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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 436094, 6 pages
Power Increasing Sequences and Their Some New Applications
P.O. Box 121, Bahçelievler, 06502 Ankara, Turkey
Received 10 August 2012; Revised 29 August 2012; Accepted 12 September 2012
Academic Editor: Toka Diagana
Copyright © 2012 Hüseyin Bor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the work of Bor (2008), we have proved a result dealing with summability factors by using a quasi--power increasing sequence. In this paper, we prove that result under less and more weaker conditions. Some new results have also been obtained.
A positive sequence is said to be almost increasing if there exists a positive increasing sequence and two positive constants and such that (see ). We write , where , = and being the space of all real or complex-valued sequences. A positive sequence is said to be a quasi--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--power increasing sequence for any nonnegative , but the converse is not true for . Moreover, for any positive there exists a quasi--power increasing sequence tending to infinity, but it is not almost increasing (see ). Let be a given infinite series with partial sums . Let be a sequence of positive numbers such that Let be any sequence of positive real constants. The series is said to be summable , if (see ) and it is said to be summable , if (see ) where If we take then summability reduces to summability. Also if we take and for all values of , then we get summability (see ). Furthermore, if we take , then summability reduces to summability (see ).
2. Known Result
In , we have proved the following theorem dealing with summability factors of infinite series.
Theorem 2.1. Let , be a quasi--power increasing sequence for some , and let be a nonincreasing sequence. Suppose also there exists sequences and such that If are satisfied, then the series is summable ,, where is the th mean of the sequence .
Remark 2.2. It should be noticed that, if we take as an almost increasing sequence and , then we obtain a theorem of Mazhar (see ), in this case the condition “” is not needed.
3. The Main Result
The aim of this paper is to prove Theorem 2.1 under less and more weaker conditions. Now, we prove the following theorem.
Theorem 3.1. Let be a quasi--power increasing sequence for some , and let be a nonincreasing sequence. Suppose also there exists sequences and such that conditions (2.1) of Theorem 2.1 are satisfied. If are satisfied, then the series is summable , .
Remark 3.2. It should be noted that conditions (3.1) and (3.2) are the same as conditions (2.2) and (2.3), respectively, when . When , conditions (3.1) and (3.2) are weaker than conditions (2.2) and (2.3), respectively. But the converses are not true. In fact, if (2.2) is satisfied, then we get that
If (3.1) is satisfied, then for , we obtain that
The similar argument is also valid for the conditions (2.3) and (3.2). Also it should be noted that condition “” has been removed.
We need following lemma for the proof of our theorem.
Lemma 3.3 (see ). Under the conditions on the sequences and as expressed in the statement of the theorem, one has the following:
4. Proof of the Theorem
Let denote the mean of the series . Then, for , we have By Abel's transformation, we have To complete the proof of the theorem, by Minkowski's inequality, it is enough to show that Firstly, we have that by virtue of the hypotheses of the theorem and lemma. Now, when applying Hölder's inequality with indices and , where , as in , we have that Again we have that by virtue of the hypotheses of the theorem and lemma. Finally, we have that by virtue of the hypotheses of the theorem and lemma. This completes the proof of the theorem. If we take for all values of and , then we get a result dealing with summability factors. Also, if we take for all values of , then we have a new result for summability. Finally, if we take , then we have another new result for summability factors.
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