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A new application of quasi power increasing sequences. II

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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.

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 nm1 (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 =0for n>0.
(3)

The series a n is said to be summable φ | C , α | k , k1 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 , k1, 0<α1 and kα+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 , k1, 0<α1 and k(α1)+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 nm1, where f=( f n )=[ n σ ( log n ) η ,η0,0<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 , k1, 0<α1 and k(α1)+ϵ>1.

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

Lemma 1 ([12])

If 0<α1 and 1vn, 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<α1, 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.

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Acknowledgements

Dedicated to Professor Hari M. Srivastava.

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

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Correspondence to Hüseyin Bor.

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The author declares that he has no competing interests.

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Keywords

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