Home | About | Journals | Submit | Contact Us | Français |

**|**Springer Open Choice**|**PMC5509829

Formats

Article sections

Authors

Related links

Journal of Inequalities and Applications

J Inequal Appl. 2017; 2017(1): 165.

Published online 2017 July 14. doi: 10.1186/s13660-017-1438-4

PMCID: PMC5509829

Independent Consultant, P.O. Box 121, TR-06502 Bahçelievler, Ankara, Turkey

Hüseyin Bor, Email: moc.liamg@33robh.

Received 2017 May 11; Accepted 2017 June 26.

Copyright © The Author(s) 2017

In the present paper, we obtained a main theorem related to factored infinite series. Some new results are also deduced.

Let ∑ *a*_{n} be a given infinite series with (*s*_{n}) as the sequence of partial sums. In [1], Borwein introduced the (*C*, *α*, *β*) methods in the following form: Let *α* + *β* ≠ −1, −2, …. Then the (*C*, *α*, *β*) mean is defined by

1

where

2

The series ∑ *a*_{n} is said to be summable |*C*, *α*, *β*, *σ*; *δ*|_{k}, *k* ≥ 1, *δ* ≥ 0, *α* + *β* > −1, and *σ* ∈ *R*, if (see [2])

3

where

is the (4

Here, we shall prove the following theorem.

*If*
(*λ*_{n})
*is a convex sequence* (*see* [5]) *such that the series*

5

*holds*, *then the series*
∑ *a*_{n}*λ*_{n}
*is summable*
|*C*, *α*, *β*, *σ*; *δ*|_{k}, *k* ≥ 1, 0 ≤ *δ* < *α* ≤ 1, *σ* ∈ *R*, *and*
(*α* + *β* + 1)*k* − *σ*(*δ**k* + *k* − 1) > 1.

*One should note that*, *if we set*
*σ* = 1, *then we obtain a well*-*known result of Bor* (*see* [6]).

We will use the following lemmas for the proof of the theorem given above.

[7]

*If*
(*λ*_{n})
*is a convex sequence such that the series*

Let

be theFirst applying Abel’s transformation and then using Lemma 1, we have

In order to complete the proof of the theorem by using Minkowski’s inequality, it is sufficient to show that

For *k* > 1, we can apply Hölder’s inequality with indices *k* and *k*^{′}, where

by virtue of hypotheses of the theorem and Lemma 2. Similarly, we have

in view of hypotheses of the theorem and Lemma 2. This completes the proof of the theorem.

By selecting proper values for *α*, *β*, *δ*, and *σ*, we have some new results concerning the |*C*, 1|_{k}, |*C*, *α*|_{k}, and |*C*, *α*; *δ*|_{k} summability methods.

**Competing interests**

The author declares that he has no competing interests.

**Author’s contributions**

The author carried out all work of this article and the main theorem. The author read and approved the final manuscript.

**Publisher’s Note**

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1. Borwein D. Theorems on some methods of summability. Quart. J. Math., Oxford, Ser. (2) 1958;9:310–316. doi: 10.1093/qmath/9.1.310. [Cross Ref]

2. Bor H. On the generalized absolute Cesàro summability. Pac. J. Appl. Math. 2010;2:217–222.

3. Tuncer AN. On generalized absolute Cesàro summability factors. Ann. Pol. Math. 2002;78:25–29. doi: 10.4064/ap78-1-3. [Cross Ref]

4. Bor H. On a new application of power increasing sequences. Proc. Est. Acad. Sci. 2008;57:205–209. doi: 10.3176/proc.2008.4.01. [Cross Ref]

5. Zygmund A. Trigonometric Series. Warsaw: Inst. Mat. Polskiej Akademi Nauk; 1935.

6. Bor H. A new application of convex sequences. J. Class. Anal. 2012;1:31–34. doi: 10.7153/jca-01-04. [Cross Ref]

7. Chow HC. On the summability factors of Fourier series. J. Lond. Math. Soc. 1941;16:215–220. doi: 10.1112/jlms/s1-16.4.215. [Cross Ref]

Articles from Springer Open Choice are provided here courtesy of **Springer**

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |