Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 849250 | https://doi.org/10.1155/2011/849250

G. Murugusundaramoorthy, K. Uma, M. Darus, "Partial Sums of Generalized Class of Analytic Functions Involving Hurwitz-Lerch Zeta Function", Abstract and Applied Analysis, vol. 2011, Article ID 849250, 9 pages, 2011. https://doi.org/10.1155/2011/849250

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

Academic Editor: Yoshikazu Giga
Received29 Jan 2011
Accepted07 Apr 2011
Published12 Jun 2011

Abstract

Let 𝑓𝑛∑(𝑧)=𝑧+𝑛𝑘=2ğ‘Žğ‘˜ğ‘§ğ‘˜ be the sequence of partial sums of the analytic function ∑𝑓(𝑧)=𝑧+âˆžğ‘˜=2ğ‘Žğ‘˜ğ‘§ğ‘˜. In this paper, we determine sharp lower bounds for ℜ{𝑓(𝑧)/𝑓𝑛(𝑧)},ℜ{𝑓𝑛(𝑧)/𝑓(𝑧)},ℜ{ğ‘“î…ž(𝑧)/ğ‘“î…žğ‘›(𝑧)}, and ℜ{ğ‘“î…žğ‘›(𝑧)/ğ‘“î…ž(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+âˆžî“ğ‘˜=2ğ‘Žğ‘˜ğ‘§ğ‘˜(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)=ğ‘§âˆ’âˆžî“ğ‘˜=2ğ‘Žğ‘˜ğ‘§ğ‘˜,𝑧∈𝑈.(1.2)

For functions 𝑓∈𝒜 given by (1.1) and 𝑔∈𝒜 given by ∑𝑔(𝑧)=𝑧+âˆžğ‘˜=2𝑏𝑘𝑧𝑘, we define the Hadamard product (or convolution) of 𝑓 and 𝑔 by(𝑓∗𝑔)(𝑧)=𝑧+âˆžî“ğ‘˜=2ğ‘Žğ‘˜ğ‘ğ‘˜ğ‘§ğ‘˜,𝑧∈𝑈.(1.3) We recall here a general Hurwitz-Lerch zeta function Φ(𝑧,𝑠,ğ‘Ž) defined in [2] byΦ(𝑧,𝑠,ğ‘Ž)∶=âˆžî“ğ‘˜=0𝑧𝑘(𝑘+ğ‘Ž)𝑠(1.4) (ğ‘Žâˆˆâ„‚â§µâ„¤âˆ’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,ğ‘Ž) 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⧵ℤ−0,;𝜇∈ℂ;𝑓∈𝒜(1.5) where, for convenience,𝐺𝜇,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):𝒥𝜇𝑏𝑓(𝑧)=𝑧+âˆžî“ğ‘˜=21+𝑏𝑘+ğ‘ğœ‡ğ‘Žğ‘˜ğ‘§ğ‘˜.(1.7) Motivated essentially by the Srivastava-Attiya operator [8], we introduce the generalized integral operator𝒥𝑚,𝜂𝜇,𝑏𝑓(𝑧)=𝑧+âˆžî“ğ‘˜=2𝐶𝑚𝑘(𝑏,𝜇)ğ‘Žğ‘˜ğ‘§ğ‘˜=𝐹(𝑧),(1.8) where𝐶𝑚𝑘||||(𝑏,𝜇)=1+𝑏𝑘+𝑏𝜇𝑚!(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 𝐽1,2𝜇,𝑏 is the Srivastava-Attiya operator [8] and 𝐽𝑚,𝜂0,𝑏 is the well-known Choi-Saigo-Srivastava operator (see [11, 12]). Suitably specializing the parameters 𝑚, 𝜂, 𝜇, and 𝑏 in 𝒥𝑚,𝜂𝜇,𝑏𝑓(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 𝒥𝑚,𝜂𝜇,𝑏, we define the following new subclass of analytic functions with negative coefficients.

