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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 849250, 9 pages
http://dx.doi.org/10.1155/2011/849250
Research Article

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

1School of Advanced Sciences, VIT University, Vellore 632014, India
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia

Received 29 January 2011; Accepted 7 April 2011

Academic Editor: YoshikazuΒ Giga

Copyright Β© 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.

Abstract

Let π‘“π‘›βˆ‘(𝑧)=𝑧+π‘›π‘˜=2π‘Žπ‘˜π‘§π‘˜ be the sequence of partial sums of the analytic function βˆ‘π‘“(𝑧)=𝑧+βˆžπ‘˜=2π‘Žπ‘˜π‘§π‘˜. In this paper, we determine sharp lower bounds for β„œ{𝑓(𝑧)/𝑓𝑛(𝑧)},β„œ{𝑓𝑛(𝑧)/𝑓(𝑧)},β„œ{π‘“ξ…ž(𝑧)/π‘“ξ…žπ‘›(𝑧)}, and β„œ{π‘“ξ…žπ‘›(𝑧)/π‘“ξ…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 π’œ denote the class of functions of the form𝑓(𝑧)=𝑧+βˆžξ“π‘˜=2π‘Žπ‘˜π‘§π‘˜(1.1) which are analytic and univalent in the open disc π‘ˆ={π‘§βˆΆ|𝑧|<1}. We also consider 𝑇 a subclass of π’œ introduced and studied by Silverman [1], consisting of functions of the form𝑓(𝑧)=π‘§βˆ’βˆžξ“π‘˜=2π‘Žπ‘˜π‘§π‘˜,π‘§βˆˆπ‘ˆ.(1.2)

For functions π‘“βˆˆπ’œ given by (1.1) and π‘”βˆˆπ’œ given by βˆ‘π‘”(𝑧)=𝑧+βˆžπ‘˜=2π‘π‘˜π‘§π‘˜, we define the Hadamard product (or convolution) of 𝑓 and 𝑔 by(π‘“βˆ—π‘”)(𝑧)=𝑧+βˆžξ“π‘˜=2π‘Žπ‘˜π‘π‘˜π‘§π‘˜,π‘§βˆˆπ‘ˆ.(1.3) We recall here a general Hurwitz-Lerch zeta function Ξ¦(𝑧,𝑠,π‘Ž) defined in [2] byΞ¦(𝑧,𝑠,π‘Ž)∢=βˆžξ“π‘˜=0π‘§π‘˜(π‘˜+π‘Ž)𝑠(1.4) (π‘Žβˆˆβ„‚β§΅β„€βˆ’0; π‘ βˆˆβ„‚, when |𝑧|<1; β„œ(𝑠)>1, when |𝑧|=1), where, as usual, β„€βˆ’0∢=℀⧡ℕ, (β„€βˆΆ={0,Β±1,Β±2,Β±3,…}); β„•βˆΆ={1,2,3,…}. Several interesting properties and characteristics of the Hurwitz-Lerch Zeta function Ξ¦(𝑧,𝑠,π‘Ž) can be found in the recent investigations by Choi and Srivastava [3], Ferreira and LΓ³pez [4], Garg et al. [5], Lin and Srivastava [6], Lin et al. [7], and others. Srivastava and Attiya [8] (see also RΔƒducanu and Srivastava [9] and Prajapat and Goyal [10]) introduced and investigated the linear operator π’₯πœ‡,π‘βˆΆπ’œβ†’π’œ defined in terms of the Hadamard product byπ’₯πœ‡,𝑏𝑓(𝑧)=𝒒𝑏,πœ‡ξ€·βˆ—π‘“(𝑧)π‘§βˆˆπ‘ˆ;π‘βˆˆβ„‚β§΅β„€βˆ’0ξ€Έ,;πœ‡βˆˆβ„‚;π‘“βˆˆπ’œ(1.5) where, for convenience,πΊπœ‡,𝑏(𝑧)∢=(1+𝑏)πœ‡ξ€ΊΞ¦(𝑧,πœ‡,𝑏)βˆ’π‘βˆ’πœ‡ξ€»(π‘§βˆˆπ‘ˆ).(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):π’₯πœ‡π‘π‘“(𝑧)=𝑧+βˆžξ“π‘˜=2ξ‚€1+π‘ξ‚π‘˜+π‘πœ‡π‘Žπ‘˜π‘§π‘˜.(1.7) Motivated essentially by the Srivastava-Attiya operator [8], we introduce the generalized integral operatorπ’₯π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)=𝑧+βˆžξ“π‘˜=2πΆπ‘šπ‘˜(𝑏,πœ‡)π‘Žπ‘˜π‘§π‘˜=𝐹(𝑧),(1.8) whereπΆπ‘šπ‘˜||||ξ‚€(𝑏,πœ‡)=1+π‘ξ‚π‘˜+π‘πœ‡π‘š!(π‘˜+πœ‚βˆ’2)!||||(πœ‚βˆ’2)!(π‘˜+π‘šβˆ’1)!,(1.9) and (throughout this paper unless otherwise mentioned) the parameters πœ‡, 𝑏 are constrained as π‘βˆˆβ„‚β§΅β„€βˆ’0; πœ‡βˆˆβ„‚, πœ‚β‰₯2 and π‘š>βˆ’1. It is of interest to note that 𝐽1,2πœ‡,𝑏 is the Srivastava-Attiya operator [8] and π½π‘š,πœ‚0,𝑏 is the well-known Choi-Saigo-Srivastava operator (see [11, 12]). Suitably specializing the parameters π‘š, πœ‚, πœ‡, and 𝑏 in π’₯π‘š,πœ‚πœ‡,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 π’₯π‘š,πœ‚πœ‡,𝑏, we define the following new subclass of analytic functions with negative coefficients.

