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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

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

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Object name is 13660_2017_1438_Article_Equ1.gif
1

where

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Object name is 13660_2017_1438_Article_Equ2.gif
2

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

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Object name is 13660_2017_1438_Article_Equ3.gif
3

where

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Object name is 13660_2017_1438_Article_IEq12.gif
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
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which is defined by (see [4])

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Object name is 13660_2017_1438_Article_Equ4.gif
4

The main result

Here, we shall prove the following theorem.

Theorem

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

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Object name is 13660_2017_1438_Article_IEq20.gif
is convergent and let
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be a sequence defined as in (4). If the condition

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Object name is 13660_2017_1438_Article_Equ5.gif
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

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Object name is 13660_2017_1438_Article_Equ6.gif
6

Lemma 2

[7]

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

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Object name is 13660_2017_1438_Article_IEq33.gif
is convergent, then nΔλn → 0 as n → ∞ and
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Object name is 13660_2017_1438_Article_IEq35.gif
is convergent.

Proof of the theorem

Let

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Object name is 13660_2017_1438_Article_IEq36.gif
be the nth (Cαβ) mean of the sequence (nanλn). Then, by (1), we have

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Object name is 13660_2017_1438_Article_Equa.gif

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

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Object name is 13660_2017_1438_Article_Equb.gif

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

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Object name is 13660_2017_1438_Article_Equc.gif

For k > 1, we can apply Hölder’s inequality with indices k and k, where

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Object name is 13660_2017_1438_Article_IEq41.gif
, and we obtain

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Object name is 13660_2017_1438_Article_Equd.gif

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

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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

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

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. [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]

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