A new application of quasi power increasing sequences. II
© Bor; licensee Springer. 2013
Received: 24 December 2012
Accepted: 18 March 2013
Published: 28 March 2013
In this paper, we prove a general theorem dealing with absolute Cesàro summability factors of infinite series by using a quasi-f-power increasing sequence instead of a quasi-σ-power increasing sequence. This theorem also includes several new results.
MSC:26D15, 40D15, 40F05, 40G99, 46A45.
Keywordsabsolute summability increasing sequences sequence spaces Hölder inequality Minkowski inequality infinite series
2 The known results
Theorem A ()
then the series is summable , , and k.
Remark 1 Here, in the hypothesis of Theorem A, we have added the condition ‘’ because it is necessary.
Theorem B ()
is satisfied, then the series is summable , , and .
Also, it should be noted that the condition ‘’ has been removed.
3 The main result
The aim of this paper is to extend Theorem B by using a general class of quasi power increasing sequence instead of a quasi-σ-power increasing sequences. For this purpose, we need the concept of quasi-f-power increasing sequence. A positive sequence is said to be a quasi-f-power increasing sequence, if there exists a constant such that , holds for , where (see ). It should be noted that if we take , then we get a quasi-σ-power increasing sequence. Now, we will prove the following theorem.
Theorem Let be a quasi-f-power increasing sequence. If there exists an such that the sequence is non-increasing and if the conditions from (5) to (8) and (11) are satisfied, then the series is summable , , and .
We need the following lemmas for the proof of our theorem.
Lemma 1 ()
Lemma 2 ()
4 Proof of the theorem
by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take and (resp. , and ), then we get a new result dealing with (resp. ) summability factors of infinite series. Also, if we take and , then we get another new result concerning the summability factors of infinite series. Furthermore, if we take as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see ). Finally, if we take , then we obtain Theorem B.
Dedicated to Professor Hari M. Srivastava.
The author express his thanks the referees for their useful comments and suggestions.
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