎_𝑘𝑧^𝑘−1+1−𝜌_𝑛+1/((𝑛+1)(1−𝛾))∑^{∞𝑘=𝑛+1}𝑘𝑎_𝑘𝑧^𝑘−1.(2.26) Now |𝑤(𝑧)|≤1 if and only if ^𝑛_𝑘=2𝑘||𝑎_𝑘||+𝜌^𝑛+1(𝑛+1)(1−𝛾)^∞_𝑘=𝑛+1𝑘||𝑎_𝑘||≤1.(2.27) It suffices to show that the left hand side of (2.27) is bounded previously by the condition ^∞_𝑘=2𝜌_𝑘1−𝛾||𝑎^𝑘||,(2.28) which is equivalent to ^𝑛_𝑘=21𝜌−^𝑘𝛾−𝑘||𝑎^𝑘||+^∞_𝑘_=𝑛+11𝜌−^𝑘𝛾−(𝑛+1𝜌)(1^𝑛+1−𝛾)𝑘||𝑎^𝑘||≥0.(2.29)

Remark 2.4. As a special case of the previous theorems, we can determine new sharp lower bounds forℜ{𝑓(𝑧)/𝑓_𝑛(𝑧)}, ℜ{𝑓_𝑛(𝑧)/𝑓(𝑧)}, ℜ{𝑓^(𝑧)/𝑓^𝑛(𝑧)}, and ℜ{𝑓^𝑛(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.