For πœ†β‰₯0, βˆ’1≀𝛾<1, and 𝛽β‰₯0, let π‘ƒπœ†πœ‡(𝛾,𝛽) be the subclass of π’œ consisting of functions of the form (1.1) and satisfying the analytic criterionβ„œβŽ§βŽͺ⎨βŽͺβŽ©π‘§ξ‚€π’₯π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)ξ…ž+πœ†π‘§2ξ‚€π’₯π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)ξ…žξ…ž(1βˆ’πœ†)π’₯π‘š,πœ‚πœ‡,𝑏𝑓π’₯(𝑧)+πœ†π‘§π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)ξ…žβŽ«βŽͺ⎬βŽͺ⎭||||||𝑧π’₯βˆ’π›Ύ>π›½π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)ξ…ž+πœ†π‘§2ξ‚€π’₯π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)ξ…žξ…ž(1βˆ’πœ†)π’₯π‘š,πœ‚πœ‡,𝑏𝑓π’₯(𝑧)+πœ†π‘§π‘š,πœ‚πœ‡,𝑏𝑓(𝑧)ξ…ž||||||βˆ’1,(1.10) where π‘§βˆˆπ‘ˆ. Shortly we can state this condition byξ‚»βˆ’β„œπ‘§πΊξ…ž(𝑧)ξ‚Ό||||𝐺(𝑧)βˆ’π›Ύ>π›½π‘§πΊξ…ž(𝑧)||||,𝐺(𝑧)βˆ’1(1.11) where𝐺(𝑧)=(1βˆ’πœ†)𝐹(𝑧)+πœ†π‘§πΉξ…ž(𝑧)=𝑧+βˆžξ“π‘˜=2[]||𝐢1+πœ†(π‘˜βˆ’1)π‘šπ‘˜(||π‘Žπ‘,πœ‡)π‘˜π‘§π‘˜,(1.12) and 𝐹(𝑧)=π’₯π‘š,πœ‚πœ‡,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+π‘›ξ“π‘˜=2π‘Žπ‘˜π‘§π‘˜,(1.13) when the coefficients of 𝑓(𝑧) satisfy the condition (1.14). Also, we will determine sharp lower bounds for β„œ{𝑓(𝑧)/𝑓𝑛(𝑧)}, β„œ{𝑓𝑛(𝑧)/𝑓(𝑧)}, β„œ{π‘“ξ…ž(𝑧)/π‘“ξ…žπ‘›(𝑧)}, and β„œ{π‘“ξ…žπ‘›(𝑧)/π‘“ξ…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 π‘ƒπœ†πœ‡(𝛾,𝛽) if βˆžξ“π‘˜=2([]||π‘Ž1+πœ†(π‘˜βˆ’1))π‘˜(1+𝛽)βˆ’(𝛾+𝛽)π‘˜||πΆπ‘šπ‘˜(𝑏,πœ‡)≀1βˆ’π›Ύ,(1.14) where, for convenience, πœŒπ‘˜=πœŒπ‘˜([]πΆπœ†,𝛾,πœ‚)=(1+πœ†(π‘˜βˆ’1))𝑛(1+𝛽)βˆ’(𝛾+𝛽)π‘šπ‘˜(𝑏,πœ‡),(1.15)  0β‰€πœ†β‰€1, βˆ’1≀𝛾<1, 𝛽β‰₯0, and πΆπ‘šπ‘˜(𝑏,πœ‡), 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 β„œξ‚»π‘“(𝑧)𝑓𝑛β‰₯𝜌(𝑧)𝑛+1(πœ†,𝛾,πœ‚)βˆ’1+π›ΎπœŒπ‘›+1(πœ†,𝛾,πœ‚)(π‘§βˆˆπ‘ˆ),(2.1) where πœŒπ‘˜=πœŒπ‘˜ξƒ―πœŒ(πœ†,𝛾,πœ‚)β‰₯1βˆ’π›Ύifπ‘˜=2,3,…,𝑛,𝑛+1ifπ‘˜=𝑛+1,𝑛+2,….(2.2) The result (2.1) is sharp with the function given by 𝑓(𝑧)=𝑧+1βˆ’π›ΎπœŒπ‘›+1𝑧𝑛+1.(2.3)