For 𝜆≥0, −1≤𝛾<1, and 𝛽≥0, let 𝑃𝜆𝜇(𝛾,𝛽) be the subclass of 𝒜 consisting of functions of the form (1.1) and satisfying the analytic criterionâ„œâŽ§âŽªâŽ¨âŽªâŽ©ğ‘§î‚€ğ’¥ğ‘š,𝜂𝜇,𝑏𝑓(𝑧)+𝜆𝑧2𝒥𝑚,𝜂𝜇,𝑏𝑓(𝑧)(1−𝜆)𝒥𝑚,𝜂𝜇,𝑏𝑓𝒥(𝑧)+𝜆𝑧𝑚,𝜂𝜇,𝑏𝑓(𝑧)⎫⎪⎬⎪⎭||||||𝑧𝒥−𝛾>𝛽𝑚,𝜂𝜇,𝑏𝑓(𝑧)+𝜆𝑧2𝒥𝑚,𝜂𝜇,𝑏𝑓(𝑧)(1−𝜆)𝒥𝑚,𝜂𝜇,𝑏𝑓𝒥(𝑧)+𝜆𝑧𝑚,𝜂𝜇,𝑏𝑓(𝑧)||||||−1,(1.10) where 𝑧∈𝑈. Shortly we can state this condition byî‚»âˆ’â„œğ‘§ğºî…ž(𝑧)||||𝐺(𝑧)−𝛾>ğ›½ğ‘§ğºî…ž(𝑧)||||,𝐺(𝑧)−1(1.11) where𝐺(𝑧)=(1−𝜆)𝐹(𝑧)+ğœ†ğ‘§ğ¹î…ž(𝑧)=𝑧+âˆžî“ğ‘˜=2[]||𝐶1+𝜆(𝑘−1)𝑚𝑘(||ğ‘Žğ‘,𝜇)𝑘𝑧𝑘,(1.12) and 𝐹(𝑧)=𝒥𝑚,𝜂𝜇,𝑏𝑓(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+𝑛𝑘=2ğ‘Žğ‘˜ğ‘§ğ‘˜,(1.13) 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{ğ‘“î…ž(𝑧)/ğ‘“î…žğ‘›(𝑧)}, and ℜ{ğ‘“î…žğ‘›(𝑧)/ğ‘“î…ž(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 𝑃𝜆𝜇(𝛾,𝛽) if âˆžî“ğ‘˜=2([]||ğ‘Ž1+𝜆(𝑘−1))𝑘(1+𝛽)−(𝛾+𝛽)𝑘||𝐶𝑚𝑘(𝑏,𝜇)≤1−𝛾,(1.14) where, for convenience, 𝜌𝑘=𝜌𝑘([]𝐶𝜆,𝛾,𝜂)=(1+𝜆(𝑘−1))𝑛(1+𝛽)−(𝛾+𝛽)𝑚𝑘(𝑏,𝜇),(1.15)  0≤𝜆≤1, −1≤𝛾<1, 𝛽≥0, and 𝐶𝑚𝑘(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)𝑓𝑛≥𝜌(𝑧)𝑛+1(𝜆,𝛾,𝜂)−1+𝛾𝜌𝑛+1(𝜆,𝛾,𝜂)(𝑧∈𝑈),(2.1) where 𝜌𝑘=𝜌𝑘𝜌(𝜆,𝛾,𝜂)≥1−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−𝛾𝜌𝑛+1𝑧𝑛+1.(2.3)

