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Erschienen in: Journal of Inequalities and Applications 1/2013

Open Access 01.12.2013 | Research

Almost increasing sequences and their new applications

verfasst von: Hüseyin Bor

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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Abstract

In this paper, we generalize a known theorem dealing with | C , 1 | k summability factors to the | C , α | k summability factors of infinite series using an almost increasing sequence. This theorem also includes some known and new results.
MSC:26D15, 40D15, 40F05, 40G05.
Hinweise

Competing interests

The author declares that he has no competing interests.

1 Introduction

A positive sequence ( b n ) is said to be an almost increasing sequence if there exists a positive increasing sequence ( c n ) and two positive constants A and B such that A c n b n B c n (see [1]). Let a n be a given infinite series with the sequence of partial sums ( s n ) . By t n α we denote the n th Cesàro mean of order α, with α > 1 , of the sequence ( n a n ) , that is,
t n α = 1 A n α v = 0 n A n v α 1 v a v ,
(1)
where
A n α = ( n + α n ) = ( α + 1 ) ( α + 2 ) ( α + n ) n ! = O ( n α ) , A n α = 0 for  n > 0 .
(2)
The series a n is said to be summable | C , α | k , k 1 , if (see [2])
n = 1 1 n | t n α | k < .
(3)
If we take α = 1 , then | C , α | k summability reduces to | C , 1 | k 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 [311]). Among them, in [10], the following main theorem dealing with | C , 1 | k summability factors has been proved.
Theorem A Let ( φ n ) be a positive sequence and ( X n ) be an almost increasing sequence. If the conditions
n = 1 n | Δ 2 λ n | X n < ,
(4)
| λ n | X n = O ( 1 ) as n ,
(5)
φ n = O ( 1 ) as n ,
(6)
n Δ φ n = O ( 1 ) as n ,
(7)
v = 1 n | t v | k v X v k 1 = O ( X n ) as n
(8)
are satisfied, then the series a n λ n φ n is summable | C , 1 | k , k 1 .

3 The main result

The aim of this paper is to generalize Theorem A to the | C , α | k summability in the following form.
Theorem Let ( φ n ) be a positive sequence and let ( X n ) be an almost increasing sequence.
If the conditions (4), (5), (6) and (7) are satisfied, and the sequence ( w n α ) defined by (see [12])
w n α = { | t n α | , α = 1 , max 1 v n | t v α | , 0 < α < 1 ,
(9)
satisfies the condition
v = 1 n ( w v α ) k v X v k 1 = O ( X n ) as n ,
(10)
then the series a n λ n φ n is summable | C , α | k , 0 < α 1 , ( α 1 ) k > 1 and k 1 .
Remark It should be noted that if we take α = 1 , then we get Theorem A. In this case, condition (10) reduces to condition (8) and the condition ‘ ( α 1 ) k > 1 ’ is trivial.
We need the following lemmas for the proof of our theorem.
Lemma 1 [13]
If 0 < α 1 and 1 v n , then
| p = 0 v A n p α 1 a p | max 1 m v | p = 0 m A m p α 1 a p | .
(11)
Lemma 2 [14]
Under the conditions (4) and (5), we have
n X n | Δ λ n | = O ( 1 ) as n ,
(12)
n = 1 X n | Δ λ n | < .
(13)