Proof. Define the function 𝑀(𝑧) by 1+𝑀(𝑧)=𝜌1βˆ’π‘€(𝑧)𝑛+1ξ‚Έ1βˆ’π›Ύπ‘“(𝑧)π‘“π‘›βˆ’πœŒ(𝑧)𝑛+1βˆ’1+π›ΎπœŒπ‘›+1ξ‚Ή=βˆ‘1+π‘›π‘˜=2π‘Žπ‘˜π‘§π‘˜βˆ’1+ξ€·πœŒπ‘›+1ξ€Έβˆ‘/(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1π‘Žπ‘˜π‘§π‘˜βˆ’1βˆ‘1+π‘›π‘˜=2π‘Žπ‘˜π‘§π‘˜βˆ’1.(2.4)
It suffices to show that |𝑀(𝑧)|≀1. Now, from (2.4) we can write π‘€ξ€·πœŒ(𝑧)=𝑛+1ξ€Έβˆ‘/(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1π‘Žπ‘˜π‘§π‘˜βˆ’1βˆ‘2+2π‘›π‘˜=2π‘Žπ‘˜π‘§π‘˜βˆ’1+ξ€·πœŒπ‘›+1ξ€Έβˆ‘/(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1π‘Žπ‘˜π‘§π‘˜βˆ’1.(2.5) Hence we obtain ||||β‰€ξ€·πœŒπ‘€(𝑧)𝑛+1/ξ€Έβˆ‘(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1||π‘Žπ‘˜||βˆ‘2βˆ’2π‘›π‘˜=2||π‘Žπ‘˜||βˆ’ξ€·πœŒπ‘›+1ξ€Έβˆ‘/(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1||π‘Žπ‘˜||.(2.6) Now |𝑀(𝑧)|≀1 if and only if 2ξ‚΅πœŒπ‘›+1ξ‚Ά1βˆ’π›Ύβˆžξ“π‘˜=𝑛+1||π‘Žπ‘˜||≀2βˆ’2π‘›ξ“π‘˜=2||π‘Žπ‘˜||(2.7) or, equivalently, π‘›ξ“π‘˜=2||π‘Žπ‘˜||+βˆžξ“π‘˜=𝑛+1πœŒπ‘›+1||π‘Ž1βˆ’π›Ύπ‘˜||≀1.(2.8)
From the condition (1.14), it is sufficient to show that π‘›ξ“π‘˜=2||π‘Žπ‘˜||+βˆžξ“π‘˜=𝑛+1πœŒπ‘›+1||π‘Ž1βˆ’π›Ύπ‘˜||β‰€βˆžξ“π‘˜=2πœŒπ‘˜||π‘Ž1βˆ’π›Ύπ‘˜||(2.9) which is equivalent to π‘›ξ“π‘˜=2ξ‚΅πœŒπ‘˜βˆ’1+𝛾||π‘Ž1βˆ’π›Ύπ‘˜||+βˆžξ“π‘˜=𝑛+1ξ‚΅πœŒπ‘˜βˆ’πœŒπ‘›+1ξ‚Ά||π‘Ž1βˆ’π›Ύπ‘˜||β‰₯0.(2.10) To see that the function given by (2.3) gives the sharp result, we observe that for 𝑧=π‘Ÿπ‘’π‘–πœ‹/𝑛, 𝑓(𝑧)𝑓𝑛(𝑧)=1+1βˆ’π›ΎπœŒπ‘›+1π‘§π‘›βŸΆ1βˆ’1βˆ’π›ΎπœŒπ‘›+1=πœŒπ‘›+1βˆ’1+π›ΎπœŒπ‘›+1whenπ‘ŸβŸΆ1βˆ’.(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 β„œξ‚»π‘“π‘›(𝑧)ξ‚Όβ‰₯πœŒπ‘“(𝑧)𝑛+1πœŒπ‘›+1+1βˆ’π›Ύ(π‘§βˆˆπ‘ˆ),(2.12) where πœŒπ‘›+1β‰₯1βˆ’π›Ύ and πœŒπ‘˜β‰₯ξƒ―πœŒ1βˆ’π›Ύifπ‘˜=2,3,…,𝑛,𝑛+1ifπ‘˜=𝑛+1,𝑛+2,….(2.13) The result (2.12) is sharp with the function given by (2.3).

