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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 793548, 6 pages
Quasimonotone and Almost Increasing Sequences and Their New Applications
P.O. Box 121, 06502 Bah####^~^~^~^~^~^####xe7;elievler, Ankara, Turkey
Received 11 August 2012; Accepted 4 September 2012
Academic Editor: Sung Guen Kim
Copyright © 2012 H####^~^~^~^~^~^####xfc;seyin 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.
Recently, we have proved a main theorem dealing with the absolute N####^~^~^~^~^~^####xf6;rlund summability factors of infinite series by using -quasimonotone sequences. In this paper, we prove that result under weaker conditions. A new result has also been obtained.
A positive sequence is said to be almost increasing if there exist a positive increasing sequence and two positive constants and such that (see ). Obviously, every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking an example, say . A sequence is said to be -quasimonotone if ultimately and , where is a sequence of positive numbers (see ). Let be a given infinite series with the sequence of partial sums and . By and , we denote the Ces####^~^~^~^~^~^####xe0;ro means of order , with , of the sequences and , respectively, that is, where The series is said to be summable , , if (see ) If we take , then summability reduces to summability.
Let be a sequence of constants, real or complex, and let us write The sequence-to-sequence transformation defines the sequence of the N####^~^~^~^~^~^####xf6;rlund mean of the sequence , generated by the sequence of coefficients . The series is said to be summable , , if (see ) In the special case when the N####^~^~^~^~^~^####xf6;rlund mean reduces to the mean and summability becomes summability. For , we get the mean and then summability becomes summability. Also, if we take , then we get summability. For any sequence , we write . Quite recently, in , we have proved the following theorem dealing with the absolute N####^~^~^~^~^~^####xf6;rlund summability factors of infinite series.
Theorem A. Let , , and be a nonincreasing sequence. Let be an almost increasing sequence such that and is a sequence such that Suppose also that there exists a sequence of numbers such that it is -quasimonotone with , is convergent, and for all . If the sequence defined by (see ) satisfies the condition then the series is summable , .
2. The Main Results
The aim of this paper is to prove Theorem A under weaker conditions. We will prove the following theorems.
Theorem 2.1. If the sequences , , and are as in Theorem A and if conditions (1.9) and are satisfied, then the series is summable , and .
Remark 2.3. The following sequences satisfy the conditions of the theorems:
Remark 2.4. It should be noted that condition (2.1) is the same as condition (1.11) when . When , condition (2.1) is weaker than condition (1.11), but the converse is not true. In fact, if (1.11) is satisfied, then we get that
To show that the converse is false when , the following example is sufficient. We can take , , and then construct a sequence such that
This is because for .
This shows that, when , (1.11) implies (2.1) but not conversely.
We need the following lemmas for the proof of our theorem.
Lemma 2.5 (see ). If and , then
Lemma 2.6 (see ). If , and the series is summable , then it is also summable .
Lemma 2.7 (see ). Let be an almost increasing sequence such that .
If is a -quasimonotone with , is convergent, then
Lemma 2.8 (see ). Let , , and be a nonincreasing sequence. If the series is summable , then the series is summable , .
3. Proof of Theorem 2.1
Let be the th , with , mean of the sequence . Then, by (1.2), we have First applying Abel's transformation and then using Lemma 2.5, we have that To complete the proof of Theorem 2.1, by Minkowski's inequality, it is sufficient to show that Whenever , we can apply H####^~^~^~^~^~^####xf6;lder's inequality with indices and , where , we get that by virtue of the hypotheses of Theorem 2.1 and Lemma 2.7. Again, we have that by virtue of the hypotheses of Theorem 2.1. This completes the proof of Theorem 2.1. If we take , then we get a new result dealing with summability factors.
Proof of Theorem 2.2. In order to prove Theorem 2.2, we need to consider only the special case in which is . Therefore, Theorem 2.2 will then follow by means of Theorem 2.1, Lemma 2.6 (for ), and Lemma 2.8. If we take , then we get a new result for the absolute N####^~^~^~^~^~^####xf6;rlund summability factors of infinite series.
The author wishes his sincerest thanks to the referee for invaluable suggestions for the improvement of this paper.
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