Almost increasing sequences and their new applications
Journal of Inequalities and Applications volume 2013, Article number: 207 (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.
A positive sequence is said to be an almost increasing sequence if there exists a positive increasing sequence and two positive constants A and B such that (see ). Let be a given infinite series with the sequence of partial sums . By we denote the n th Cesàro mean of order α, with , of the sequence , that is,
The series is said to be summable , , if (see )
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.
Theorem A Let be a positive sequence and be an almost increasing sequence. If the conditions
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.
If the conditions (4), (5), (6) and (7) are satisfied, and the sequence defined by (see )
satisfies the condition
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 
If and , then
Lemma 2 
Under the conditions (4) and (5), we have
4 Proof of the Theorem
Let be the n th mean, with , of the sequence .
Then, by (1), we find that
Thus, applying Abel’s transformation first and then using Lemma 1, we have that
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
Now, when , applying Hölder’s inequality with indices k and , where , we get that
by virtue of the hypotheses of the theorem and Lemma 2. Again, we get that
by hypotheses of the theorem and Lemma 2. Finally, as in , we have that
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.
Bari NK, Stečkin SB: Best approximation and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483–522. (in Russian)
Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 1957, 7: 113–141.
Bor H: An application of almost increasing sequences. Math. Inequal. Appl. 2002, 5(1):79–83.
Bor H, Srivastava HM: Almost increasing sequences and their applications. Int. J. Pure Appl. Math. 2002, 3: 29–35.
Bor H: A study on almost increasing sequences. JIPAM. J. Inequal. Pure Appl. Math. 2003., 4(5): Article ID 97
Bor H, Leindler L: A note on δ -quasi-monotone and almost increasing sequences. Math. Inequal. Appl. 2005, 8(1):129–134.
Bor H, Özarslan HS: On the quasi-monotone and almost increasing sequences. J. Math. Inequal. 2007, 1(4):529–534.
Bor H: An application of almost increasing sequences. Appl. Math. Lett. 2011, 24(3):298–301. 10.1016/j.aml.2010.10.009
Bor H: On a new application of almost increasing sequences. Math. Comput. Model. 2011, 53(1–2):230–233. 10.1016/j.mcm.2010.08.011
Sulaiman WT: On a new application of almost increasing sequences. Bull. Math. Anal. Appl. 2012, 4(3):29–33.
Bor H, Srivastava HM, Sulaiman WT: A new application of certain generalized power increasing sequences. Filomat 2012, 26(4):871–879. 10.2298/FIL1204871B
Pati T: The summability factors of infinite series. Duke Math. J. 1954, 21: 271–284. 10.1215/S0012-7094-54-02127-4
Bosanquet LS: A mean value theorem. J. Lond. Math. Soc. 1941, 16: 146–148.
Mazhar SM: Absolute summability factors of infinite series. Kyungpook Math. J. 1999, 39: 67–73.
Dedicated to Professor Hari M Srivastava.
The author expresses his thanks to the referees for their useful comments and suggestions.
The author declares that he has no competing interests.
About this article
Cite this article
Bor, H. Almost increasing sequences and their new applications. J Inequal Appl 2013, 207 (2013). https://doi.org/10.1186/1029-242X-2013-207