A new note on absolute matrix summability

Abstract

In the present paper, we have proved theorems dealing with matrix summability factors by using quasi β-power increasing sequences. Some new results have also been obtained.

MSC:40D15, 40F05, 40G99.

1 Introduction

A positive sequence $({\gamma }_n)$ is said to be quasi β-power increasing sequence if there exists a constant $K=K(\beta ,\gamma )\ge 1$ such that $K{n}^{\beta }{\gamma }_n\ge m^{\beta }{\gamma }_m$ holds for all $n\ge m\ge 1$ [1]. A sequence $({\lambda }_n)$ is said to be of bounded variation, denote by $({\lambda }_n)\in \mathcal{BV}$, if ${\sum }_{n=1}^{\mathrm{\infty }}|\mathrm{\Delta }{\lambda }_n|={\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_n-{\lambda }_{n+1}|<\mathrm{\infty }$. Let $\sum a_n$ be a given infinite series with the partial sums $(s_n)$. Let $(p_n)$ be a sequence of positive numbers such that

$$P_n=\sum _{v=0}^n{p}_v\to \mathrm{\infty }\text{as }n\to \mathrm{\infty }(P_{-i}=p_{-i}=0,i\ge 1).$$

(1)

The sequence-to-sequence transformation

$${\sigma }_n=\frac{1}{P_n}\sum _{v=0}^n{p}_v{s}_v$$

(2)

defines the sequence $({\sigma }_n)$ of the $(\overline{N},p_n)$ mean of the sequence $(s_n)$, generated by the sequence of coefficients $(p_n)$ [2].

The series $\sum a_n$ is said to be summable $|\overline{N},p_n{|}_k$, $k\ge 1$ if [3]

$$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{P_n}{p_n}\right)}^{k-1}{|{\sigma }_n-{\sigma }_{n-1}|}^k<\mathrm{\infty }.$$

(3)

Let $A=(a_{nv})$ 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

$$A_n(s)=\sum _{v=0}^n{a}_{nv}{s}_v,n=0,1,\dots .$$

(4)

The series $\sum a_n$ is said to be summable $|A,p_n{|}_k$, $k\ge 1$ if [4]

$$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{P_n}{p_n}\right)}^{k-1}{|\overline{\mathrm{\Delta }}{A}_n(s)|}^k<\mathrm{\infty },$$

(5)

where

$$\overline{\mathrm{\Delta }}{A}_n(s)=A_n(s)-A_{n-1}(s).$$

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

Given a normal matrix $A=(a_{nv})$, we associate two lower semimatrices $\overline{A}=({\overline{a}}_{nv})$ and $\hat{A}=({\hat{a}}_{nv})$ as follows:

$${\overline{a}}_{nv}=\sum _{i=v}^n{a}_{ni},n,v=0,1,\dots$$

(6)

and

$${\hat{a}}_{00}={\overline{a}}_{00}=a_{00},{\hat{a}}_{nv}={\overline{a}}_{nv}-{\overline{a}}_{n-1,v},n=1,2,\dots .$$

(7)

It may be noted that $\overline{A}$ and $\hat{A}$ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have

$$A_n(s)=\sum _{v=0}^n{a}_{nv}{s}_v=\sum _{v=0}^n{\overline{a}}_{nv}{a}_v$$

(8)

and

$$\overline{\mathrm{\Delta }}{A}_n(s)=\sum _{v=0}^n{\hat{a}}_{nv}{a}_v.$$

(9)

2 Known result

Recently, many authors have come up with theorems dealing with the applications of power increasing sequences [1, 5–7]. Among them, Bor and Özarslan have proved two theorems for $|\overline{N},p_n{|}_k$ summability method by using quasi β-power increasing sequence [5]. Their theorems are as follows.

Theorem A Let $(X_n)$ be a quasi β-power increasing sequence for some $0<\beta <1$, and let there be sequences $({\beta }_n)$ and $({\lambda }_n)$ such that

$$|\mathrm{\Delta }{\lambda }_n|\le {\beta }_n,$$

(10)

$${\beta }_n\to 0\mathit{\text{as }}n\to 0,$$

(11)

$$\sum _{n=1}^{\mathrm{\infty }}n|\mathrm{\Delta }{\beta }_n|X_n<\mathrm{\infty },$$

(12)

$$|{\lambda }_n|X_n=O(1)\mathit{\text{as }}n\to \mathrm{\infty }.$$

(13)

If

$$\sum _{v=1}^n\frac{{|s_v|}^k}{v}=O(X_n),$$

(14)

$$\sum _{n=1}^m\frac{p_n}{P_n}{|s_n|}^k=O(X_m),m\to \mathrm{\infty },$$

(15)

then $\sum a_n{\lambda }_n$ is summable $|\overline{N},p_n{|}_k$, $k\ge 1$.