Proof. We write 1+𝑀(𝑧)=𝜌1βˆ’π‘€(𝑧)𝑛+1+1βˆ’π›Ύξ‚Έπ‘“1βˆ’π›Ύπ‘›(𝑧)βˆ’πœŒπ‘“(𝑧)𝑛+1πœŒπ‘›+1ξ‚Ή=βˆ‘+1βˆ’π›Ύ1+π‘›π‘˜=2π‘Žπ‘˜π‘§π‘˜βˆ’1βˆ’ξ€·πœŒπ‘›+1ξ€Έβˆ‘/(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1π‘Žπ‘˜π‘§π‘˜βˆ’1βˆ‘1+βˆžπ‘˜=2π‘Žπ‘˜π‘§π‘˜βˆ’1,(2.14) where ||||β‰€πœŒπ‘€(𝑧)𝑛+1ξ€Έ/ξ€Έβˆ‘+1βˆ’π›Ύ(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1||π‘Žπ‘˜||βˆ‘2βˆ’2π‘›π‘˜=2||π‘Žπ‘˜||βˆ’πœŒξ€·ξ€·π‘›+1ξ€Έξ€Έβˆ‘βˆ’1+𝛾/(1βˆ’π›Ύ)βˆžπ‘˜=𝑛+1||π‘Žπ‘˜||≀1.(2.15) This last inequality is equivalent to π‘›ξ“π‘˜=2||π‘Žπ‘˜||+βˆžξ“π‘˜=𝑛+1πœŒπ‘›+1||π‘Ž1βˆ’π›Ύπ‘˜||≀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 β„œξ‚»π‘“ξ…ž(𝑧)π‘“ξ…žπ‘›ξ‚Όβ‰₯𝜌(𝑧)𝑛+1βˆ’(𝑛+1)(1βˆ’π›Ύ)πœŒπ‘›+1β„œξ‚»π‘“(π‘§βˆˆπ‘ˆ),(2.17)ξ…žπ‘›(𝑧)π‘“ξ…žξ‚Όβ‰₯𝜌(𝑧)𝑛+1πœŒπ‘›+1+(𝑛+1)(1βˆ’π›Ύ)(π‘§βˆˆπ‘ˆ),(2.18) where πœŒπ‘›+1β‰₯(𝑛+1)(1βˆ’π›Ύ) and πœŒπ‘˜β‰₯⎧βŽͺ⎨βŽͺβŽ©π‘˜ξ‚€πœŒπ‘˜(1βˆ’π›Ύ)ifπ‘˜=2,3,…,𝑛,𝑛+1𝑛+1ifπ‘˜=𝑛+1,𝑛+2,….(2.19) The results are sharp with the function given by (2.3).

