Open Access

A new application of quasi power increasing sequences. II

Fixed Point Theory and Applications20132013:75

https://doi.org/10.1186/1687-1812-2013-75

Received: 24 December 2012

Accepted: 18 March 2013

Published: 28 March 2013

Abstract

In this paper, we prove a general theorem dealing with absolute Cesàro summability factors of infinite series by using a quasi-f-power increasing sequence instead of a quasi-σ-power increasing sequence. This theorem also includes several new results.

MSC:26D15, 40D15, 40F05, 40G99, 46A45.

Keywords

absolute summability increasing sequences sequence spaces Hölder inequality Minkowski inequality infinite series

1 Introduction

A positive sequence ( b n ) is said to be almost increasing 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]). A sequence ( λ n ) is said to be of bounded variation, denoted by ( λ n ) BV , if n = 1 | Δ λ n | = n = 1 | λ n λ n + 1 | < . A positive sequence X = ( X n ) is said to be a quasi-σ-power increasing sequence if there exists a constant K = K ( σ , X ) 1 such that K n σ X n m σ X m holds for all n m 1 (see [2]). It should be noted that every almost increasing sequence is a quasi-σ-power increasing sequence for any nonnegative σ, but the converse may not be true as can be seen by taking an example, say X n = n σ for σ > 0 . Let ( φ n ) be a sequence of complex numbers and let a n be a given infinite series with partial sums ( s n ) . We denote by z n α and t n α the n th Cesàro means of order α, with α > 1 , of the sequences ( s n ) and ( n a n ) , respectively, that is,
(1)
(2)
where
A n α = ( n + α n ) = ( α + 1 ) ( α + 2 ) ( α + n ) n ! = O ( n α ) , A n α = 0 for  n > 0 .
(3)
The series a n is said to be summable φ | C , α | k , k 1 and α > 1 , if (see [3, 4])
n = 1 | φ n ( z n α z n 1 α ) | k = n = 1 n k | φ n t n α | k < .
(4)

In the special case if we take φ n = n 1 1 k , then φ | C , α | k summability is the same as | C , α | k summability (see [5]). Also, if we take φ n = n δ + 1 1 k , then φ | C , α | k summability reduces to | C , α ; δ | k summability (see [6]).

2 The known results

Theorem A ([7])

Let ( λ n ) BV and let ( X n ) be a quasi-σ-power increasing sequence for some σ ( 0 < σ < 1 ). Suppose also that there exist sequences ( β n ) and ( λ n ) such that
(5)
(6)
(7)
(8)
If there exists an ϵ > 0 such that the sequence ( n ϵ k | φ n | k ) is nonincreasing and if the sequence ( w n α ) defined by (see [8])
w n α = { | t n α | , α = 1 , max 1 v n | t v α | , 0 < α < 1 ,
(9)
satisfies the condition
n = 1 m ( | φ n | w n α ) k n k = O ( X m ) as m ,
(10)

then the series a n λ n is summable φ | C , α | k , k 1 , 0 < α 1 and k α + ϵ > 1 .

Remark 1 Here, in the hypothesis of Theorem A, we have added the condition ‘ ( λ n ) BV ’ because it is necessary.

Theorem B ([9])

Let ( X n ) be a quasi-σ-power increasing sequence for some σ ( 0 < σ < 1 ). If there exists an ϵ > 0 such that the sequence ( n ϵ k | φ n | k ) is nonincreasing and if the conditions from (5) to (8) are satisfied and if the condition
n = 1 m ( | φ n | w n α ) k n k X n k 1 = O ( X m ) as m ,
(11)

is satisfied, then the series a n λ n is summable φ | C , α | k , k 1 , 0 < α 1 and k ( α 1 ) + ϵ > 1 .

Remark 2 It should be noted that condition (11) is the same as condition (10) when k = 1 . When k > 1 , condition (11) is weaker than condition (10) but the converse is not true. As in [10], we can show that if (10) is satisfied, then we get
n = 1 m ( | φ n | w n α ) k n k X n k 1 = O ( 1 X 1 k 1 ) n = 1 m ( | φ n | w n α ) k n k = O ( X m ) .
If (11) is satisfied, then for k > 1 we obtain that
n = 1 m ( | φ n | w n α ) k n k = n = 1 m X n k 1 ( | φ n | w n α ) k n k X n k 1 = O ( X m k 1 ) n = 1 m ( | φ n | w n α ) k n k X n k 1 = O ( X m k ) O ( X m ) .

Also, it should be noted that the condition ‘ ( λ n ) BV ’ has been removed.