Theorem B Let $(X_n)$ be a quasi β-power increasing sequence for some $0<\beta <1$, and let sequences $({\beta }_n)$ and $({\lambda }_n)$ satisfy conditions (10)-(13) and (15). If

$$\sum _{n=1}^{\mathrm{\infty }}{P}_n|\mathrm{\Delta }{\beta }_n|X_n<\mathrm{\infty },$$

(16)

$$\sum _{n=1}^m\frac{{|s_n|}^k}{P_n}=O(X_m),$$

(17)

then $\sum a_n{\lambda }_n$ is summable $|\overline{N},p_n{|}_k$, $k\ge 1$.

3 The main result

The aim of this paper is to generalize Theorem A and Theorem B to $|A,p_n{|}_k$ summability. Now, we shall prove the following two theorems.

Theorem 1 Let $A=(a_{nv})$ be a positive normal matrix such that

$${\overline{a}}_{n0}=1,n=0,1,\dots ,$$

(18)

$$a_{n-1,v}\ge a_{nv},\mathit{\text{for }}n\ge v+1,$$

(19)

$$a_{nn}=O\left(\frac{p_n}{P_n}\right),$$

(20)

and $(X_n)$ is a quasi β-power increasing sequence for some $0<\beta <1$. If all the conditions of Theorem A and

$$({\lambda }_n)\in \mathcal{BV}$$

(21)

are satisfied, then the series $\sum a_n{\lambda }_n$ is summable ${|A,p_n|}_k$, $k\ge 1$.

In the special case of $a_{nv}=\frac{p_v}{P_n}$, this theorem reduces to Theorem A.

Theorem 2 Let $A=(a_{nv})$ be a positive normal matrix as in Theorem 1, and let $(X_n)$ is a quasi β-power increasing sequence for some $0<\beta <1$. If all the conditions of Theorem B and (21) are satisfied, then the series $\sum a_n{\lambda }_n$ is summable ${|A,p_n|}_k$, $k\ge 1$.

We need following lemmas for the proof of our theorems.

Lemma 1 [1]

Let $(X_n)$ be a quasi β-power increasing sequence for some $0<\beta <1$. If conditions (11) and (12) satisfied, then

$$n{X}_n{\beta }_n=O(1)\mathit{\text{as }}n\to \mathrm{\infty },$$

(22)

$$\sum _{n=1}^{\mathrm{\infty }}{X}_n{\beta }_n<\mathrm{\infty }.$$

(23)

Lemma 2 Let $(X_n)$ be a quasi β-power increasing sequence for some $0<\beta <1$. If conditions (11) and (16) are satisfied, then

$$P_n{\beta }_n{X}_n=O(1),$$

(24)

$$\sum _{n=1}^{\mathrm{\infty }}{p}_n{\beta }_n{X}_n<\mathrm{\infty }.$$

(25)

The proof of Lemma 2 is similar to that of Bor in [8] and hence is omitted.

4 Proof of Theorem 1

Let $(T_n)$ denote A-transform of the series $\sum a_n{\lambda }_n$. Then by (8), (9) and applying Abel’s transformation, we have

$$\begin{array}{rcl}\overline{\mathrm{\Delta }}{T}_n& =& \sum _{v=1}^n{\hat{a}}_{nv}{a}_v{\lambda }_v\\ =& \sum _{v=1}^{n-1}{\mathrm{\Delta }}_v({\hat{a}}_{nv}{\lambda }_v)\sum _{k=1}^v{a}_k+{\hat{a}}_{nn}{\lambda }_n\sum _{v=1}^n{a}_v\\ =& \sum _{v=1}^{n-1}({\hat{a}}_{nv}{\lambda }_v-{\hat{a}}_{n,v+1}{\lambda }_{v+1})s_v+a_{nn}{\lambda }_n{s}_n\\ =& \sum _{v=1}^{n-1}({\hat{a}}_{nv}{\lambda }_v-{\hat{a}}_{n,v+1}{\lambda }_{v+1}-{\hat{a}}_{n,v+1}{\lambda }_v+{\hat{a}}_{n,v+1}{\lambda }_v)s_v+a_{nn}{\lambda }_n{s}_n\\ =& \sum _{v=1}^{n-1}{\mathrm{\Delta }}_v({\hat{a}}_{nv}){\lambda }_v{s}_v+\sum _{v=1}^{n-1}{\hat{a}}_{n,v+1}\mathrm{\Delta }{\lambda }_v{s}_v+a_{nn}{\lambda }_n{s}_n\\ =& T_{n,1}+T_{n,2}+T_{n,3}\text{say}.\end{array}$$

Since

$$|T_{n,1}+T_{n,2}+T_{n,3}{|}^k\le 3^k({|T_{n,1}|}^k+T_{n,2}{|}^k+T_{n,3}{|}^k),$$

to complete the proof of the Theorem 1, it is sufficient to show that

$$\sum _{n=1}^{\mathrm{\infty }}{(P_n/p_n)}^{k-1}|T_{n,r}{|}^k<\mathrm{\infty },\text{for }r=1,2,3.$$

(26)

First, applying Hölder’s inequality with indices k and $k^{\prime }$, where $k>1$ and $\frac{1}{k}+\frac{1}{k^{\prime }}=1$, we get that

