- Open Access
A new application of quasi power increasing sequences. II
Fixed Point Theory and Applications volume 2013, Article number: 75 (2013)
In this paper, we prove a general theorem dealing with absolute Cesàro summability factors of infinite series by using a quasi-f-power increasing sequence instead of a quasi-σ-power increasing sequence. This theorem also includes several new results.
MSC:26D15, 40D15, 40F05, 40G99, 46A45.
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 ). A sequence is said to be of bounded variation, denoted by , if . A positive sequence is said to be a quasi-σ-power increasing sequence if there exists a constant such that holds for all (see ). It should be noted that every almost increasing sequence is a quasi-σ-power increasing sequence for any nonnegative σ, but the converse may not be true as can be seen by taking an example, say for . Let be a sequence of complex numbers and let be a given infinite series with partial sums . We denote by and the n th Cesàro means of order α, with , of the sequences and , respectively, that is,
The series is said to be summable , and , if (see [3, 4])
In the special case if we take , then summability is the same as summability (see ). Also, if we take , then summability reduces to summability (see ).
2 The known results
Theorem A ()
Let and let be a quasi-σ-power increasing sequence for some σ (). Suppose also that there exist sequences and such that
If there exists an such that the sequence is nonincreasing and if the sequence defined by (see )
satisfies the condition
then the series is summable , , and k.
Remark 1 Here, in the hypothesis of Theorem A, we have added the condition ‘’ because it is necessary.
Theorem B ()
Let be a quasi-σ-power increasing sequence for some σ (). If there exists an such that the sequence is nonincreasing and if the conditions from (5) to (8) are satisfied and if the condition
is satisfied, then the series is summable , , and .
Remark 2 It should be noted that condition (11) is the same as condition (10) when . When , condition (11) is weaker than condition (10) but the converse is not true. As in , we can show that if (10) is satisfied, then we get
If (11) is satisfied, then for we obtain that
Also, it should be noted that the condition ‘’ has been removed.
3 The main result
The aim of this paper is to extend Theorem B by using a general class of quasi power increasing sequence instead of a quasi-σ-power increasing sequences. For this purpose, we need the concept of quasi-f-power increasing sequence. A positive sequence is said to be a quasi-f-power increasing sequence, if there exists a constant such that , holds for , where (see ). It should be noted that if we take , then we get a quasi-σ-power increasing sequence. Now, we will prove the following theorem.
Theorem Let be a quasi-f-power increasing sequence. If there exists an such that the sequence is non-increasing and if the conditions from (5) to (8) and (11) are satisfied, then the series is summable , , and .
We need the following lemmas for the proof of our theorem.
Lemma 1 ()
If and , then
Lemma 2 ()
Under the conditions on , , and as expressed in the statement of the theorem, we have the following:
4 Proof of the theorem
Let be the n th , with , mean of the sequence . Then, by (2), we have
First, applying Abel’s transformation and then using Lemma 1, 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 get that
by virtue of the hypotheses of the theorem and Lemma 2. Finally, we have that
by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take and (resp. , and ), then we get a new result dealing with (resp. ) summability factors of infinite series. Also, if we take and , then we get another new result concerning the summability factors of infinite series. Furthermore, if we take as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see ). Finally, if we take , then we obtain Theorem B.
Bari NK, Stečkin SB: Best approximation and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483–522. (in Russian)
Leindler L: A new application of quasi power increasing sequences. Publ. Math. (Debr.) 2001, 58: 791–796.
Balcı M: Absolute φ -summability factors. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 1980, 29: 63–80.
Kogbetliantz E: Sur lés series absolument sommables par la méthode des moyennes arithmétiques. Bull. Sci. Math. 1925, 49: 234–256.
Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 1957, 7: 113–141.
Flett TM: Some more theorems concerning the absolute summability of Fourier series. Proc. Lond. Math. Soc. 1958, 8: 357–387.
Bor H, Özarslan HS: A study on quasi power increasing sequences. Rocky Mt. J. Math. 2008, 38: 801–807.
Pati T: The summability factors of infinite series. Duke Math. J. 1954, 21: 271–284.
Bor H: A new application of quasi power increasing sequences. I. J. Inequal. Appl. 2013, 2013: 69.
Sulaiman WT:A note on summability factors of infinite series. Appl. Math. Comput. 2010, 216(9):2645–2648.
Sulaiman WT: Extension on absolute summability factors of infinite series. J. Math. Anal. Appl. 2006, 322: 1224–1230.
Bosanquet LS: A mean value theorem. J. Lond. Math. Soc. 1941, 16: 146–148.
Bor H, Seyhan H: A note on almost increasing sequences. Ann. Soc. Math. Pol., 1 Comment. Math. 1999, 39: 37–42.
Dedicated to Professor Hari M. Srivastava.
The author express his thanks the referees for their useful comments and suggestions.
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Bor, H. A new application of quasi power increasing sequences. II. Fixed Point Theory Appl 2013, 75 (2013). https://doi.org/10.1186/1687-1812-2013-75
- absolute summability
- increasing sequences
- sequence spaces
- Hölder inequality
- Minkowski inequality
- infinite series