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A new application of quasi power increasing sequences. I

Journal of Inequalities and Applications20132013:69

https://doi.org/10.1186/1029-242X-2013-69

  • Received: 22 December 2012
  • Accepted: 9 February 2013
  • Published:

Abstract

In (Rocky Mt. J. Math. 38:801-807, 2008), we proved a theorem dealing with an application of quasi-σ-power increasing sequences. In the present paper, we prove that theorem under less and more weaker conditions. This theorem also includes some new and known results.

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

Keywords

  • Hölder’s inequality
  • sequence spaces
  • absolute summability
  • increasing sequences

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 . 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 (see [2]). 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,
z n α = 1 A n α v = 0 n A n v α 1 s v ,
(1)
t n α = 1 A n α v = 1 n A n v α 1 v a v ,
(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])
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 [4]). 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 [5]).

2 Known result

In [6], we have proved the following theorem.

Theorem A 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
| Δ λ n | β n ,
(5)
β n 0 as  n ,
(6)
n = 1 n | Δ β n | X n < ,
(7)
| λ n | X n = O ( 1 ) as  n .
(8)
If there exists an ϵ > 0 such that the sequence ( n ϵ k | φ n | k ) is non-increasing and if the sequence ( w n α ) defined by (see [7])
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 .

It should be remarked that we have added the condition ‘ ( λ n ) BV ’ in the statement of Theorem A because it is necessary.

3 The main result

The aim of this paper is to prove Theorem A under less and weaker conditions. Now, we will prove the following theorem.

Theorem 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 non-increasing 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 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 [8] we can show that if (10) is satisfied, then we get that
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.

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

Lemma 1 [9]

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 [2]

Under the conditions on ( X n ) , ( β n ) and ( λ n ) , as expressed in the statement of the theorem, we have the following:
n β n X n = O ( 1 ) as  n ,
(13)
n = 1 β n X n < .
(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
T n α = 1 A n α v = 1 n 1 Δ λ v p = 1 v A n p α 1 p a p + λ 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 | Δ λ v | | p = 1 v A n p α 1 p a p | + | λ n | A n α | v = 1 n A n v α 1 v a v | 1 A n α v = 1 n 1 A v α w v α | Δ λ v | + | λ n | w n α = T n , 1 α + T n , 2 α .
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 ( β v ) k 1 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 ( β v ) k 1 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 | n = 1 m ( | φ n | w n α ) k n k X n 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. Also, if we set ϵ = 1 and φ n = n δ + 1 1 k , then we get another new result concerning the | C , α ; δ | k summability factors. Finally, if we take ( X n ) as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see [10]).

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
P.O. Box 121, Bahçelievler, Ankara, TR-06502, Turkey

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