Absolute \(\varphi- \vert C, \alpha, \beta; \delta\vert _{k}\) summability of infinite series
- Smita Sonker1 and
- Alka Munjal1Email author
https://doi.org/10.1186/s13660-017-1445-5
© The Author(s) 2017
Received: 8 March 2017
Accepted: 30 June 2017
Published: 17 July 2017
Abstract
In this paper, we established a generalized theorem on a minimal set of sufficient conditions for absolute summability factors by applying a sequence of a wider class (quasi-power increasing sequence) and the absolute Cesàro \(\varphi-\vert C, \alpha, \beta; \delta \vert _{k}\) summability for an infinite series. We further obtained well-known applications of the above theorem as corollaries, under suitable conditions.
Keywords
MSC
1 Introduction
Definition 1
Definition 2
[1]
Definition 3
[2]
Definition 4
Bor gave a number of theorems on absolute summability. In 2002, Bor found the sufficient conditions for an infinite series to be \(\vert C, \alpha \vert _{k}\) summable [3] and \(\vert C, \alpha;\delta \vert _{k}\) summable [4]. In 2011, he generalized his previous results for \(\vert C, \alpha, \beta \vert _{k}\) summability [5] and \(\vert C, \alpha, \beta; \delta \vert _{k}\) summability [6], respectively. In 2014, Bor [7] generalized the \(\vert C, \alpha \vert _{k}\) summability factor to the \(\vert C, \alpha, \beta;\delta \vert _{k}\) summability of an infinite series and in [8], he discussed a general class of power increasing sequences and absolute Riesz summability factors of an infinite series. In [9], Bor applied \(\vert C, \alpha, \gamma; \beta \vert _{k} \) summability to obtain the sufficient conditions for an infinite series to be absolute summable.
Bor [10] gave a new application of quasi-power increasing sequence by applying absolute Cesáro \(\varphi -\vert C, \alpha \vert _{k}\) summability for an infinity series. Özarslan [11] generalized the result on \(\varphi -\vert C, 1\vert _{k} \) by a more general absolute \(\varphi -\vert C, \alpha \vert _{k}\) summability. In 2016, Sonker and Munjal [12] determined a theorem on generalized absolute Cesáro summability with the sufficient conditions for an infinite series and in [13], they used the concept of triangle matrices for obtaining the minimal set of sufficient conditions of an infinite series to be bounded.
2 Known results
By using \(\vert C, \alpha \vert _{k}\) summability, Bor [14] gave a minimal set of sufficient conditions for an infinite series to be absolute summable.
Theorem 2.1
3 Main results
A positive sequence \(X = (X_{n})\) is said to be a quasi-f-power increasing sequence if there exists a constant \(K = K(X, f) \geq 1\) such that \(K f_{n} X_{n} \geq f_{m} X_{m}\) for all \(n \geq m \geq 1\), where \(f = [f_{n}(\eta, \zeta)] = \lbrace n^{\eta }(\log n)^{\zeta },\zeta \geq 0,0 < \eta < 1\rbrace \) [15]. If we set \(\zeta =0\), then we get a quasi-η-power increasing sequence [16].
With the help of generalized Cesáro \(\varphi -\vert C, \alpha, \beta; \delta \vert _{k}\) summability, we modernized the results of Bor [14] and established the following theorem.
Theorem 3.1
4 Lemmas
We need the following lemmas for the proof of our theorem.
Lemma 4.1
[18]
Lemma 4.2
[19]
5 Proof of the theorem
6 Corollaries
Corollary 6.1
Proof
Corollary 6.2
Proof
On putting \(\delta =0 \) in Theorem 3.1, we will get (34) and (35). We omit the details as the proof is similar to that of Theorem 3.1 using the conditions (34) and (35) instead of (18) and (19). □
Corollary 6.3
[14]
Proof
7 Conclusion
The aim of our paper is to obtain the minimal set of sufficient conditions for an infinite series to be absolute Cesáro \(\varphi -\vert C, \alpha, \beta; \delta \vert _{k}\) summable. Through the investigation, we may conclude that our theorem is a generalized version which can be reduced for several well-known summabilities as shown in the corollaries. Further, our theorem has been validated through Corollary 6.3, which is a result of Bor [14].
Declarations
Acknowledgements
The authors would like to thank the anonymous learned referee for his/her valuable suggestions which improved the paper considerably. The authors are also thankful to all the Editorial board members and reviewers of Journal of Inequalities and Applications.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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