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On generalized absolute Cesàro summability factors

Journal of Inequalities and Applications20122012:253

https://doi.org/10.1186/1029-242X-2012-253

Received: 23 December 2011

Accepted: 17 October 2012

Published: 30 October 2012

Abstract

In this paper, a known theorem dealing with | C , α , γ ; δ | k summability factors has been generalized for | C , α , β , γ ; δ | k summability factors. Some results have also been obtained.

MSC:40D15, 40F05, 40G99.

Keywords

Hölder’s inequalityquasi-monotone sequencesummability factors

1 Introduction

A sequence ( b n ) of positive numbers is said to be quasi-monotone if n Δ b n ρ b n for some ρ > 0 and is said to be δ-quasi-monotone, if b n 0 , b n > 0 ultimately and Δ b n δ n , where ( δ n ) is a sequence of positive numbers (see [1]). Let a n be a given infinite series with partial sums ( s n ) . We denote by u n α , β and t n α , β the n th Cesàro means of order ( α , β ) , with α + β > 1 , of the sequences ( s n ) and ( n a n ) , respectively, that is (see [2]),
(1)
(2)
where
A n α + β = O ( n α + β ) , α + β > 1 , A 0 α + β = 1 and A n α + β = 0 for  n > 0 .
(3)
The series a n is said to be summable | C , α , β | k , k 1 and α + β > 1 , if (see [3])
n = 1 n k 1 | u n α , β u n 1 α , β | k < .
(4)
Since t n α , β = n ( u n α , β u n 1 α , β ) (see [3]), condition (4) can also be written as
n = 1 1 n | t n α , β | k < .
(5)
The series a n is said to be summable | C , α , β , γ ; δ | k , k 1 , α + β > 1 , δ 0 and γ is a real number, if (see [4])
n = 1 n γ ( δ k + k 1 ) | u n α , β u n 1 α , β | k = n = 1 n γ ( δ k + k 1 ) k | t n α , β | k < .
(6)

If we take β = 0 , then | C , α , β , γ ; δ | k summability reduces to | C , α , γ ; δ | k summability (see [5]).

2 Known result

In [6], we have proved the following theorem dealing with | C , α , γ ; δ | k summability factors of infinite series.

Theorem A Let k 1 , 0 δ < α 1 , and γ be a real number such that γ ( δ k + k 1 ) + ( α + 1 ) k > 1 . Suppose that there exists a sequence of numbers ( B n ) such that it is δ-quasi-monotone with | Δ λ n | | B n | , λ n 0 as n , n = 1 n δ n log n < and n = 1 n B n log n is convergent. If the sequence ( w n α ) defined by (see [7])
(7)
(8)
satisfies the condition
n = 1 m n γ ( δ k + k 1 ) k ( w n α ) k = O ( log m ) as m ,
(9)

then the series a n λ n is summable | C , α , γ ; δ | k .

3 The main result

The aim of this paper is to generalize Theorem A for | C , α , β , γ ; δ | k summability. We shall prove the following theorem.

Theorem Let k 1 , 0 δ < α 1 , and γ be a real number such that ( α + β + 1 γ ( δ + 1 ) ) k > 1 , and let there be sequences ( B n ) and ( λ n ) such that the conditions of Theorem A are satisfied. If the sequence ( w n α , β ) defined by
(10)
(11)
satisfies the condition
n = 1 m n γ ( δ k + k 1 ) k ( w n α , β ) k = O ( log m ) as m ,
(12)

then the series a n λ n is summable | C , α , β , γ ; δ | k . It should be noted that if we take β = 0 , then we get Theorem A.

We need the following lemmas for the proof of our theorem.

Lemma 1 ([8])

Under the conditions on ( B n ) , as taken in the statement of the theorem, we have the following:
(13)
(14)

Lemma 2 ([9])

If 0 < α 1 , β > 1 , and 1 v n , then
| p = 0 v A n p α 1 A p β a p | max 1 m v | p = 0 m A m p α 1 A p β a p | .
(15)

4 Proof of the theorem

Let ( T n α , β ) be the n th ( C , α , β ) mean of the sequence ( n a n λ n ) . Then by (2), we have
T n α , β = 1 A n α + β v = 1 n A n v α 1 A v β v a v λ v .
Firstly applying Abel’s transformation and then using Lemma 2, we have that
since
| T n , 1 α , β + T n , 2 α , β | k 2 k ( | T n , 1 α , β | k + | T n , 2 α , β | k ) .
(16)
In order to complete the proof of the theorem, by (6), it is sufficient to show that for r = 1 , 2 ,
n = 1 n γ ( δ k + k 1 ) k | T n , r α , β | k < .
Whenever k > 1 , we can apply Hölder’s inequality with indices k and k , where 1 k + 1 k = 1 , we get that
in view of the hypotheses of the theorem and Lemma 1. Similarly, we have that
n = 2 m + 1 n γ ( δ k + k 1 ) k | T n , 2 α , β | k = O ( 1 ) n = 1 m | λ n | n γ ( δ k + k 1 ) k ( w n α , β ) k = O ( 1 ) n = 1 m 1 | Δ λ n | v = 1 n v γ ( δ k + k 1 ) k ( w v α , β ) k + O ( 1 ) | λ m | v = 1 m v γ ( δ k + k 1 ) k ( w v α , β ) k = O ( 1 ) n = 1 m 1 | Δ λ n | log n + O ( 1 ) | λ m | log m = O ( 1 ) n = 1 m 1 | B n | log n + O ( 1 ) | λ m | log m = O ( 1 ) as  m ,
by virtue of the hypotheses of the theorem and Lemma 1. Therefore, by (6), we get that for r = 1 , 2 ,
n = 1 n γ ( δ k + k 1 ) k | T n , r α , β | k < .

This completes the proof of the theorem. If we take δ = 0 and γ = 1 , then we get a result for | C , α , β | k summability factors. Also, if we take β = 0 , δ = 0 , and α = 1 , then we get a result for | C , 1 | k summability.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Erciyes University, Kayseri, Turkey

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Copyright

© Tuncer; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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