On a new application of almost increasing sequences

Abstract

In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with ${|\overline{N},p_n|}_k$ summability factors of an infinite series. In the present paper, we have generalized this theorem on the $\phi -{|A,p_n|}_k$ 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 $\sum a_n$ 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 $\sum a_n$ is said to be summable ${|C,1|}_k$, $k\ge 1$, if (see [1])

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{k-1}{|t_n-t_{n-1}|}^k<\mathrm{\infty }.$$

(1)

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

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

(2)

The sequence-to-sequence transformation

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

(3)

defines the sequence $({\sigma }_n)$ of the $(\overline{N},p_n)$ mean of the sequence $(s_n)$, generated by the sequence of coefficients $(p_n)$ (see [2]). The series $\sum a_n$ is said to be summable $|\overline{N},p_n{|}_k$, $k\ge 1$, if (see [3])

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

(4)

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

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

(5)

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

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

(6)

where

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

Let $({\phi }_n)$ be any sequence of positive real numbers. The series $\sum a_n$ is said to be summable $\phi -|A,p_n{|}_k$, $k\ge 1$, if (see [5])

$$\sum _{n=1}^{\mathrm{\infty }}{\phi }_n^{k-1}|\overline{\mathrm{\Delta }}{A}_n(s){|}^k<\mathrm{\infty }.$$

(7)

If we take ${\phi }_n=\frac{P_n}{p_n}$, then $\phi -|A,p_n{|}_k$ summability reduces to ${|A,p_n|}_k$ summability. Also, if we take ${\phi }_n=\frac{P_n}{p_n}$ and $a_{nv}=\frac{p_v}{P_n}$, then we get $|\overline{N},p_n{|}_k$ summability. Furthermore, if we take ${\phi }_n=n$, $a_{nv}=\frac{p_v}{P_n}$ and $p_n=1$ for all values of n, $\phi -|A,p_n{|}_k$ reduces to ${|C,1|}_k$ summability. Finally, if we take ${\phi }_n=n$ and $a_{nv}=\frac{p_v}{P_n}$, then we get ${|R,p_n|}_k$ summability (see [6]).

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:

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

(8)

and

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

(9)

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

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

(10)

and

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

(11)

2 Known result

Many works have been done dealing with $|\overline{N},p_n{|}_k$ summability factors of infinite series (see [7–22]). Among them, in [21], the following main theorem has been proved.

Theorem A Let $(X_n)$ be a positive non-decreasing sequence and let there be sequences $({\beta }_n)$ and $({\lambda }_n)$ 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 $\sum a_n{\lambda }_n$ is summable ${|\overline{N},p_n|}_k$, $k\ge 1$.

3 The main result

The aim of this paper is to generalize Theorem A for $\phi -{|A,p_n|}_k$ 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 $A{b}_n\le c_n\le B{b}_n$ (see [23]). 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 [24]). 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 $A=(a_{nv})$ be a positive normal matrix such that

(18)

(19)

(20)

(21)

Let $(X_n)$ be an almost increasing sequence and $(\frac{{\phi }_n{p}_n}{P_n})$ be a non-increasing sequence. If conditions (12)-(16) and

$$\sum _{n=1}^m{\phi }_n^{k-1}{\left(\frac{p_n}{P_n}\right)}^k{|s_n|}^k=O(X_m)\mathit{\text{as}}m\to \mathrm{\infty },$$

(22)

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

Remark It should be noted that if we take $(X_n)$ as a positive non-decreasing sequence, ${\phi }_n=\frac{P_n}{p_n}$ and $a_{nv}=\frac{p_v}{P_n}$, then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘$(\frac{{\phi }_n{p}_n}{P_n})$ is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.

Lemma [22]

Under the conditions on $(X_n)$, $({\beta }_n)$ and $({\lambda }_n)$ 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 $\sum a_n{\lambda }_n$. Then we have, by (10) and (11),

$$\overline{\mathrm{\Delta }}{T}_n=\sum _{v=1}^n{\hat{a}}_{nv}{\lambda }_v{a}_v.$$

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

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

To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that

$$\sum _{n=1}^{\mathrm{\infty }}{\phi }_n^{k-1}|T_n(r){|}^k<\mathrm{\infty }\text{for }r=1,2,3.$$

Now, when $k>1$, applying Hölder’s inequality with indices k and $\acute{k}$, where $1/k+1/\acute{k}=1$, we have that

