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.
Similar content being viewed by others
1 Introduction
Let \(\sum_{n=0}^{\infty }a_{n}\) be an infinite series with sequence of partial sums \(\lbrace {s_{n}}\rbrace \) and the nth sequence to sequence transformation (mean) of \(\lbrace {s_{n}}\rbrace \) be given by \(u_{n}\) s.t.
Before discussing \(\varphi - \vert C, \alpha, \beta; \delta \vert _{k}\) summability, let us introduce some well-known basic summabilities which are helpful in understanding the \(\varphi - \vert C, \alpha, \beta;\delta \vert _{k}\) summability.
Definition 1
The series \(\sum_{n=0}^{\infty }a _{n}\) is said to be absolute summable, if
and \(\sum_{n=1}^{\infty }\vert u_{n}-u_{n-1}\vert <\infty\).
Definition 2
[1]
Let \(t_{n}\) represent the nth \((C, 1)\) means of the sequence \((na_{n})\), then the series \(\sum_{n=0}^{\infty }a_{n}\) is said to be \(\vert C, 1\vert _{k}\) summable for \(k\geq 1\), if
Definition 3
[2]
The nth Cesáro means of order \((\alpha, \beta)\), with \(\alpha +\beta >-1\), of the sequence \((n a_{n}) \) is denoted by \(t_{n}^{\alpha, \beta }\), i.e.
where
If the sequence \(\lbrace t_{n}^{\alpha, \beta } \rbrace \) satisfies
then the series \(\sum_{n=0}^{\infty }a_{n}\) is said to be \(\varphi -\vert C, \alpha, \beta \vert _{k}\) summable.
Definition 4
For the following condition:
the series \(\sum_{n=0}^{\infty }a_{n}\) is said to be \(\varphi -\vert C, \alpha, \beta; \delta \vert _{k}\) summable, where \(k\geq 1\), \(\delta \geq 0\) and \((\varphi_{n})\) is a sequence of positive real numbers.
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
Let \(X_{n}\) be a quasi-f-power increasing sequence for some η (\(0<\eta <1 \)). Suppose also that there exists a sequence of numbers \((A_{n}) \) such that it is ξ-quasi-monotone satisfying the following:
If the conditions
are satisfied, then the series \({\sum a_{n}\lambda_{n}} \) is \(\vert C, \alpha \vert _{k}\) summable, \(0<\alpha \leq 1\) and \(k \geq 1\).
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
Let \(X_{n}\) be a quasi-f-power increasing sequence for some η (\(0<\eta <1\)). Suppose also that there exists a ξ-quasi-monotone sequence of numbers \((A_{n}) \) such that
Then the series \({\sum a_{n}\lambda_{n}} \) is \(\varphi -\vert C, \alpha,\beta; \delta \vert _{k}\) summable for \(k\geq 1\), \(0<\alpha \leq 1\), \(\beta > -1\), \(\alpha +\beta >0 \) and \(\delta \geq 0\), if the following conditions are satisfied:
where \(w_{n}^{\alpha, \beta } \) is given by [17]
4 Lemmas
We need the following lemmas for the proof of our theorem.
Lemma 4.1
[18]
If \(0<\alpha \leq 1\), \(\beta > -1 \) and \(1\leq v\leq n\), then
Lemma 4.2
[19]
Let \((X_{n}) \) be a quasi-f-power increasing sequence for some η (\(0<\eta <1\)). If \((A_{n}) \) is a ξ-quasi-monotone sequence with \(\Delta A_{n} \leq \xi_{n} \) and \(\sum n \xi_{n} X_{n} <\infty\), then
5 Proof of the theorem
Let \(t_{n}^{\alpha, \beta }\) be the nth \((C, \alpha, \beta)\) mean of the sequence \((n a_{n} \lambda_{n} )\). Then the series will be \(\varphi -\vert C, \alpha, \beta; \delta \vert _{k}\) summable (by Definition 4), if
Applying Abel’s transformation and Lemma 4.1, we have
We use Minkowski’s inequality,
In order to complete the proof of the theorem, it is sufficient to show that
By using Hölder’s inequality, Abel’s transformation and the conditions of Lemma 4.2 [19], we have
Hence the proof of the theorem is completed.
6 Corollaries
Corollary 6.1
Let \(X_{n}\) be a quasi-f-power increasing sequence for some η (\(0<\eta <1\)) and there exists a sequence of numbers \((A_{n}) \) such that it is ξ-quasi-monotone satisfying (13)-(17) and the following condition:
then the series \({\sum a_{n}\lambda_{n}} \) is \(\vert C, \alpha, \beta;\delta \vert _{k} \) summable, \(\alpha +\beta >\delta\), \(0<\alpha \leq 1\), \(\beta >-1\), \(\delta \geq 0\), \(k\geq 1\), where \(w_{n}^{ \alpha, \beta } \) is given by (20).