Proof. We write 1+𝑀(𝑧)=𝜌1βˆ’π‘€(𝑧)𝑛+1𝑓(𝑛+1)(1βˆ’π›Ύ)ξ…ž(𝑧)π‘“ξ…žπ‘›βˆ’ξ‚΅πœŒ(𝑧)𝑛+1βˆ’(𝑛+1)(1βˆ’π›Ύ)πœŒπ‘›+1,ξ‚Άξ‚Ή(2.20) where π‘€ξ€·πœŒ(𝑧)=𝑛+1ξ€Έβˆ‘/((𝑛+1)(1βˆ’π›Ύ))βˆžπ‘˜=𝑛+1π‘Žπ‘˜π‘§π‘˜βˆ’1βˆ‘2+2π‘›π‘˜=2π‘˜π‘Žπ‘˜π‘§π‘˜βˆ’1+ξ€·πœŒπ‘›+1ξ€Έβˆ‘/((𝑛+1)(1βˆ’π›Ύ))βˆžπ‘˜=𝑛+1π‘˜π‘Žπ‘˜π‘§π‘˜βˆ’1.(2.21) Now |𝑀(𝑧)|≀1 if and only if π‘›ξ“π‘˜=2π‘˜||π‘Žπ‘˜||+πœŒπ‘›+1(𝑛+1)(1βˆ’π›Ύ)βˆžξ“π‘˜=𝑛+1π‘˜||π‘Žπ‘˜||≀1.(2.22) From the condition (1.14), it is sufficient to show that π‘›ξ“π‘˜=2π‘˜||π‘Žπ‘˜||+πœŒπ‘›+1(𝑛+1)(1βˆ’π›Ύ)βˆžξ“π‘˜=𝑛+1π‘˜||π‘Žπ‘˜||β‰€βˆžξ“π‘˜=2πœŒπ‘˜||π‘Ž1βˆ’π›Ύπ‘˜||(2.23) which is equivalent to π‘›ξ“π‘˜=2ξ‚΅πœŒπ‘˜βˆ’(1βˆ’π›Ύ)π‘˜ξ‚Ά||π‘Ž1βˆ’π›Ύπ‘˜||+βˆžξ“π‘˜=𝑛+1(𝑛+1)πœŒπ‘˜βˆ’π‘˜πœŒπ‘›+1||π‘Ž(𝑛+1)(1βˆ’π›Ύ)π‘˜||β‰₯0.(2.24)
To prove the result (2.18), define the function 𝑀(𝑧) by 1+𝑀(𝑧)=1βˆ’π‘€(𝑧)(𝑛+1)(1βˆ’π›Ύ)+πœŒπ‘›+1𝑓(1βˆ’π›Ύ)(𝑛+1)ξ…žπ‘›(𝑧)π‘“ξ…žβˆ’πœŒ(𝑧)𝑛+1(𝑛+1)(1βˆ’π›Ύ)+πœŒπ‘›+1ξ‚Ή,(2.25) where π‘€βˆ’ξ€·(𝑧)=1+πœŒπ‘›+1ξ€Έβˆ‘/((𝑛+1)(1βˆ’π›Ύ))βˆžπ‘˜=𝑛+1π‘˜π‘Žπ‘˜π‘§π‘˜βˆ’1βˆ‘2+2π‘›π‘˜=2π‘˜π‘Žπ‘˜π‘§π‘˜βˆ’1+ξ€·1βˆ’πœŒπ‘›+1ξ€Έβˆ‘/((𝑛+1)(1βˆ’π›Ύ))βˆžπ‘˜=𝑛+1π‘˜π‘Žπ‘˜π‘§π‘˜βˆ’1.(2.26) Now |𝑀(𝑧)|≀1 if and only if π‘›ξ“π‘˜=2π‘˜||π‘Žπ‘˜||+ξ‚΅πœŒπ‘›+1ξ‚Ά(𝑛+1)(1βˆ’π›Ύ)βˆžξ“π‘˜=𝑛+1π‘˜||π‘Žπ‘˜||≀1.(2.27) It suffices to show that the left hand side of (2.27) is bounded previously by the condition βˆžξ“π‘˜=2πœŒπ‘˜||π‘Ž1βˆ’π›Ύπ‘˜||,(2.28) which is equivalent to π‘›ξ“π‘˜=2ξ‚΅πœŒπ‘˜ξ‚Ά||π‘Ž1βˆ’π›Ύβˆ’π‘˜π‘˜||+βˆžξ“π‘˜=𝑛+1ξ‚΅πœŒπ‘˜βˆ’πœŒ1βˆ’π›Ύπ‘›+1ξ‚Άπ‘˜||π‘Ž(𝑛+1)(1βˆ’π›Ύ)π‘˜||β‰₯0.(2.29)

Remark 2.4. As a special case of the previous theorems, we can determine new sharp lower bounds forβ„œ{𝑓(𝑧)/𝑓𝑛(𝑧)}, β„œ{𝑓𝑛(𝑧)/𝑓(𝑧)}, β„œ{π‘“ξ…ž(𝑧)/π‘“ξ…žπ‘›(𝑧)}, and β„œ{π‘“ξ…žπ‘›(𝑧)/π‘“ξ…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 πœ‚, π‘š, πœ‡, 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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