4 Proof of the Theorem

Let ( T n α ) be the n th ( C , α ) mean, with 0 < α 1 , of the sequence ( n a n λ n φ n ) .
Then, by (1), we find that
T n α = 1 A n α v = 1 n A n v α 1 v a v λ v φ n .
(14)
Thus, applying Abel’s transformation first and then using Lemma 1, we have that
T n α = 1 A n α v = 1 n 1 Δ ( λ v φ n ) p = 1 v A n p α 1 p a p + λ n φ n A n α v = 1 n A n v α 1 v a v = 1 A n α v = 1 n 1 ( λ v Δ φ v + φ v + 1 Δ λ v ) p = 1 v A n p α 1 p a p + λ n φ n A n α v = 1 n A n v α 1 v a v , | T n α | 1 A n α v = 1 n 1 | λ v Δ φ v | | p = 1 v A n p α 1 p a p | + 1 A n α v = 1 n 1 | φ v + 1 Δ λ v | | p = 1 v A n p α 1 p a p | + | λ n φ n | A n α | v = 1 v A n v α 1 v a v | 1 A n α v = 1 n 1 A v α w v α | λ v | | Δ φ v | + 1 A n α v = 1 n 1 A v α w v α | φ v + 1 | | Δ λ v | + | λ n | | φ n | w n α = T n , 1 α + T n , 2 α + T n , 3 α .
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
n = 1 n 1 | T n , r α | k < for  r = 1 , 2 , 3 .
Now, when k > 1 , applying Hölder’s inequality with indices k and k , where 1 k + 1 k = 1 , we get that
n = 2 m + 1 n 1 | T n , 1 α | k n = 2 m + 1 n 1 ( A n α ) k { v = 1 n 1 A v α w v α | Δ φ v | | λ v | } k = O ( 1 ) n = 2 m + 1 1 n 1 + α k v = 1 n 1 ( v α ) k ( w v α ) k | Δ φ v | k | λ v | k { v = 1 n 1 1 } k 1 = O ( 1 ) n = 2 m + 1 1 n 2 + ( α 1 ) k v = 1 n 1 v α k ( w v α ) k | λ v | k 1 v k = O ( 1 ) v = 1 m v α k ( w v α ) k v k | λ v | k n = v + 1 m + 1 1 n 2 + ( α 1 ) k = O ( 1 ) v = 1 m v α k ( w v α ) k v k | λ v | k v d x x 2 + ( α 1 ) k = O ( 1 ) v = 1 m ( w v α ) k | λ v | | λ v | k 1 1 v = O ( 1 ) v = 1 m ( w v α ) k | λ v | 1 v X v k 1 = O ( 1 ) v = 1 m 1 Δ | λ v | r = 1 v ( w r α ) k r X r k 1 + O ( 1 ) | λ m | v = 1 m ( w v α ) k v X v k 1 = O ( 1 ) v = 1 m | Δ λ v | X v + O ( 1 ) | λ m | X m = O ( 1 ) as  m
by virtue of the hypotheses of the theorem and Lemma 2. Again, we get that
n = 2 m + 1 n 1 | T n , 2 α | k n = 2 m + 1 n 1 ( A n α ) k { v = 1 n 1 A v α w v α | φ v + 1 | | Δ λ v | } k = O ( 1 ) n = 2 m + 1 1 n 1 + α k { v = 1 n v α ( w v α ) | Δ λ v | } k = O ( 1 ) n = 2 m + 1 1 n 1 + α k v = 1 n 1 v α k ( w v α ) k | Δ λ v | k { v = 1 n 1 1 } k 1 = O ( 1 ) n = 2 m + 1 1 n 2 + ( α 1 ) k v = 1 n 1 v α k ( w v α ) k | Δ λ v | k = O ( 1 ) v = 1 m v α k ( w v α ) k | Δ λ v | | Δ λ v | k 1 n = v + 1 m + 1 1 n 2 + ( α 1 ) k = O ( 1 ) v = 1 m v α k ( w v α ) k | Δ λ v | v k 1 X v k 1 v d x x 2 + ( α 1 ) k = O ( 1 ) v = 1 m v | Δ λ v | ( w v α ) k v X v k 1 = O ( 1 ) v = 1 m Δ ( v | Δ λ v | ) r = 1 v ( w r α ) k r X r k 1 + O ( 1 ) m | Δ λ m | v = 1 m ( w v α ) k v X v k 1 = O ( 1 ) v = 1 m 1 v | Δ 2 λ v | X v + O ( 1 ) v = 1 m 1 X v | Δ λ v | + O ( 1 ) m | Δ λ m | X m = O ( 1 ) as  m
by hypotheses of the theorem and Lemma 2. Finally, as in T n , 1 α , we have that
n = 1 m n 1 | T n , 3 α | k = n = 1 m n 1 | λ n φ n w n α | k = O ( 1 ) n = 1 m ( w n α ) k | λ n | n X n k 1 = O ( 1 ) as  m
by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. Also, if we take k = 1 , then we get a new result concerning the | C , α | summability factors of infinite series.

Acknowledgements

Dedicated to Professor Hari M Srivastava.
The author expresses his thanks to the referees for their useful comments and suggestions.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The author declares that he has no competing interests.
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Metadaten
Titel
Almost increasing sequences and their new applications
verfasst von
Hüseyin Bor
Publikationsdatum
01.12.2013
Verlag
Springer International Publishing
Erschienen in
Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI
https://doi.org/10.1186/1029-242X-2013-207

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