Skip to main content

A generalization of a theorem of Bor

Abstract

In this paper, a general theorem concerning absolute matrix summability is established by applying the concepts of almost increasing and δ-quasi-monotone sequences.

1 Introduction

A positive sequence \((y_{n})\) is said to be almost increasing if there is a positive increasing sequence \((u_{n})\) and two positive constants K and M such that \(Ku_{n} \leq y_{n} \leq Mu_{n}\) (see [1]). A sequence \((c_{n})\) is said to be δ-quasi-monotone, if \(c_{n}\rightarrow 0\), \(c_{n}>0\) ultimately and \(\Delta c_{n}\geq-\delta_{n}\), where \(\Delta c_{n}=c_{n}-c_{n+1}\) and \(\delta=(\delta_{n})\) is a sequence of positive numbers (see [2]). Let \(\sum a_{n}\) be a given infinite series with partial sums \((s_{n})\). Let \(T=(t_{nv})\) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. At that time T describes the sequence-to-sequence transformation, mapping the sequence \(s=(s_{n})\) to \(Ts=(T_{n}(s))\), where

$$ T_{n}(s)=\sum_{v=0}^{n}t_{nv}s_{v}, \quad n=0,1,\ldots $$
(1)

Let \((\varphi_{n})\) be any sequence of positive real numbers. The series \(\sum{a_{n}}\) is said to be summable \({\varphi}-|T,p_{n}|_{k}\), \(k\geq 1\), if (see [3])

$$ \sum_{n=1}^{\infty} \varphi_{n}^{k-1}\big| \bar{\Delta}T_{n}(s)\big|^{k}< \infty, $$
(2)

where

$$ \bar{\Delta}T_{n}(s)=T_{n}(s)-T_{n-1}(s). $$

If we take \(\varphi_{n}=\frac{P_{n}}{p_{n}}\), then \({\varphi }-|T,p_{n}|_{k}\) summability reduces to \(|T,p_{n}|_{k}\) summability (see [4]). If we set \(\varphi_{n}=n\) for all n, \(\varphi- \vert T,p_{n} \vert _{k}\) summability is the same as \(\vert T \vert _{k}\) summability (see [5]). Also, if we take \(\varphi_{n}=\frac{P_{n}}{p_{n}}\) and \(t_{nv}=\frac {p_{v}}{P_{n}}\), then we get \(|\bar{N},p_{n}|_{k}\) summability (see [6]).

2 Known result

In [7, 8], Bor has established the following theorem dealing with \(\vert \bar{N},p_{n} \vert _{k}\) summability factors of infinite series.

Theorem 2.1

Let \((Y_{n})\) be an almost increasing sequence such that \(|\Delta{Y_{n}}|=O(Y_{n}/n)\) and \(\lambda_{n}\rightarrow0\) as \(n\rightarrow\infty\). Assume that there is a sequence of numbers \((B_{n})\) such that it is δ-quasi-monotone with \(\sum nY_{n}\delta_{n}<\infty\), \(\sum B_{n}Y_{n}\) is convergent and \(|\Delta\lambda_{n}|\leq|B_{n}|\) for all n. If

$$\begin{aligned}& \sum_{n=1}^{m} \frac{1}{n}|\lambda_{n}|= O(1)\quad \textit{as }m\rightarrow\infty, \end{aligned}$$
(3)
$$\begin{aligned}& \sum_{n=1}^{m} \frac{1}{n}|z_{n}|^{k}= O(Y_{m})\quad \textit{as }m \rightarrow\infty, \end{aligned}$$
(4)

and

$$ \sum_{n=1}^{m} \frac{p_{n}}{P_{n}}|z_{n}|^{k}= O(Y_{m})\quad \textit{as }m \rightarrow\infty, $$
(5)

where \((z_{n})\) is the nth \((C,1)\) mean of the sequence \((n a_{n})\), then the series \(\sum a_{n}\lambda_{n}\) is summable \(|\bar{N},p_{n}|_{k}\), \(k\geq1\).

