- Open Access
A new application of quasi power increasing sequences. I
© Bor; licensee Springer 2013
- Received: 22 December 2012
- Accepted: 9 February 2013
- Published: 26 February 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.
- Hölder’s inequality
- sequence spaces
- absolute summability
- increasing sequences
In , we have proved the following theorem.
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.
The aim of this paper is to prove Theorem A under less and weaker conditions. Now, we will prove the following theorem.
is satisfied, then the series is summable , , , and .
Also, it should be noted that the condition ‘’ has been removed.
We need the following lemmas for the proof of our theorem.
Lemma 1 
Lemma 2 
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 ).
Dedicated to Professor Hari M Srivastava.
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