Proof. Define the function 𝑤(𝑧) by 1+𝑤(𝑧)=𝜌1−𝑤(𝑧)𝑛+11−𝛾𝑓(𝑧)𝑓𝑛−𝜌(𝑧)𝑛+1−1+𝛾𝜌𝑛+1=∑1+𝑛𝑘=2ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1+𝜌𝑛+1∑/(1−𝛾)âˆžğ‘˜=𝑛+1ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1∑1+𝑛𝑘=2ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1.(2.4)
It suffices to show that |𝑤(𝑧)|≤1. Now, from (2.4) we can write 𝑤𝜌(𝑧)=𝑛+1∑/(1−𝛾)âˆžğ‘˜=𝑛+1ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1∑2+2𝑛𝑘=2ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1+𝜌𝑛+1∑/(1−𝛾)âˆžğ‘˜=𝑛+1ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1.(2.5) Hence we obtain ||||≤𝜌𝑤(𝑧)𝑛+1/∑(1−𝛾)âˆžğ‘˜=𝑛+1||ğ‘Žğ‘˜||∑2−2𝑛𝑘=2||ğ‘Žğ‘˜||−𝜌𝑛+1∑/(1−𝛾)âˆžğ‘˜=𝑛+1||ğ‘Žğ‘˜||.(2.6) Now |𝑤(𝑧)|≤1 if and only if 2𝜌𝑛+11âˆ’ğ›¾âˆžî“ğ‘˜=𝑛+1||ğ‘Žğ‘˜||≤2−2𝑛𝑘=2||ğ‘Žğ‘˜||(2.7) or, equivalently, 𝑛𝑘=2||ğ‘Žğ‘˜||+âˆžî“ğ‘˜=𝑛+1𝜌𝑛+1||ğ‘Ž1−𝛾𝑘||≤1.(2.8)
From the condition (1.14), it is sufficient to show that 𝑛𝑘=2||ğ‘Žğ‘˜||+âˆžî“ğ‘˜=𝑛+1𝜌𝑛+1||ğ‘Ž1−𝛾𝑘||â‰¤âˆžî“ğ‘˜=2𝜌𝑘||ğ‘Ž1−𝛾𝑘||(2.9) which is equivalent to 𝑛𝑘=2𝜌𝑘−1+𝛾||ğ‘Ž1−𝛾𝑘||+âˆžî“ğ‘˜=𝑛+1𝜌𝑘−𝜌𝑛+1||ğ‘Ž1−𝛾𝑘||≥0.(2.10) 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−.(2.11)

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)≥𝜌𝑓(𝑧)𝑛+1𝜌𝑛+1+1−𝛾(𝑧∈𝑈),(2.12) where 𝜌𝑛+1≥1−𝛾 and 𝜌𝑘≥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)=𝜌1−𝑤(𝑧)𝑛+1+1−𝛾𝑓1−𝛾𝑛(𝑧)−𝜌𝑓(𝑧)𝑛+1𝜌𝑛+1=∑+1−𝛾1+𝑛𝑘=2ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1−𝜌𝑛+1∑/(1−𝛾)âˆžğ‘˜=𝑛+1ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1∑1+âˆžğ‘˜=2ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1,(2.14) where ||||≤𝜌𝑤(𝑧)𝑛+1/∑+1−𝛾(1−𝛾)âˆžğ‘˜=𝑛+1||ğ‘Žğ‘˜||∑2−2𝑛𝑘=2||ğ‘Žğ‘˜||−𝜌𝑛+1∑−1+𝛾/(1−𝛾)âˆžğ‘˜=𝑛+1||ğ‘Žğ‘˜||≤1.(2.15) This last inequality is equivalent to 𝑛𝑘=2||ğ‘Žğ‘˜||+âˆžî“ğ‘˜=𝑛+1𝜌𝑛+1||ğ‘Ž1−𝛾𝑘||≤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 â„œî‚»ğ‘“î…ž(𝑧)ğ‘“î…žğ‘›î‚¼â‰¥ğœŒ(𝑧)𝑛+1−(𝑛+1)(1−𝛾)𝜌𝑛+1ℜ𝑓(𝑧∈𝑈),(2.17)î…žğ‘›(𝑧)ğ‘“î…žî‚¼â‰¥ğœŒ(𝑧)𝑛+1𝜌𝑛+1+(𝑛+1)(1−𝛾)(𝑧∈𝑈),(2.18) where 𝜌𝑛+1≥(𝑛+1)(1−𝛾) and ğœŒğ‘˜â‰¥âŽ§âŽªâŽ¨âŽªâŽ©ğ‘˜î‚€ğœŒğ‘˜(1−𝛾)if𝑘=2,3,…,𝑛,𝑛+1𝑛+1if𝑘=𝑛+1,𝑛+2,….(2.19) The results are sharp with the function given by (2.3).