3 Main result

The purpose of this paper is to generalize Theorem 2.1 to the \(\varphi - \vert T,p_{n} \vert _{k}\) summability. Before giving main theorem, let us introduce some well-known notations. Let \(T= (t_{nv} )\) be a normal matrix. Lower semimatrices \(\bar{T}=(\bar{t}_{nv})\) and \(\hat{T}=(\hat{t}_{nv})\) are defined as follows:

$$ \bar{t}_{nv}=\sum_{i=v}^{n}t_{ni}, \quad n,v=0,1,\ldots $$
(6)

and

$$ \hat{t}_{00}=\bar{t}_{00}=t_{00} , \qquad \hat{t}_{nv}=\bar {t}_{nv}-\bar{t}_{n-1,v},\quad n=1,2,\ldots $$
(7)

Here, and are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we write

$$ T_{n} (s ) = \sum_{v=0}^{n}t_{nv}s_{v}= \sum_{v=0}^{n}\bar {t}_{nv}a_{v} $$
(8)

and

$$ \bar{\Delta}T_{n} (s ) = \sum _{v=0}^{n}\hat{t}_{nv}a_{v}. $$
(9)

By taking the definition of general absolute matrix summability, we established the following theorem.

Theorem 3.1

Let \(T=(t_{nv})\) be a positive normal matrix such that

$$\begin{aligned}& \bar{t}_{n0}=1, \quad n=0,1,\ldots, \end{aligned}$$
(10)
$$\begin{aligned}& t_{n-1,v} \geq t_{nv}, \quad \textit{for }n \geq v+1, \end{aligned}$$
(11)
$$\begin{aligned}& t_{nn}=O \biggl( \frac{p_{n}}{P_{n}} \biggr), \end{aligned}$$
(12)

and \((\frac{\varphi_{n}p_{n}}{P_{n}} )\) be a non-increasing sequence. If all conditions of Theorem 2.1 with conditions (4) and (5) are replaced by

$$ \sum_{n=1}^{m} \varphi_{n}^{k-1} \biggl( \frac{p_{n}}{P_{n}} \biggr) ^{k-1}\frac{1}{n}|z_{n}|^{k} = O(Y_{m}) \quad\textit{as } {m\rightarrow\infty} $$
(13)

and

$$ \sum_{n=1}^{m} \varphi_{n}^{k-1} \biggl( \frac{p_{n}}{P_{n}} \biggr) ^{k}|z_{n}|^{k} = O(Y_{m}) \quad\textit{as } {m \rightarrow\infty}, $$
(14)

then the series \(\sum a_{n} \lambda_{n}\) is \({\varphi}-|T,p_{n}|_{k}\) summable, \(k\geq1\).

We need the following lemmas for the proof of Theorem 3.1.

Lemma 3.2

[7]

Let \((Y_{n})\) be an almost increasing sequence and \(\lambda_{n}\rightarrow0\) as \(n\rightarrow\infty\). If \((B_{n})\) is δ-quasi-monotone with \(\sum B_{n}Y_{n}\) is convergent and \(|\Delta\lambda_{n}|\leq|B_{n}|\) for all n, then we have

$$ \vert \lambda_{n} \vert Y_{n}=O (1 ) \quad \textit{as }n\rightarrow\infty. $$
(15)

Lemma 3.3

[8]

Let \((Y_{n})\) be an almost increasing sequence such that \(n| {\Delta Y_{n}}|=O (Y_{n} )\). If \((B_{n})\) is δ-quasi monotone with \(\sum nY_{n}\delta_{n}<\infty\), and \(\sum B_{n}Y_{n}\) is convergent, then

$$\begin{aligned}& n B_{n} Y_{n}=O (1 ) \quad \textit{as }n\rightarrow \infty, \end{aligned}$$
(16)
$$\begin{aligned}& \sum_{n=1}^{\infty}nY_{n}| \Delta B_{n}|< \infty. \end{aligned}$$
(17)

4 Proof of Theorem 3.1

Let \((I_{n})\) indicate the T-transform of the series \(\sum a_{n}\lambda_{n}\). Then we obtain

$$ \bar{\Delta}I_{n} = \sum _{v=0}^{n}\hat{t}_{nv}a_{v} \lambda_{v}=\sum_{v=1}^{n} \frac{\hat{t}_{nv}\lambda_{v}}{v} v a_{v} $$
(18)

by means of (8) and (9).

