- Research Article
- Open Access
A Recent Note on Quasi-Power Increasing Sequence for Generalized Absolute Summability
© E. Savaş and H. Şevli. 2009
- Received: 15 May 2009
- Accepted: 30 July 2009
- Published: 19 August 2009
We prove two theorems on , , summability factors for an infinite series by using quasi-power increasing sequences. We obtain sufficient conditions for to be summable , , by using quasi-f-increasing sequences.
- Real Number
- Natural Number
- Triangular Matrix
- Infinite Series
- Positive Sequence
Quite recently, Savaş  obtained sufficient conditions for to be summable , The purpose of this paper is to obtain the corresponding result for quasi- -increasing sequence. Our result includes and moderates the conditions of his theorem with the special case .
A sequence is said to be of bounded variation if Let where denotes the set of all null sequences.
The concept of absolute summability of order was defined by Flett  as follows. Let denote a series with partial sums and a lower triangular matrix. Then is said to be absolutely -summable of order written that is summable if
In , Flett considered further extension of absolute summability in which he introduced a further parameter The series is said to be summable , , if
A positive sequence is said to be an almost increasing sequence if there exist an increasing sequence and 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 the example, say .
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 need not be true as can be seen by taking an example, say for (see ). If (1.4) stays with then is simply called a quasi-increasing sequence. It is clear that if is quasi- -power increasing, then is quasi-increasing.
A positive sequence is said to be a quasi- -power increasing sequence, if there exists a constant such that holds for all , .
We may associate two lower triangular matrices and as follows:
Given any sequence the notation means and For any matrix entry
Quite recently, Savaş  obtained sufficient conditions for to be summable , as follows.
then the series is summable ,
Theorem 1.1 enhanced a theorem of Savas  by replacing an almost increasing sequence with a quasi- -power increasing sequence for some . It should be noted that if is an almost increasing sequence, then (1.15) implies that the sequence is bounded. However, when is a quasi- -power increasing sequence or a quasi- -increasing sequence, (1.15) does not imply For example, since is a quasi- -power increasing sequence for and if we take then holds but (see ). Therefore, we remark that condition should be added to the statement of Theorem 1.1.
The goal of this paper is to prove the following theorem by using quasi- -increasing sequences. Our main result includes the moderated version of Theorem 1.1. We will show that the crucial condition of our proof, can be deduced from another condition of the theorem. Also, we shall eliminate condition (1.15) in our theorem; however we shall deduce this condition from the conditions of our theorem.
We now shall prove the following theorems.
are satisfied, then the series is summable where and
Theorem 2.1 includes the following theorem with the special case . Theorem 2.2 moderates the hypotheses of Theorem 1.1.
are satisfied, where then the series is summable ,
The crucial condition, and condition (1.15) do not appear among the conditions of Theorems 2.1 and 2.2. By Lemma 3.3, under the conditions on and as taken in the statement of Theorem 2.1, also in the statement of Theorem 2.2 with the special case conditions and (1.15) hold.
We shall need the following lemmas for the proof of our main Theorem 2.1.
Lemma 3.1 (see ).
Lemma 3.2 (see ).
If is a quasi- -increasing sequence, where then, under conditions (1.13), (1.14), (2.1), and (2.2), conditions (1.15) and (3.5) are satisfied.
It is clear that (1.13) and (1.14) (3.3). Also, (1.13) and (2.2) (3.4). By Lemma 3.2, under conditions (1.13)-(1.14) and (2.1)–(2.2), we have (1.15) and (3.5).
Let denote the th term of the -transform of the series Then, by definition, we have
Then, for , we have
Applying Abel's transformation, we may write
to complete the proof, it is sufficient to show that
Since is bounded by Lemma 3.3, using (1.9), we have
Using properties (1.15), in view of Lemma 3.3, and (3.7), from (1.9), (1.13), and (1.17),
Applying Hölder's inequality,
Using (1.9) and (1.11) and boundedness of
as in the proof of
Finally, again using Hölder's inequality, from (1.9), (1.10), and (1.12),
By Lemma 3.1, condition (3.3), in view of Lemma 3.3, implies that
holds. Thus, by Lemma 3.3, (3.4) implies that is bounded. Therefore, from (1.9) and (1.13),
Using Abel transformation and (1.17),
by virtue of (2.2) and properties (3.6) and (3.7) of Lemma 3.4.
So we obtain (4.7). This completes the proof.
Setting in Theorems 2.1 and 2.2 yields the following two corollaries, respectively.
are satisfied, then the series is summable
If we take in Theorem 2.1, then condition (1.17) reduces condition (5.1). In this case conditions (1.11) and (1.12) are obtained by conditions (1.7)–(1.10).
Let satisfy conditions (1.7)–(1.10), and let and be sequences satisfying conditions (1.13), (1.14), and (2.1). If is a quasi- -power increasing sequence for some and conditions (2.3) and (5.1) are satisfied, then the series is summable
A weighted mean matrix, denoted by is a lower triangular matrix with entries where is nonnegative sequence with and as
and let and be sequences satisfying conditions (1.13), (1.14), and (2.1). If is a quasi- -increasing sequence, where and conditions (1.17) and (2.2) are satisfied, then the series, is summable for and
In Theorem 2.1 set . It is clear that conditions (1.7), (1.8), and (1.10) are automatically satisfied. Condition (1.9) becomes condition (5.2), and conditions (1.11) and (1.12) become condition (5.3) for weighted mean method.
Corollary 5.3 includes the following result with the special case
Let be a positive sequence satisfying (5.2) and (5.3), and let be a quasi- -power increasing sequence for some Then under conditions (1.13), (1.14), (1.17), (2.1), and (2.3), is summable ,
- Savaş E: Quasi-power increasing sequence for generalized absolute summability. Nonlinear Analysis: Theory, Methods & Applications 2008,68(1):170–176. 10.1016/j.na.2006.10.039MathSciNetView ArticleMATHGoogle Scholar
- Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proceedings of the London Mathematical Society 1957, 7: 113–141. 10.1112/plms/s3-7.1.113MathSciNetView ArticleMATHGoogle Scholar
- Flett TM: Some more theorems concerning the absolute summability of Fourier series and power series. Proceedings of the London Mathematical Society 1958, 8: 357–387. 10.1112/plms/s3-8.3.357MathSciNetView ArticleMATHGoogle Scholar
- Alijancic S, Arendelovic D: -regularly varying functions. Publications de l'Institut Mathématique 1977,22(36):5–22.MathSciNetGoogle Scholar
- Leindler L: A new application of quasi power increasing sequences. Publicationes Mathematicae Debrecen 2001,58(4):791–796.MathSciNetMATHGoogle Scholar
- Sulaiman WT: Extension on absolute summability factors of infinite series. Journal of Mathematical Analysis and Applications 2006,322(2):1224–1230. 10.1016/j.jmaa.2005.09.019MathSciNetView ArticleMATHGoogle Scholar
- Savaş E: On almost increasing sequences for generalized absolute summability. Mathematical Inequalities & Applications 2006,9(4):717–723.MathSciNetMATHGoogle Scholar
- Şevli H, Leindler L: On the absolute summability factors of infinite series involving quasi-power-increasing sequences. Computers & Mathematics with Applications 2009,57(5):702–709. 10.1016/j.camwa.2008.11.007MathSciNetView ArticleMATHGoogle Scholar
- Leindler L: A note on the absolute Riesz summability factors. Journal of Inequalities in Pure and Applied Mathematics 2005,6(4, article 96):-5.MathSciNetMATHGoogle Scholar
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