Almost increasing sequences and their new applications
© Bor; licensee Springer 2013
Received: 9 January 2013
Accepted: 12 April 2013
Published: 25 April 2013
In this paper, we generalize a known theorem dealing with summability factors to the summability factors of infinite series using an almost increasing sequence. This theorem also includes some known and new results.
MSC:26D15, 40D15, 40F05, 40G05.
If we take , then summability reduces to summability.
2 Known result
Many works dealing with an application of almost increasing sequences to the absolute Cesàro summability factors of infinite series have been done (see [3–11]). Among them, in , the following main theorem dealing with summability factors has been proved.
are satisfied, then the series is summable , .
3 The main result
The aim of this paper is to generalize Theorem A to the summability in the following form.
Theorem Let be a positive sequence and let be an almost increasing sequence.
then the series is summable , , and .
Remark It should be noted that if we take , then we get Theorem A. In this case, condition (10) reduces to condition (8) and the condition ‘’ is trivial.
We need the following lemmas for the proof of our theorem.
Lemma 1 
Lemma 2 
4 Proof of the Theorem
Let be the n th mean, with , of the sequence .
by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. Also, if we take , then we get a new result concerning the summability factors of infinite series.
Dedicated to Professor Hari M Srivastava.
The author expresses his thanks to the referees for their useful comments and suggestions.
- Bari NK, Stečkin SB: Best approximation and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483–522. (in Russian)Google Scholar
- Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 1957, 7: 113–141.MathSciNetView ArticleMATHGoogle Scholar
- Bor H: An application of almost increasing sequences. Math. Inequal. Appl. 2002, 5(1):79–83.MathSciNetMATHGoogle Scholar
- Bor H, Srivastava HM: Almost increasing sequences and their applications. Int. J. Pure Appl. Math. 2002, 3: 29–35.MathSciNetMATHGoogle Scholar
- Bor H: A study on almost increasing sequences. JIPAM. J. Inequal. Pure Appl. Math. 2003., 4(5): Article ID 97Google Scholar
- Bor H, Leindler L: A note on δ -quasi-monotone and almost increasing sequences. Math. Inequal. Appl. 2005, 8(1):129–134.MathSciNetMATHGoogle Scholar
- Bor H, Özarslan HS: On the quasi-monotone and almost increasing sequences. J. Math. Inequal. 2007, 1(4):529–534.MathSciNetView ArticleMATHGoogle Scholar
- Bor H: An application of almost increasing sequences. Appl. Math. Lett. 2011, 24(3):298–301. 10.1016/j.aml.2010.10.009MathSciNetView ArticleMATHGoogle Scholar
- Bor H: On a new application of almost increasing sequences. Math. Comput. Model. 2011, 53(1–2):230–233. 10.1016/j.mcm.2010.08.011MathSciNetView ArticleMATHGoogle Scholar
- Sulaiman WT: On a new application of almost increasing sequences. Bull. Math. Anal. Appl. 2012, 4(3):29–33.MathSciNetMATHGoogle Scholar
- Bor H, Srivastava HM, Sulaiman WT: A new application of certain generalized power increasing sequences. Filomat 2012, 26(4):871–879. 10.2298/FIL1204871BMathSciNetView ArticleMATHGoogle Scholar
- Pati T: The summability factors of infinite series. Duke Math. J. 1954, 21: 271–284. 10.1215/S0012-7094-54-02127-4MathSciNetView ArticleMATHGoogle Scholar
- Bosanquet LS: A mean value theorem. J. Lond. Math. Soc. 1941, 16: 146–148.MathSciNetView ArticleMATHGoogle Scholar
- Mazhar SM: Absolute summability factors of infinite series. Kyungpook Math. J. 1999, 39: 67–73.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.