Using Abel’s formula for (18), we obtain

$$\begin{aligned} \bar{\Delta}I_{n} ={} & \sum_{v=1}^{n-1} \Delta_{v} \biggl( \frac{\hat{t}_{nv}\lambda _{v}}{v} \biggr) \sum _{r=1}^{v} ra_{r}+\frac{\hat{t}_{nn}\lambda _{n}}{n}\sum _{r=1}^{n} ra_{r} \\ ={} & \sum_{v=1}^{n-1}\frac{v+1}{v} \Delta_{v} (\hat{t}_{nv} )\lambda_{v}z_{v}+ \sum_{v=1}^{n-1}\frac{v+1}{v} \hat{t}_{n,v+1}\Delta \lambda_{v}z_{v} \\ &+\sum_{v=1}^{n-1}\hat{t}_{n,v+1} \lambda _{v+1}\frac{z_{v}}{v}+\frac{n+1}{n}{t}_{nn} \lambda_{n}z_{n} \\ = {}& I_{n,1}+I_{n,2}+I_{n,3}+I_{n,4}. \end{aligned}$$

For the proof of Theorem 3.1, it suffices to prove that

$$ \sum_{n=1}^{\infty}\varphi_{n}^{k-1} \vert I_{n,r} \vert ^{k}< \infty $$

for \(r=1,2,3,4\).

By Hölder’s inequality, we have

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}| I_{n,1}|^{k} & = O(1) \sum_{n=2}^{m+1} \varphi_{n}^{k-1} \Biggl(\sum_{v=1}^{n-1} \bigl\vert \Delta_{v}(\hat{t}_{nv}) \bigr\vert \vert \lambda_{v} \vert \vert z_{v} \vert \Biggr)^{k} \\ &= O(1) \sum_{n=2}^{m+1} \varphi_{n}^{k-1} \Biggl(\sum_{v=1}^{n-1} \bigl\vert \Delta_{v}(\hat{t}_{nv}) \bigr\vert \vert \lambda_{v} \vert ^{k} \vert z_{{v}} \vert ^{k} \Biggr) \Biggl(\sum_{v=1}^{n-1}\big| \Delta_{v}(\hat{t}_{nv})\big| \Biggr) ^{k-1}. \end{aligned}$$

By (6) and (7), we have

$$ \begin{aligned}[b]\Delta_{v}(\hat{t}_{nv})&= \hat{t}_{nv}-\hat{t}_{n,v+1} \\ &=\bar{t}_{nv}-\bar{t}_{n-1,v}-\bar{t}_{n,v+1}+\bar {t}_{n-1,v+1} \\ &= {t}_{nv}-{t}_{n-1,v}. \end{aligned}$$
(19)

Thus using (6), (10) and (11)

$$ \sum_{v=1}^{n-1} \big|\Delta_{v}( \hat{t}_{nv})\big|=\sum_{v=1}^{n-1}(t_{n-1,v}-t_{nv}) \leq t_{nn}. $$

Hence, we get

$$ \sum_{n=2}^{m+1}\varphi_{n}^{k-1}|I_{n,1}|^{k} = O(1)\sum_{n=2}^{m+1} \varphi_{n}^{k-1} {t}_{nn}^{k-1} \Biggl(\sum _{v=1}^{n-1} \bigl\vert \Delta_{v}( \hat{t}_{nv}) \bigr\vert \vert \lambda_{v} \vert ^{k} \vert z_{{v}} \vert ^{k} \Biggr) $$

by using (12)

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}| I_{n,1}|^{k} & =O(1)\sum_{n=2}^{m+1} \biggl( \frac{\varphi _{n}{p_{n}}}{P_{n}} \biggr) ^{k-1} \Biggl(\sum _{v=1}^{n-1} \bigl\vert \Delta_{v}( \hat{t}_{nv}) \bigr\vert \vert \lambda_{v} \vert ^{k} \vert z_{{v}} \vert ^{k} \Biggr) \\ &= O(1)\sum_{v=1}^{m} | \lambda_{v} |^{k}| z_{v} |^{k} \sum _{n=v+1}^{m+1} \biggl( \frac{\varphi_{n}{p_{n}}}{P_{n}} \biggr) ^{k-1} \big|\Delta_{v}(\hat{t}_{nv})\big| \\ &= O(1)\sum_{v=1}^{m} \biggl( \frac{\varphi_{v}{p_{v}}}{P_{v}} \biggr) ^{k-1} |\lambda_{v} |^{k}| z_{v} |^{k} \sum_{n=v+1}^{m+1} \big|\Delta_{v}(\hat{t}_{nv})\big|. \end{aligned}$$

Now, using (11) and (19), we obtain

$$ \sum_{n=v+1}^{m+1} \big|\Delta_{v}( \hat{t}_{nv})\big|=\sum_{n=v+1}^{m+1}(t_{n-1,v}-t_{nv}) \leq t_{vv}. $$

