# On a new application of almost increasing sequences

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## Abstract

In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with $| N ¯ , p n | k$ summability factors of an infinite series. In the present paper, we have generalized this theorem on the $φ− | A , p n | k$ summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence.

MSC:40D15, 40F05, 40G99.

## 1 Introduction

Let $∑ a n$ be a given infinite series with the partial sums $( s n )$. We denote by $t n$ the n th $(C,1)$ mean of the sequence $( s n )$. The series $∑ a n$ is said to be summable $| C , 1 | k$, $k≥1$, if (see )

$∑ n = 1 ∞ n k − 1 | t n − t n − 1 | k <∞.$
(1)

Let $( p n )$ be a sequence of positive numbers such that

(2)

The sequence-to-sequence transformation

$σ n = 1 P n ∑ v = 0 n p v s v$
(3)

defines the sequence $( σ n )$ of the $( N ¯ , p n )$ mean of the sequence $( s n )$, generated by the sequence of coefficients $( p n )$ (see ). The series $∑ a n$ is said to be summable $| N ¯ , p n | k$, $k≥1$, if (see )

$∑ n = 1 ∞ ( P n p n ) k − 1 | σ n − σ n − 1 | k <∞.$
(4)

Let $A=( a n v )$ be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence $s=( s n )$ to $As=( A n (s))$, where

$A n (s)= ∑ v = 0 n a n v s v ,n=0,1,….$
(5)

The series $∑ a n$ is said to be summable $|A, p n | k$, $k≥1$, if (see )

$∑ n = 1 ∞ ( P n p n ) k − 1 | Δ ¯ A n ( s ) | k <∞,$
(6)

where

$Δ ¯ A n (s)= A n (s)− A n − 1 (s).$

Let $( φ n )$ be any sequence of positive real numbers. The series $∑ a n$ is said to be summable $φ−|A, p n | k$, $k≥1$, if (see )

$∑ n = 1 ∞ φ n k − 1 | Δ ¯ A n (s) | k <∞.$
(7)

If we take $φ n = P n p n$, then $φ−|A, p n | k$ summability reduces to $| A , p n | k$ summability. Also, if we take $φ n = P n p n$ and $a n v = p v P n$, then we get $| N ¯ , p n | k$ summability. Furthermore, if we take $φ n =n$, $a n v = p v P n$ and $p n =1$ for all values of n, $φ−|A, p n | k$ reduces to $| C , 1 | k$ summability. Finally, if we take $φ n =n$ and $a n v = p v P n$, then we get $| R , p n | k$ summability (see ).

Before stating the main theorem, we must first introduce some further notations.

Given a normal matrix $A=( a n v )$, we associate two lower semimatrices $A ¯ =( a ¯ n v )$ and $A ˆ =( a ˆ n v )$ as follows:

$a ¯ n v = ∑ i = v n a n i ,n,v=0,1,…$
(8)

and

$a ˆ 00 = a ¯ 00 = a 00 , a ˆ n v = a ¯ n v − a ¯ n − 1 , v ,n=1,2,….$
(9)

It may be noted that $A ¯$ and $A ˆ$ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we have

$A n (s)= ∑ v = 0 n a n v s v = ∑ v = 0 n a ¯ n v a v$
(10)

and

$Δ ¯ A n (s)= ∑ v = 0 n a ˆ n v a v .$
(11)

## 2 Known result

Many works have been done dealing with $| N ¯ , p n | k$ summability factors of infinite series (see ). Among them, in , the following main theorem has been proved.

Theorem A Let $( X n )$ be a positive non-decreasing sequence and let there be sequences $( β n )$ and $( λ n )$ such that (12) (13) (14) (15)

are satisfied. Furthermore, if $( p n )$ is a sequence of positive numbers such that (16) (17)

then the series $∑ a n λ n$ is summable $| N ¯ , p n | k$, $k≥1$.

## 3 The main result

The aim of this paper is to generalize Theorem A for $φ− | A , p n | k$ summability under weaker conditions. For this, we need the concept of an almost increasing sequence. A positive sequence $( c n )$ is said to be almost increasing if there exists a positive increasing sequence $( b n )$ and two positive constants A and B such that $A b n ≤ c n ≤B b n$ (see ). Obviously, every increasing sequence is an almost increasing sequence but the converse need not be true as can be seen from the example $b n =n e ( − 1 ) n$. Also, one can find some results dealing with absolute almost convergent sequences (see ). So, we are weakening the hypotheses of Theorem A replacing the increasing sequence by an almost increasing sequence. Now, we shall prove the following theorem.

Theorem Let $A=( a n v )$ be a positive normal matrix such that (18) (19) (20) (21)

Let $( X n )$ be an almost increasing sequence and $( φ n p n P n )$ be a non-increasing sequence. If conditions (12)-(16) and

$∑ n = 1 m φ n k − 1 ( p n P n ) k | s n | k =O( X m ) as m→∞,$
(22)

are satisfied, then the series $∑ a n λ n$ is summable $φ− | A , p n | k$, $k≥1$.

Remark It should be noted that if we take $( X n )$ as a positive non-decreasing sequence, $φ n = P n p n$ and $a n v = p v P n$, then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘$( φ n p n P n )$ is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.

Lemma 

Under the conditions on $( X n )$, $( β n )$ and $( λ n )$ as taken in the statement of the theorem, we have the following: (23) (24)

Proof of the Theorem Let $( T n )$ denote A-transform of the series $∑ a n λ n$. Then we have, by (10) and (11),

$Δ ¯ T n = ∑ v = 1 n a ˆ n v λ v a v .$

Applying Abel’s transformation to this sum, we get that

$Δ ¯ T n = ∑ v = 1 n − 1 Δ v ( a ˆ n v λ v ) s v + a ˆ n n λ n s n = ∑ v = 1 n − 1 ( a ˆ n v λ v − a ˆ n , v + 1 λ v + 1 ) s v + a ˆ n n λ n s n = ∑ v = 1 n − 1 Δ v ( a ˆ n v ) λ v s v + ∑ v = 1 n − 1 a ˆ n , v + 1 s v Δ λ v + a n n λ n s n = T n ( 1 ) + T n ( 2 ) + T n ( 3 ) .$

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 have that

by virtue of the hypotheses of the theorem and the lemma. Again, applying Hölder’s inequality and using the fact that $v β v =O( 1 X v )=O(1)$ by (23), we get that

by virtue of the hypotheses of the theorem and the lemma. Finally, as in $T n (1)$, we have that

This completes the proof of the theorem. If we take $φ n = P n p n$, then we get a result concerning the $| A , p n | k$ summability factors. If we take $a n v = p v P n$, then we have another result dealing with $| N ¯ , p n , φ n | k$ summability. If we take $a n v = p v P n$ and $p n =1$ for all values of n, then we get a result dealing with $| C , 1 , φ n | k$ summability. If we take $φ n =n$, $a n v = p v P n$ and $p n =1$ for all values of n, then we get a result for $| C , 1 | k$ summability. □

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## Author information

Correspondence to A Keten.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

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Özarslan, H., Keten, A. On a new application of almost increasing sequences. J Inequal Appl 2013, 13 (2013) doi:10.1186/1029-242X-2013-13

• #### DOI

https://doi.org/10.1186/1029-242X-2013-13

### Keywords

• absolute matrix summability
• almost increasing sequences
• infinite series 