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}\right)}^{k-1}|T_{n,1}{|}^k& \le & \sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}\right)}^{k-1}{(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v{\hat{a}}_{nv}||{\lambda }_v||s_v|)}^k\\ =& O(1)\sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}\right)}^{k-1}(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v{\hat{a}}_{nv}|{|{\lambda }_v|}^k{|s_v|}^k)\times {(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v{\hat{a}}_{nv}|)}^{k-1}\\ =& O(1)\sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}{a}_{nn}\right)}^{k-1}(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v{\hat{a}}_{nv}|{|{\lambda }_v|}^k{|s_v|}^k)\\ =& O(1)\sum _{v=1}^m{|{\lambda }_v|}^k{|s_v|}^k\sum _{n=v+1}^{m+1}|{\mathrm{\Delta }}_v{\hat{a}}_{nv}|\\ =& O(1)\sum _{v=1}^m\frac{p_v}{P_v}{|{\lambda }_v|}^{k-1}|{\lambda }_v|{|s_v|}^k=O(1)\sum _{v=1}^m\frac{p_v}{P_v}|{\lambda }_v|{|s_v|}^k\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }|{\lambda }_v|\sum _{i=1}^v\frac{p_i}{P_i}{|s_i|}^k+O(1)|{\lambda }_m|\sum _{v=1}^m\frac{p_v}{P_v}{|s_v|}^k\\ =& O(1)\sum _{v=1}^{m-1}{\beta }_v{X}_v+O(1)|{\lambda }_m|X_m\\ =& O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of Theorem 1 and Lemma 1.

Since $({\lambda }_n)\in \mathcal{BV}$ by (21), applying Hölder’s inequality with the same indices as those above, we have

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}\right)}^{k-1}{|T_n(2)|}^k& \le & \sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}\right)}^{k-1}{(\sum _{v=1}^{n-1}|\mathrm{\Delta }{\lambda }_v||{\hat{a}}_{n,v+1}||s_v|)}^k\\ =& O(1)\sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}\right)}^{k-1}(\sum _{v=1}^{n-1}|\mathrm{\Delta }{\lambda }_v||{\hat{a}}_{n,v+1}|{|s_v|}^k)\\ \times {(\sum _{v=1}^{n-1}|\mathrm{\Delta }{\lambda }_v||{\hat{a}}_{n,v+1}|)}^{k-1}\\ =& O(1)\sum _{n=2}^{m+1}{\left(\frac{P_n}{p_n}{a}_{nn}\right)}^{k-1}(\sum _{v=1}^{n-1}{\beta }_v|{\hat{a}}_{n,v+1}|{|s_v|}^k)\times {(\sum _{v=1}^{n-1}|\mathrm{\Delta }{\lambda }_v|)}^{k-1}\\ =& O(1)\sum _{v=1}^m{\beta }_v{|s_v|}^k\sum _{n=v+1}^{m+1}|{\hat{a}}_{n,v+1}|\\ =& O(1)\sum _{v=1}^m{\beta }_v{|s_v|}^k\\ =& O(1)\sum _{v=1}^m(v{\beta }_v)\frac{{|s_v|}^k}{v}\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }(v{\beta }_v)\sum _{i=1}^v\frac{{|s_i|}^k}{i}+O(1)m{\beta }_m\sum _{v=1}^m\frac{{|s_v|}^k}{v}\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }(v{\beta }_v)X_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\sum _{v=1}^{m-1}v|\mathrm{\Delta }{\beta }_v|X_v+O(1)\sum _{v=1}^{m-1}{\beta }_{v+1}{X}_{v+1}+O(1)m{\beta }_m{X}_m\\ =& O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of Theorem 1 and Lemma 1.

Finally, by following the similar process as in $T_{n,1}$, we have that

$$\begin{array}{rcl}\sum _{n=1}^m{\left(\frac{P_n}{p_n}\right)}^{k-1}{|T_n(3)|}^k& \le & \sum _{n=1}^m{\left(\frac{P_n}{p_n}\right)}^{k-1}{|a_{nn}|}^k{|{\lambda }_n|}^k{|s_n|}^k\\ =& O(1)\sum _{n=1}^m\frac{p_n}{P_n}|{\lambda }_n|{|s_n|}^k\\ =& O(1)\text{as }m\to \mathrm{\infty }.\end{array}$$

So, we get

$$\sum _{n=1}^{\mathrm{\infty }}{(P_n/p_n)}^{k-1}|T_{n,r}{|}^k<\mathrm{\infty },\text{for }r=1,2,3.$$

This completes the proof of Theorem 1.

5 Proof of Theorem 2

Using Lemma 2 and proceeding as in the proof of Theorem 1, replacing ${\sum }_{v=1}^m{\beta }_v{|s_v|}^k$ by ${\sum }_{v=1}^m{\beta }_v{P}_v(\frac{{|s_v|}^k}{P_v})$, we can easily prove Theorem 2.

If we take $p_n=1$ in these theorems, then we have two new results dealing with ${|A|}_k$ summability factors of infinite series. Also, if we take $k=1$, then we obtain another two new results concerning $|A|$ summability. Finally, by taking $(X_n)$ as almost increasing sequence in the theorems, we get new results dealing with $|A,p_n{|}_k$ summability factors of infinite series.