Thus, by using Abel’s formula, we obtain

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}| I_{n,1}|^{k}&= O(1)\sum_{v=1}^{m} \biggl( \frac{\varphi _{v}{p_{v}}}{P_{v}} \biggr) ^{k-1} |\lambda_{v} |^{k-1}|\lambda_{v} || z_{v} |^{k} t_{vv} \\ &= O(1)\sum_{v=1}^{m}\varphi_{v}^{k-1} \biggl( \frac{p_{v}}{P_{v}} \biggr) ^{k} \vert \lambda_{v} \vert \vert z_{v} \vert ^{k} \\ & = O(1) \sum_{v=1}^{m-1}\Delta| \lambda_{v} |\sum_{r=1}^{v} \varphi _{r}^{k-1} \biggl( \frac{p_{r}}{P_{r}} \biggr) ^{k}| z_{r} |^{k}+O(1)| \lambda_{m} | \sum_{v=1}^{m} \varphi_{v}^{k-1} \biggl( \frac {p_{v}}{P_{v}} \biggr) ^{k}|z_{v}|^{k} \\ &= O(1)\sum_{v=1}^{m-1}| \Delta \lambda_{v} | Y_{v} + O(1)| \lambda_{m} | Y_{m} \\ &= O(1)\sum_{v=1}^{m-1}| B_{v}| Y_{v}+O(1)|\lambda_{m}|Y_{m} \\ & = O(1) \quad \text{as }m\rightarrow\infty, \end{aligned}$$

in view of (14) and (15).

Again, using Hölder’s inequality, we have

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}| I_{n,2}|^{k} ={} & O(1) \sum_{n=2}^{m+1} \varphi_{n}^{k-1} \Biggl(\sum_{v=1}^{n-1}| \hat {t}_{n,v+1}||\Delta\lambda_{v}||z_{v}| \Biggr)^{k} \\ ={} & O(1) \sum_{n=2}^{m+1} \varphi_{n}^{k-1} \Biggl(\sum_{v=1}^{n-1}| \hat {t}_{n,v+1}||B_{v}||z_{v}|^{k} \Biggr) \\ &\times \Biggl(\sum_{v=1}^{n-1}|\hat {t}_{n,v+1}||B_{v}| \Biggr)^{k-1}. \end{aligned}$$

By means of (6), (7) and (11), we have

$$\begin{aligned} \hat{t}_{n,v+1}&= \bar{t}_{n,v+1}-\bar{t}_{n-1,v+1} \\ &= \sum_{i=v+1}^{n} t_{ni}-\sum _{i=v+1}^{n-1} t_{n-1,i} \\ &= t_{nn}+\sum_{i=v+1}^{n-1} ( t_{ni}- t_{n-1,i} ) \\ &\leq t_{nn}. \end{aligned}$$

In this way, we have

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}| I_{n,2}|^{k}& = O(1) \sum_{n=2}^{m+1} \varphi_{n}^{k-1}{t}_{nn}^{k-1} \Biggl( \sum _{v=1}^{n-1}|\hat{t}_{n,v+1}||B_{v}||z_{v}|^{k} \Biggr) \\ & = O(1) \sum_{n=2}^{m+1} \biggl( \frac{\varphi_{n}{p_{n}}}{P_{n}} \biggr) ^{k-1} \Biggl( \sum _{v=1}^{n-1}|\hat{t}_{n,v+1}||B_{v}||z_{v}|^{k} \Biggr) \\ & = O(1)\sum_{v=1}^{m}|B_{v}||z_{v}|^{k} \sum_{n=v+1}^{m+1} \biggl( \frac{\varphi_{n}{p_{n}}}{P_{n}} \biggr) ^{k-1}|\hat {t}_{n,v+1}| \\ &= O(1)\sum_{v=1}^{m} \biggl( \frac{\varphi _{v}{p_{v}}}{P_{v}} \biggr) ^{k-1}|B_{v}||z_{v}|^{k} \sum_{n=v+1}^{m+1}|\hat {t}_{n,v+1}|. \end{aligned}$$

