- Open Access
On a new application of almost increasing sequences
© Özarslan and Keten; licensee Springer 2013
- Received: 3 September 2012
- Accepted: 6 November 2012
- Published: 9 January 2013
In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with summability factors of an infinite series. In the present paper, we have generalized this theorem on the summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence.
MSC:40D15, 40F05, 40G99.
- absolute matrix summability
- almost increasing sequences
- infinite series
If we take , then summability reduces to summability. Also, if we take and , then we get summability. Furthermore, if we take , and for all values of n, reduces to summability. Finally, if we take and , then we get summability (see ).
Before stating the main theorem, we must first introduce some further notations.
then the series is summable , .
The aim of this paper is to generalize Theorem A for summability under weaker conditions. For this, we need the concept of an almost increasing sequence. A positive sequence is said to be almost increasing if there exists a positive increasing sequence and two positive constants A and B such that (see ). Obviously, every increasing sequence is an almost increasing sequence but the converse need not be true as can be seen from the example . Also, one can find some results dealing with absolute almost convergent sequences (see ). So, we are weakening the hypotheses of Theorem A replacing the increasing sequence by an almost increasing sequence. Now, we shall prove the following theorem.
are satisfied, then the series is summable , .
Remark It should be noted that if we take as a positive non-decreasing sequence, and , then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘ is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.
This completes the proof of the theorem. If we take , then we get a result concerning the summability factors. If we take , then we have another result dealing with summability. If we take and for all values of n, then we get a result dealing with summability. If we take , and for all values of n, then we get a result for summability. □
- Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 1957, 7: 113–141. 10.1112/plms/s3-7.1.113MATHMathSciNetView ArticleGoogle Scholar
- Hardy GH: Divergent Series. Oxford University Press, Oxford; 1949.MATHGoogle Scholar
- Bor H: On two summability methods. Math. Proc. Camb. Philos. Soc. 1985, 97: 147–149. 10.1017/S030500410006268XMATHMathSciNetView ArticleGoogle Scholar
- Sulaiman WT: Inclusion theorems for absolute matrix summability methods of an infinite series (IV). Indian J. Pure Appl. Math. 2003, 34(11):1547–1557.MATHMathSciNetGoogle Scholar
- Özarslan, HS, Keten, A: A new application of almost increasing sequences. An. ştiinţ. Univ. “Al.I. Cuza” Iaşi, Mat. (2012, in press)Google Scholar
- Bor H: On the relative strength of two absolute summability methods. Proc. Am. Math. Soc. 1991, 113: 1009–1012.MATHMathSciNetView ArticleGoogle Scholar
- Bor H:On summability factors. Proc. Am. Math. Soc. 1985, 94: 419–422.MathSciNetGoogle Scholar
- Bor H:A note on summability factors of infinite series. Indian J. Pure Appl. Math. 1987, 18: 330–336.MATHMathSciNetGoogle Scholar
- Bor H: On absolute summability factors. Analysis 1987, 7: 185–193.MATHMathSciNetGoogle Scholar
- Bor H: Absolute summability factors for infinite series. Indian J. Pure Appl. Math. 1988, 19: 664–671.MATHMathSciNetGoogle Scholar
- Bor H, Kuttner B: On the necessary conditions for absolute weighted arithmetic mean summability factors. Acta Math. Hung. 1989, 54: 57–61. 10.1007/BF01950709MATHMathSciNetView ArticleGoogle Scholar
- Bor H:A note on summability factors. Bull. Calcutta Math. Soc. 1990, 82: 357–362.MATHMathSciNetGoogle Scholar
- Bor H: Absolute summability factors for infinite series. Math. Jpn. 1991, 36: 215–219.MATHMathSciNetGoogle Scholar
- Bor H:Factors for summability of infinite series. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 1991, 16: 151–154.MATHMathSciNetView ArticleGoogle Scholar
- Bor H:On absolute summability factors for summability. Comment. Math. Univ. Carol. 1991, 32(3):435–439.MATHMathSciNetGoogle Scholar
- Bor H:On the summability factors for infinite series. Proc. Indian Acad. Sci. Math. Sci. 1991, 101: 143–146. 10.1007/BF02868023MATHMathSciNetView ArticleGoogle Scholar
- Bor H:A note on summability factors. Rend. Mat. Appl. (7) 1992, 12: 937–942.MATHMathSciNetGoogle Scholar
- Bor H: On absolute summability factors. Proc. Am. Math. Soc. 1993, 118: 71–75. 10.1090/S0002-9939-1993-1155594-4MATHMathSciNetView ArticleGoogle Scholar
- Bor H: On the absolute Riesz summability factors. Rocky Mt. J. Math. 1994, 24: 1263–1271. 10.1216/rmjm/1181072337MATHMathSciNetView ArticleGoogle Scholar
- Bor H:On summability factors. Kuwait J. Sci. Eng. 1996, 23: 1–5.MATHMathSciNetGoogle Scholar
- Bor H: A note on absolute summability factors. Int. J. Math. Math. Sci. 1994, 17: 479–482. 10.1155/S0161171294000700MATHMathSciNetView ArticleGoogle Scholar
- Mazhar SM: A note on absolute summability factors. Bull. Inst. Math. Acad. Sin. 1997, 25(3):233–242.MATHMathSciNetGoogle Scholar
- Bari NK, Stečkin SB: Best approximation and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483–522. in RussianMATHGoogle Scholar
- Çakalli H, Çanak G:-absolute almost convergent sequences. Indian J. Pure Appl. Math. 1997, 28(4):525–532.MATHMathSciNetGoogle 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.