A Summability Factor Theorem for Quasi-Power-Increasing Sequences
© E. Savaş. 2010
Received: 23 June 2010
Accepted: 15 September 2010
Published: 20 September 2010
Recently, Rhoades and Savaş  obtained sufficient conditions for to be summable , by using almost increasing sequence. The purpose of this paper is to obtain the corresponding result for quasi -increasing sequence.
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, ). A sequence satisfying (1.4) for is 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 , where , (see, ).
Rhoades and Savaş  proved the following theorem for summability factors of infinite series.
It should be noted that, if is an almost increasing sequence, then condition (iv) implies that the sequence is bounded. However, if is a quasi -power increasing sequence or a quasi -increasing sequence, (iv) does not imply that is bounded. For example, the sequence defined by is trivially a quasi -power increasing sequence for each If for any then but is not bounded, (see, [6, 7]).
2. The Main Results
We now will prove the following theorems.
The conditions and (iv) 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 (iv) hold.
We will 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
5. Corollaries and Applications to Weighted Means
Let satisfy conditions (v)–(viii) and let and be sequences satisfying conditions (i), (ii), 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 ,
In Theorem 2.1, set . Conditions (i) and (ii) of Corollary 5.3 are, respectively, conditions (i) and (ii) of Theorem 2.1. Condition (v) becomes condition (5.2) and conditions (ix) and (x) become condition (5.3) for weighted mean method. Conditions (vi), (vii), and (viii) of Theorem 2.1 are automatically satisfied for any weighted mean method.
- Rhoades BE, Savaş E: A summability factor theorem for generalized absolute summability. Real Analysis Exchange 2006, 31(2):355–363.MATHGoogle 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/s3-7.1.113MATHMathSciNetView ArticleGoogle Scholar
- Alijancic S, Arendelovic D: -regularly varying functions. Publications de l'Institut Mathématique 1977, 22(36):5–22.Google Scholar
- Leindler L: A new application of quasi power increasing sequences. Publicationes Mathematicae Debrecen 2001, 58(4):791–796.MATHMathSciNetGoogle 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.2005.09.019MATHMathSciNetView ArticleGoogle Scholar
- Savaş E: A note on generalized -summability factors for infinite series. Journal of Inequalities and Applications 2010, 2010:-10.Google Scholar
- Savaş E, Şevli H: A recent note on quasi-power increasing sequence for generalized absolute summability. Journal of Inequalities and Applications 2009, 2009:-10.Google Scholar
- Leindler L: A note on the absolute Riesz summability factors. Journal of Inequalities in Pure and Applied Mathematics 2005., 6(4, article 96):Google 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.2008.11.007MATHMathSciNetView ArticleGoogle Scholar
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