© Ekrem Savaş. 2010
Received: 9 September 2009
Accepted: 14 December 2009
Published: 12 January 2010
Mishra and Srivastava  obtained sufficient conditions on a sequence and a sequence for the series to be absolutely summable by the weighted mean matrix .
Recently Savaş and Rhoades  established the corresponding result for a nonnegative triangle, using the correct definition of absolute summability of order .
respectively. The motivation for these definitions will become clear as we proceed.
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 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.
Quite recently, Savaş and Rhoades  proved the following theorem for -summability factors of infinite series.
It should be noted that if is an almost increasing sequence then (viii) implies that the sequence is bounded. However, when is a quasi -power increasing sequence or a quasi -increasing sequence, (viii) does not imply , For example, since is a quasi -power increasing sequence for if we take , then , holds but (see ).
2. The Main Results
We have the following theorem:
If we take that is an almost increasing sequence instead of a quasi -power increasing sequence then our Theorem 2.2 reduces to [8, Theorem ].
The crucial condition, and condition (viii) 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 the Theorem 2.1, also in the statement of Theorem 2.2 with the special case conditions and (viii) hold.
We shall need the following lemmas for the proof of our main Theorem 2.1.
Lemma 3.1 (see ).
Lemma 3.2 (see ).
Lemma 3.3 (see ).
Lemma 3.4 (see ).
4. Proof of Theorem 2.1
It is easy to see that
Also we may write
Thus, using (iv) and (ii),
Using (iii), Hölder's inequality, and (v),
Using Abel's transformation, (vi), (2.2), and properties (3.7) and (3.6) of Lemma 3.4,
The author wishes to thank the referees for their careful reading of the manuscript and for their helpful suggestions.
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