Journal of Inequalities and Applications

J Inequal Appl. 2017; 2017(1): 165.
Published online 2017 July 14.
PMCID: PMC5509829

# Some new results on convex sequences

## Abstract

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

Keywords: absolute summability, convex sequence, Minkowsky inequality, Hölder inequality

## Introduction

Let ∑ an be a given infinite series with (sn) as the sequence of partial sums. In [1], Borwein introduced the (Cαβ) methods in the following form: Let αβ ≠ −1, −2, …. Then the (Cαβ) mean is defined by

1

where

2

The series ∑ an is said to be summable |Cαβσδ|k, k ≥ 1, δ ≥ 0, αβ > −1, and σ ∈ R, if (see [2])

3

where

is the (Cαβ) transform of the sequence (nan). It should be noted that, for β = 0, the |Cαβσδ|k summability method reduces to the |Cασδ|k summability method (see [3]). Let us consider the sequence which is defined by (see [4])

4

## The main result

Here, we shall prove the following theorem.

### Theorem

If (λn) is a convex sequence (see [5]) such that the series

is convergent and let be a sequence defined as in (4). If the condition

5

holds, then the series ∑ anλn is summable |Cαβσδ|k, k ≥ 1, 0 ≤ δ < α ≤ 1, σ ∈ R, and (αβ + 1)k − σ(δkk − 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.

### Lemma 1

[4]

If 0 < α ≤ 1, β > −1, and 1 ≤ v ≤ n, then

6

### Lemma 2

[7]

If (λn) is a convex sequence such that the series

is convergent, then nΔλn → 0 as n → ∞ and is convergent.

## Proof of the theorem

Let

be the nth (Cαβ) mean of the sequence (nanλn). Then, by (1), we have

First 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

, and we obtain

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.

## Conclusions

By selecting proper values for α, β, δ, and σ, we have some new results concerning the |C, 1|k, |Cα|k, and |Cαδ|k summability methods.

## Footnotes

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

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## References

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.
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.
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.
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.
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.

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