Research | Open | Published:
On a new application of almost increasing sequences
Journal of Inequalities and Applicationsvolume 2013, Article number: 13 (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.
Let be a given infinite series with the partial sums . We denote by the n th mean of the sequence . The series is said to be summable , , if (see )
Let be a sequence of positive numbers such that
The sequence-to-sequence transformation
Let be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence to , where
The series is said to be summable , , if (see )
Let be any sequence of positive real numbers. The series is said to be summable , , if (see )
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.
Given a normal matrix , we associate two lower semimatrices and as follows:
It may be noted that and are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we have
2 Known result
Theorem A Let be a positive non-decreasing sequence and let there be sequences and such that
are satisfied. Furthermore, if is a sequence of positive numbers such that
then the series is summable , .
3 The main result
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.
Theorem Let be a positive normal matrix such that
Let be an almost increasing sequence and be a non-increasing sequence. If conditions (12)-(16) and
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.
Under the conditions on , and as taken in the statement of the theorem, we have the following:
Proof of the Theorem Let denote A-transform of the series . Then we have, by (10) and (11),
Applying Abel’s transformation to this sum, we get that
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
Now, when , applying Hölder’s inequality with indices k and , where , we have that
by virtue of the hypotheses of the theorem and the lemma. Again, applying Hölder’s inequality and using the fact that by (23), we get that
by virtue of the hypotheses of the theorem and the lemma. Finally, as in , we have that
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/s3-7.1.113
Hardy GH: Divergent Series. Oxford University Press, Oxford; 1949.
Bor H: On two summability methods. Math. Proc. Camb. Philos. Soc. 1985, 97: 147–149. 10.1017/S030500410006268X
Sulaiman WT: Inclusion theorems for absolute matrix summability methods of an infinite series (IV). Indian J. Pure Appl. Math. 2003, 34(11):1547–1557.
Özarslan, HS, Keten, A: A new application of almost increasing sequences. An. ştiinţ. Univ. “Al.I. Cuza” Iaşi, Mat. (2012, in press)
Bor H: On the relative strength of two absolute summability methods. Proc. Am. Math. Soc. 1991, 113: 1009–1012.
Bor H:On summability factors. Proc. Am. Math. Soc. 1985, 94: 419–422.
Bor H:A note on summability factors of infinite series. Indian J. Pure Appl. Math. 1987, 18: 330–336.
Bor H: On absolute summability factors. Analysis 1987, 7: 185–193.
Bor H: Absolute summability factors for infinite series. Indian J. Pure Appl. Math. 1988, 19: 664–671.
Bor H, Kuttner B: On the necessary conditions for absolute weighted arithmetic mean summability factors. Acta Math. Hung. 1989, 54: 57–61. 10.1007/BF01950709
Bor H:A note on summability factors. Bull. Calcutta Math. Soc. 1990, 82: 357–362.
Bor H: Absolute summability factors for infinite series. Math. Jpn. 1991, 36: 215–219.
Bor H:Factors for summability of infinite series. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 1991, 16: 151–154.
Bor H:On absolute summability factors for summability. Comment. Math. Univ. Carol. 1991, 32(3):435–439.
Bor H:On the summability factors for infinite series. Proc. Indian Acad. Sci. Math. Sci. 1991, 101: 143–146. 10.1007/BF02868023
Bor H:A note on summability factors. Rend. Mat. Appl. (7) 1992, 12: 937–942.
Bor H: On absolute summability factors. Proc. Am. Math. Soc. 1993, 118: 71–75. 10.1090/S0002-9939-1993-1155594-4
Bor H: On the absolute Riesz summability factors. Rocky Mt. J. Math. 1994, 24: 1263–1271. 10.1216/rmjm/1181072337
Bor H:On summability factors. Kuwait J. Sci. Eng. 1996, 23: 1–5.
Bor H: A note on absolute summability factors. Int. J. Math. Math. Sci. 1994, 17: 479–482. 10.1155/S0161171294000700
Mazhar SM: A note on absolute summability factors. Bull. Inst. Math. Acad. Sin. 1997, 25(3):233–242.
Bari NK, Stečkin SB: Best approximation and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483–522. in Russian
Çakalli H, Çanak G:-absolute almost convergent sequences. Indian J. Pure Appl. Math. 1997, 28(4):525–532.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.