3 The main result

The aim of this paper is to extend Theorem B by using a general class of quasi power increasing sequence instead of a quasi-σ-power increasing sequences. For this purpose, we need the concept of quasi-f-power increasing sequence. A positive sequence X = ( X n ) is said to be a quasi-f-power increasing sequence, if there exists a constant K = K ( X , f ) such that K f n X n f m X m , holds for n m 1 , where f = ( f n ) = [ n σ ( log n ) η , η 0 , 0 < σ < 1 ] (see [11]). It should be noted that if we take η = 0 , then we get a quasi-σ-power increasing sequence. Now, we will prove the following theorem.

Theorem Let ( X n ) be a quasi-f-power increasing sequence. If there exists an ϵ > 0 such that the sequence ( n ϵ k | φ n | k ) is non-increasing and if the conditions from (5) to (8) and (11) are satisfied, then the series a n λ n is summable φ | C , α | k , k 1 , 0 < α 1 and k ( α 1 ) + ϵ > 1 .

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

Lemma 1 ([12])

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 | .
(12)

Lemma 2 ([11])

Under the conditions on ( X n ) , ( β n ) , and ( λ n ) as expressed in the statement of the theorem, we have the following:
(13)
(14)

4 Proof of the theorem

Let ( T n α ) be the n th ( C , α ) , with 0 < α 1 , mean of the sequence ( n a n λ n ) . Then, by (2), we have
T n α = 1 A n α v = 1 n A n v α 1 v a v λ v .
(15)
First, applying Abel’s transformation and then using Lemma 1, we get that
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
n = 1 n k | φ n T n , r α | k < for  r = 1 , 2 .
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 k | φ n T n , 1 α | k n = 2 m + 1 n k ( A n α ) k | φ n | k { v = 1 n 1 A v α w v α | Δ λ v | } k n = 2 m + 1 n k n α k | φ n | 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 ) v = 1 m v α k ( w v α ) k ( β v ) k n = v + 1 m + 1 n ϵ k | φ n | k n k ( α 1 ) + ϵ + 1 = O ( 1 ) v = 1 m v α k ( w v α ) k ( β v ) k v ϵ k | φ v | k n = v + 1 m + 1 1 n k ( α 1 ) + ϵ + 1 = O ( 1 ) v = 1 m v α k ( w v α ) k ( β v ) k v ϵ k | φ v | k v d x x k ( α 1 ) + ϵ + 1 = O ( 1 ) v = 1 m β v ( β v ) k 1 ( w v α | φ v | ) k = O ( 1 ) v = 1 m β v ( 1 v X v ) k 1 ( w v α | φ v | ) k = O ( 1 ) v = 1 m 1 Δ ( v β v ) r = 1 v ( | φ r | w r α ) k r k X r k 1 + O ( 1 ) m β m v = 1 m ( | φ v | w v α ) k v k X v k 1 = O ( 1 ) v = 1 m 1 | Δ ( v β v ) | X v + O ( 1 ) m β m X m = O ( 1 ) v = 1 m 1 | ( v + 1 ) Δ β v β v | X v + O ( 1 ) m β m X m = O ( 1 ) v = 1 m 1 v | Δ β v | X v + O ( 1 ) v = 1 m 1 β v X v + O ( 1 ) m β m X m = O ( 1 ) as  m ,
by virtue of the hypotheses of the theorem and Lemma 2. Finally, we have that
n = 1 m n k | φ n T n , 2 α | k = n = 1 m | λ n | | λ n | k 1 n k ( w n α | φ n | ) k = O ( 1 ) n = 1 m | λ n | ( 1 X n ) k 1 n k ( w n α | φ n | ) k = O ( 1 ) n = 1 m 1 Δ | λ n | v = 1 n ( | φ v | w v α ) k v k X v k 1 + O ( 1 ) | λ m | n = 1 m ( | φ n | w n α ) k n k X n k 1 = O ( 1 ) n = 1 m 1 | Δ λ n | X n + O ( 1 ) | λ m | X m = O ( 1 ) n = 1 m 1 β n X n + O ( 1 ) | λ m | X m = O ( 1 ) as  m ,

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take ϵ = 1 and φ n = n 1 1 k (resp. ϵ = 1 , α = 1 and φ n = n 1 1 k ), then we get a new result dealing with | C , α | k (resp. | C , 1 | k ) summability factors of infinite series. Also, if we take ϵ = 1 and φ n = n δ + 1 1 k , then we get another new result concerning the | C , α ; δ | k summability factors of infinite series. Furthermore, if we take ( X n ) as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see [13]). Finally, if we take η = 0 , then we obtain Theorem B.

Declarations

Acknowledgements

Dedicated to Professor Hari M. Srivastava.

The author express his thanks the referees for their useful comments and suggestions.

Authors’ Affiliations

(1)

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Copyright

© Bor; licensee Springer. 2013

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