Proof
On putting \(\varphi_{n}=n \) in Theorem 3.1, we will get (32) and the following condition:
Here, condition (33) always holds. We omit the details as the proof is similar to that of Theorem 3.1 using the conditions (33) and (32) instead of (18) and (19). □
Corollary 6.2
Let \(X_{n}\) be a quasi-f-power increasing sequence for some η (\(0<\eta <1 \)) and there exists a sequence of numbers \((A_{n}) \) such that it is ξ-quasi-monotone satisfying (13)-(17) and the following conditions:
then the series \({\sum a_{n}\lambda_{n}} \) is \(\varphi -\vert C, \alpha,\beta \vert _{k}\) summable, \(\alpha +\beta >0\), \(0<\alpha \leq 1\), \(\beta > -1\), \(k\geq 1\), where \(w_{n}^{\alpha, \beta } \) is given by (20).
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]
Let \(X_{n}\) be a quasi-f-power increasing sequence for some η (\(0<\eta <1\)) and there exists a sequence of numbers \((A_{n}) \) such that it is ξ-quasi-monotone satisfying (13)-(17) and the following conditions:
then the series \({\sum a_{n}\lambda_{n}} \) is \(\vert C, \alpha \vert _{k}\) summable, \(0<\alpha \leq 1\), \(k\geq 1\), where \(w_{n}^{\alpha } \) is given by
Proof
On putting \(\varphi_{n}=n \), \(\delta =0 \) and \(\beta =0 \) in Theorem 3.1, we will get (36) and the following condition:
Here, condition (38) always holds. We omit the details as the proof is similar to that of Theorem 3.1 using the conditions (38) and (36) instead of (18) and (19). □
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].
References
Flett, TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 3(1), 113-141 (1957)
Borwein, D: Theorems on some methods of summability. Q. J. Math. 9(1), 310-316 (1958)
Bor, H, Srivastava, HM: Almost increasing sequences and their applications. Int. J. Pure Appl. Math. 3, 29-35 (2002)
Bor, H: An application of almost increasing sequences. Math. Inequal. Appl. 5(1), 79-83 (2002)
Bor, H: Factors for generalized absolute Cesàro summability. Math. Comput. Model. 53(5), 1150-1153 (2011)
Bor, H: An application of almost increasing sequences. Appl. Math. Lett. 24(3), 298-301 (2011)
Bor, H: Almost increasing sequences and their new applications II. Filomat 28(3), 435-439 (2014)
Bor, H: A new theorem on the absolute Riesz summability factors. Filomat 28(8), 1537-1541 (2014)
Bor, H: Factors for generalized absolute Cesàro summability. Math. Commun. 13(1), 21-25 (2008)
Bor, H: A new application of quasi power increasing sequences II. Fixed Point Theory Appl. 2013, 75 (2013)
Özarslan, HS: A note on absolute summability factors. Proc. Indian Acad. Sci. Math. Sci. 113(2), 165-169 (2003)
Sonker, S, Munjal, A: Absolute summability factor \(\varphi-\vert {C}, 1, \delta \vert _{k}\) of infinite series. Int. J. Math. Anal. 10(23), 1129-1136 (2016)
Sonker, S, Munjal, A: Sufficient conditions for triple matrices to be bounded. Nonlinear Stud. 23(4), 533-542 (2016)
Bor, H: Some new results on infinite series and Fourier series. Positivity 19(3), 467-473 (2015)
Sulaiman, WT: Extension on absolute summability factors of infinite series. J. Math. Anal. Appl. 322(2), 1224-1230 (2006)
Leindler, L: A new application of quasi power increasing sequences. Publ. Math. (Debr.) 58(4), 791-796 (2001)
Bor, H: On a new application of power increasing sequences. Proc. Est. Acad. Sci. 57(4), 205-209 (2008)
Bosanquet, LS: A mean value theorem. J. Lond. Math. Soc. 1-16(3), 146-148 (1941)
Bor, H: On the quasi monotone and generalized power increasing sequences and their new applications. J. Class. Anal. 2(2), 139-144 (2013)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Sonker, S., Munjal, A. Absolute \(\varphi- \vert C, \alpha, \beta; \delta\vert _{k}\) summability of infinite series. J Inequal Appl 2017, 168 (2017). https://doi.org/10.1186/s13660-017-1445-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1445-5