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A new application of quasi power increasing sequences. I
Journal of Inequalities and Applications volume 2013, Article number: 69 (2013)
In (Rocky Mt. J. Math. 38:801-807, 2008), we proved a theorem dealing with an application of quasi-σ-power increasing sequences. In the present paper, we prove that theorem under less and more weaker conditions. This theorem also includes some new and known results.
MSC: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 . 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 (see ). 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 )
2 Known result
In , we have proved the following theorem.
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 non-increasing and if the sequence defined by (see )
satisfies the condition
then the series is summable , , and .
It should be remarked that we have added the condition ‘’ in the statement of Theorem A because it is necessary.
3 The main result
The aim of this paper is to prove Theorem A under less and weaker conditions. Now, we will prove the following theorem.
Theorem Let be a quasi-σ-power increasing sequence for some σ (). If there exists an such that the sequence is non-increasing and if the conditions from (5) to (8) are satisfied and if the condition
is satisfied, then the series is summable , , , and .
Remark 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 that
If (11) is satisfied, then for we obtain that
Also, it should be noted that the condition ‘’ has been removed.
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. Also, if we set and , then we get another new result concerning the summability factors. Finally, if we take as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see ).
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Dedicated to Professor Hari M Srivastava.
The author declares that he has no competing interests.
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Bor, H. A new application of quasi power increasing sequences. I. J Inequal Appl 2013, 69 (2013). https://doi.org/10.1186/1029-242X-2013-69
- Hölder’s inequality
- sequence spaces
- absolute summability
- increasing sequences