Proof. We write 1+𝑤(𝑧)=𝜌1−𝑤(𝑧)𝑛+1𝑓(𝑛+1)(1−𝛾)(𝑧)ğ‘“î…žğ‘›âˆ’î‚µğœŒ(𝑧)𝑛+1−(𝑛+1)(1−𝛾)𝜌𝑛+1,(2.20) where 𝑤𝜌(𝑧)=𝑛+1∑/((𝑛+1)(1−𝛾))âˆžğ‘˜=𝑛+1ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1∑2+2𝑛𝑘=2ğ‘˜ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1+𝜌𝑛+1∑/((𝑛+1)(1−𝛾))âˆžğ‘˜=𝑛+1ğ‘˜ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1.(2.21) Now |𝑤(𝑧)|≤1 if and only if 𝑛𝑘=2𝑘||ğ‘Žğ‘˜||+𝜌𝑛+1(𝑛+1)(1−𝛾)âˆžî“ğ‘˜=𝑛+1𝑘||ğ‘Žğ‘˜||≤1.(2.22) From the condition (1.14), it is sufficient to show that 𝑛𝑘=2𝑘||ğ‘Žğ‘˜||+𝜌𝑛+1(𝑛+1)(1−𝛾)âˆžî“ğ‘˜=𝑛+1𝑘||ğ‘Žğ‘˜||â‰¤âˆžî“ğ‘˜=2𝜌𝑘||ğ‘Ž1−𝛾𝑘||(2.23) which is equivalent to 𝑛𝑘=2𝜌𝑘−(1−𝛾)𝑘||ğ‘Ž1−𝛾𝑘||+âˆžî“ğ‘˜=𝑛+1(𝑛+1)𝜌𝑘−𝑘𝜌𝑛+1||ğ‘Ž(𝑛+1)(1−𝛾)𝑘||≥0.(2.24)
To prove the result (2.18), define the function 𝑤(𝑧) by 1+𝑤(𝑧)=1−𝑤(𝑧)(𝑛+1)(1−𝛾)+𝜌𝑛+1𝑓(1−𝛾)(𝑛+1)î…žğ‘›(𝑧)ğ‘“î…žâˆ’ğœŒ(𝑧)𝑛+1(𝑛+1)(1−𝛾)+𝜌𝑛+1,(2.25) where 𝑤−(𝑧)=1+𝜌𝑛+1∑/((𝑛+1)(1−𝛾))âˆžğ‘˜=𝑛+1ğ‘˜ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1∑2+2𝑛𝑘=2ğ‘˜ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1+1−𝜌𝑛+1∑/((𝑛+1)(1−𝛾))âˆžğ‘˜=𝑛+1ğ‘˜ğ‘Žğ‘˜ğ‘§ğ‘˜âˆ’1.(2.26) Now |𝑤(𝑧)|≤1 if and only if 𝑛𝑘=2𝑘||ğ‘Žğ‘˜||+𝜌𝑛+1(𝑛+1)(1−𝛾)âˆžî“ğ‘˜=𝑛+1𝑘||ğ‘Žğ‘˜||≤1.(2.27) It suffices to show that the left hand side of (2.27) is bounded previously by the condition âˆžî“ğ‘˜=2𝜌𝑘||ğ‘Ž1−𝛾𝑘||,(2.28) which is equivalent to 𝑛𝑘=2𝜌𝑘||ğ‘Ž1−𝛾−𝑘𝑘||+âˆžî“ğ‘˜=𝑛+1𝜌𝑘−𝜌1−𝛾𝑛+1𝑘||ğ‘Ž(𝑛+1)(1−𝛾)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)/𝑓𝑛(𝑧)}, ℜ{𝑓𝑛(𝑧)/𝑓(𝑧)}, ℜ{ğ‘“î…ž(𝑧)/ğ‘“î…žğ‘›(𝑧)}, and ℜ{ğ‘“î…žğ‘›(𝑧)/ğ‘“î…ž(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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