By (6), (7), (10) and (11), we obtain

$$ |\hat{t}_{n,v+1}|=\sum_{i=0}^{v} (t_{n-1,i}-t_{ni}). $$

Thus, using (6) and (10), we have

$$ \sum_{n=v+1}^{m+1} |\hat{t}_{n,v+1}|= \sum_{n=v+1}^{m+1} \sum _{i=0}^{v} (t_{n-1,i}-t_{ni})\leq1, $$

then we get

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}| I_{n,2}|^{k} ={} & O(1)\sum_{v=1}^{m} \varphi_{v}^{k-1} \biggl( \frac{p_{v}}{P_{v}} \biggr) ^{k-1} v |B_{v}|\frac{1}{v}|z_{v}|^{k} \\ ={} & O(1)\sum_{v=1}^{m-1} \Delta\bigl(v|B_{v}|\bigr)\sum_{r=1}^{v} \varphi_{r}^{k-1} \biggl( \frac {p_{r}}{P_{r}} \biggr) ^{k-1} \frac{1}{r} | z_{r} |^{k} \\ &+ O(1)m|B_{m}| \sum_{v=1}^{m} \varphi_{v}^{k-1} \biggl( \frac {p_{v}}{P_{v}} \biggr) ^{k-1} \frac{1}{v}| z_{v} |^{k} \\ ={} & O(1)\sum_{v=1}^{m-1}v|\Delta B_{v}|Y_{v}+O(1)\sum_{v=1}^{m-1}|B_{v}|Y_{v}+O(1)m|B_{m}|Y_{m} \\ ={} & O(1)\quad \text{as }m\rightarrow\infty, \end{aligned}$$

in view of (13), (16) and (17).

Also, we have

$$\begin{aligned} \sum_{n=2}^{m+1}\varphi_{n}^{k-1}|I_{n,3}|^{k} \leq{}& \sum_{n=2}^{m+1}\varphi_{n}^{k-1} \Biggl(\sum_{v=1}^{n-1}|\hat {t}_{n,v+1}||\lambda_{v+1}|\frac{|z_{v}|}{v} \Biggr)^{k} \\ \leq{}& \sum_{n=2}^{m+1}\varphi_{n}^{k-1} \Biggl(\sum_{v=1}^{n-1}|\hat {t}_{n,v+1}||\lambda_{v+1}|\frac{|z_{v}|^{k}}{v} \Biggr) \Biggl(\sum _{v=1}^{n-1}|\hat{t}_{n,v+1}| \frac{|\lambda_{v+1}|}{v} \Biggr)^{k-1} \\ \leq{}& \sum_{n=2}^{m+1} \varphi_{n}^{k-1} t_{nn}^{k-1} \Biggl(\sum _{v=1}^{n-1}|\hat{t}_{n,v+1}|| \lambda_{v+1}|\frac{|z_{v}|^{k}}{v} \Biggr) \Biggl(\sum _{v=1}^{n-1} \frac{|\lambda_{v+1}|}{v} \Biggr)^{k-1} \\ ={} & O(1)\sum_{n=2}^{m+1} \varphi_{n}^{k-1} \biggl(\frac{p_{n}}{P_{n}} \biggr) ^{k-1} \Biggl(\sum_{v=1}^{n-1}| \hat{t}_{n,v+1}||\lambda_{v+1}|\frac {|z_{v}|^{k}}{v} \Biggr) \\ ={} & O(1)\sum_{n=2}^{m+1} \biggl( \frac{\varphi_{n}{p_{n}}}{P_{n}} \biggr) ^{k-1} \Biggl(\sum _{v=1}^{n-1}| \hat{t}_{n,v+1}|| \lambda_{v+1}| \frac {|z_{v}|^{k}}{v} \Biggr) \\ ={} & O(1)\sum_{v=1}^{m}| \lambda_{v+1}|\frac {|z_{v}|}{v}^{k}\sum _{n=v+1}^{m+1} \biggl( \frac{\varphi _{n}{p_{n}}}{P_{n}} \biggr) ^{k-1}|\hat{t}_{n,v+1}| \\ ={} & O(1)\sum_{v=1}^{m} \biggl( \frac{\varphi_{v}{p_{v}}}{P_{v}} \biggr) ^{k-1}|\lambda_{v+1}| \frac{|z_{v}|}{v}^{k}\sum_{n=v+1}^{m+1}| \hat {t}_{n,v+1}| \\ ={} & O(1)\sum_{v=1}^{m} \varphi_{v}^{k-1} \biggl( \frac {{p_{v}}}{P_{v}} \biggr) ^{k-1}|\lambda_{v+1}|\frac{|z_{v}|}{v}^{k} \\ ={}& O(1) \sum_{v=1}^{m-1}|\Delta \lambda_{v+1} |\sum_{r=1}^{v} \varphi _{r}^{k-1} \biggl( \frac{p_{r}}{P_{r}} \biggr) ^{k-1} \frac{1}{r}| z_{r} |^{k} \\ &+ O(1)| \lambda_{m+1} | \sum_{v=1}^{m} \varphi _{v}^{k-1} \biggl( \frac{p_{v}}{P_{v}} \biggr)^{k-1} \frac{1}{v}|z_{v} |^{k} \\ ={} & O(1)\sum_{v=1}^{m-1}|B_{v+1}| Y_{v+1}+O(1) |\lambda_{m+1} | Y_{m+1} \\ ={} & O(1)\quad \text{as }m\rightarrow\infty, \end{aligned}$$

