Almost increasing sequences and their new applications
© Bor; licensee Springer 2013
Received: 9 January 2013
Accepted: 12 April 2013
Published: 25 April 2013
In this paper, we generalize a known theorem dealing with summability factors to the summability factors of infinite series using an almost increasing sequence. This theorem also includes some known and new results.
MSC:26D15, 40D15, 40F05, 40G05.
Keywordsincreasing sequences Cesàro mean summability factors Hölder inequality Minkowski inequality
If we take , then summability reduces to summability.
2 Known result
Many works dealing with an application of almost increasing sequences to the absolute Cesàro summability factors of infinite series have been done (see [3–11]). Among them, in , the following main theorem dealing with summability factors has been proved.
are satisfied, then the series is summable , .
3 The main result
The aim of this paper is to generalize Theorem A to the summability in the following form.
Theorem Let be a positive sequence and let be an almost increasing sequence.
then the series is summable , , and .
Remark It should be noted that if we take , then we get Theorem A. In this case, condition (10) reduces to condition (8) and the condition ‘’ is trivial.
We need the following lemmas for the proof of our theorem.
Lemma 1 
Lemma 2 
4 Proof of the Theorem
Let be the n th mean, with , of the sequence .
by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. Also, if we take , then we get a new result concerning the summability factors of infinite series.
Dedicated to Professor Hari M Srivastava.
The author expresses his thanks to the referees for their useful comments and suggestions.
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