# 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 $n≥m≥1$ (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

(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

Now, when $k>1$, applying Hölder’s inequality with indices k and $k ′$, where $1 k + 1 k ′ =1$, we get that

by virtue of the hypotheses of the theorem and Lemma 2. Finally, we have that

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.

## References

1. 1.

Bari NK, Stečkin SB: Best approximation and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 1956, 5: 483–522. (in Russian)

2. 2.

Leindler L: A new application of quasi power increasing sequences. Publ. Math. (Debr.) 2001, 58: 791–796.

3. 3.

Balcı M: Absolute φ -summability factors. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 1980, 29: 63–80.

4. 4.

Kogbetliantz E: Sur lés series absolument sommables par la méthode des moyennes arithmétiques. Bull. Sci. Math. 1925, 49: 234–256.

5. 5.

Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 1957, 7: 113–141.

6. 6.

Flett TM: Some more theorems concerning the absolute summability of Fourier series. Proc. Lond. Math. Soc. 1958, 8: 357–387.

7. 7.

Bor H, Özarslan HS: A study on quasi power increasing sequences. Rocky Mt. J. Math. 2008, 38: 801–807.

8. 8.

Pati T: The summability factors of infinite series. Duke Math. J. 1954, 21: 271–284.

9. 9.

Bor H: A new application of quasi power increasing sequences. I. J. Inequal. Appl. 2013, 2013: 69.

10. 10.

Sulaiman WT:A note on $| A | k$ summability factors of infinite series. Appl. Math. Comput. 2010, 216(9):2645–2648.

11. 11.

Sulaiman WT: Extension on absolute summability factors of infinite series. J. Math. Anal. Appl. 2006, 322: 1224–1230.

12. 12.

Bosanquet LS: A mean value theorem. J. Lond. Math. Soc. 1941, 16: 146–148.

13. 13.

Bor H, Seyhan H: A note on almost increasing sequences. Ann. Soc. Math. Pol., 1 Comment. Math. 1999, 39: 37–42.

## Acknowledgements

Dedicated to Professor Hari M. Srivastava.

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

## Author information

Correspondence to Hüseyin Bor.

### Competing interests

The author declares that he has no competing interests.