in view of (3), (12), (13) and (15).

Finally, as in \(I_{n,1}\), we have

$$\begin{aligned} \sum_{n=1}^{m}\varphi_{n}^{k-1}| I_{n,4}|^{k} & = O(1)\sum_{n=1}^{m} \varphi_{n}^{k-1} {t}_{nn}^{k} | \lambda_{n} |^{k} |z_{n}|^{k} \\ & = O(1)\sum_{n=1}^{m} \varphi_{n}^{k-1} \biggl( \frac{p_{n}}{P_{n}} \biggr) ^{k} |\lambda_{n} |^{k-1} | \lambda_{n} |{| z_{n}|^{k}} \\ & = O(1)\sum_{n=1}^{m}\varphi_{n}^{k-1} \biggl( \frac{p_{n}}{P_{n}} \biggr) ^{k} | \lambda_{n} |{| z_{n}|^{k}}=O(1)\quad \text{as }m\rightarrow\infty, \end{aligned}$$

in view of (12), (14) and (15). Finally, the proof of Theorem 3.1 is completed.

5 Corollary

If we take \(\varphi_{n}=\frac{P_{n}}{p_{n}}\) and \(t_{nv}=\frac {p_{v}}{P_{n}}\) in Theorem 3.1, then we get Theorem 2.1. In this case, conditions (13) and (14) reduce to conditions (4) and (5), respectively. Also, the condition ‘\((\frac{\varphi_{n}p_{n}}{P_{n}} )\) is a non-increasing sequence’ and the conditions (10)-(12) are clearly satisfied.

6 Conclusions

In this study, we have generalized a well-known theorem dealing with an absolute summability method to a \({\varphi}-|T,p_{n}|_{k}\) summability method of an infinite series by using almost increasing sequences and δ-quasi-monotone sequences.

References

  1. Bari, NK, Stečkin, SB: Best approximations and differential properties of two conjugate functions. Tr. Mosk. Mat. Obŝ. 5, 483-522 (1956)

    MathSciNet  Google Scholar 

  2. Boas, RP: Quasi-positive sequences and trigonometric series. Proc. Lond. Math. Soc. 14A, 38-46 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Özarslan, HS, Keten, A: A new application of almost increasing sequences. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 61, 153-160 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Sulaiman, WT: Inclusion theorems for absolute matrix summability methods of an infinite series. IV. Indian J. Pure Appl. Math. 34, 1547-1557 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Tanovic̆-Miller, N: On strong summability. Glas. Mat. Ser. III 14, 87-97 (1979)

    MathSciNet  Google Scholar 

  6. Bor, H: On two summability methods. Math. Proc. Camb. Philos. Soc. 97, 147-149 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bor, H: An application of almost increasing and δ-quasi-monotone sequences. JIPAM. J. Inequal. Pure Appl. Math. 1(2), Article ID 18 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Bor, H: Corrigendum on the paper ‘An application of almost increasing and δ-quasi-monotone sequences’ published in JIPAM, Vol.1, No.2. (2000), Article 18. JIPAM. J. Inequal. Pure Appl. Math. 3(1), Article ID 16 (2002)

    Google Scholar 

Download references

Acknowledgements

This work was supported by Research Fund of the Erciyes University, Project Number: FBA-2014-3846.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hikmet Seyhan Özarslan.

Additional information

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.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Özarslan, H.S., Kartal, B. A generalization of a theorem of Bor. J Inequal Appl 2017, 179 (2017). https://doi.org/10.1186/s13660-017-1455-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-017-1455-3

MSC

Keywords