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
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 .
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 .
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 , .
Quite recently, Savaş  obtained sufficient conditions for to be summable , as follows.
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.
2. The Main Results
We now shall prove the following theorems.
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 ).
4. Proof of Theorem 2.1
Applying Abel's transformation, we may write
to complete the proof, it is sufficient to show that
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,
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
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.
5. Corollaries and Applications to Weighted Means
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
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.
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