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{\phi }_n^{k-1}{|T_n(1)|}^k& =& O(1)\sum _{n=2}^{m+1}{\phi }_n^{k-1}{(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})||{\lambda }_v||s_v|)}^k\\ =& O(1)\sum _{n=2}^{m+1}{\phi }_n^{k-1}\left(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|{|{\lambda }_v|}^k{|s_v|}^k\right)\times {\left(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|\right)}^{k-1}\\ =& O(1)\sum _{n=2}^{m+1}{\left(\frac{{\phi }_n{p}_n}{P_n}\right)}^{k-1}\left(\sum _{v=1}^{n-1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|{|{\lambda }_v|}^k{|s_v|}^k\right)\\ =& O(1)\sum _{v=1}^m{|{\lambda }_v|}^k{|s_v|}^k\sum _{n=v+1}^{m+1}{\left(\frac{{\phi }_n{p}_n}{P_n}\right)}^{k-1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|\\ =& O(1)\sum _{v=1}^m{|{\lambda }_v|}^k{|s_v|}^k{\left(\frac{{\phi }_v{p}_v}{P_v}\right)}^{k-1}\sum _{n=v+1}^{m+1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|\\ =& O(1)\sum _{v=1}^m{|{\lambda }_v|}^{k-1}|{\lambda }_v|{|s_v|}^k{\left(\frac{{\phi }_v{p}_v}{P_v}\right)}^{k-1}\left(\frac{p_v}{P_v}\right)\\ =& O(1)\sum _{v=1}^m|{\lambda }_v|{\phi }_v^{k-1}{\left(\frac{p_v}{P_v}\right)}^k{|s_v|}^k\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }|{\lambda }_v|\sum _{r=1}^v{\phi }_r^{k-1}{\left(\frac{p_r}{P_r}\right)}^k{|s_r|}^k+O(1)|{\lambda }_m|\sum _{v=1}^m{\phi }_v^{k-1}{\left(\frac{p_v}{P_v}\right)}^k{|s_v|}^k\\ =& O(1)\sum _{v=1}^{m-1}|\mathrm{\Delta }{\lambda }_v|X_v+O(1)|{\lambda }_m|X_m\\ =& O(1)\sum _{v=1}^{m-1}{\beta }_v{X}_v+O(1)|{\lambda }_m|X_m\\ =& O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of the theorem and the lemma. Again, applying Hölder’s inequality and using the fact that $v{\beta }_v=O(\frac{1}{X_v})=O(1)$ by (23), we get that

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{\phi }_n^{k-1}|T_n(2){|}^k& =& O(1)\sum _{n=2}^{m+1}{\phi }_n^{k-1}{(\sum _{v=1}^{n-1}|{\hat{a}}_{n,v+1}||\mathrm{\Delta }{\lambda }_v||s_v|)}^k\\ =& O(1)\sum _{n=2}^{m+1}{\phi }_n^{k-1}(\sum _{v=1}^{n-1}|{\hat{a}}_{n,v+1}|{\beta }_v{|s_v|}^k)\times {(\sum _{v=1}^{n-1}|{\hat{a}}_{n,v+1}|{\beta }_v)}^{k-1}\\ =& O(1)\sum _{n=2}^{m+1}{\phi }_n^{k-1}(\sum _{v=1}^{n-1}|{\hat{a}}_{n,v+1}|{\beta }_v{|s_v|}^k)\times {\left(\sum _{v=1}^{n-1}v|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|{\beta }_v\right)}^{k-1}\\ =& O(1)\sum _{n=2}^{m+1}{\left(\frac{{\phi }_n{p}_n}{P_n}\right)}^{k-1}\left(\sum _{v=1}^{n-1}v|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|{\beta }_v{|s_v|}^k\right)\\ =& O(1)\sum _{v=1}^{m}v{\beta }_v{|s_v|}^k\sum _{n=v+1}^{m+1}{\left(\frac{{\phi }_n{p}_n}{P_n}\right)}^{k-1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|\\ =& O(1)\sum _{v=1}^{m}v{\beta }_v{|s_v|}^k{\left(\frac{{\phi }_v{p}_v}{P_v}\right)}^{k-1}\sum _{n=v+1}^{m+1}|{\mathrm{\Delta }}_v({\hat{a}}_{nv})|\\ =& O(1)\sum _{v=1}^{m}v{\beta }_v{|s_v|}^k{\left(\frac{{\phi }_v{p}_v}{P_v}\right)}^{k-1}\left(\frac{p_v}{P_v}\right)\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }(v{\beta }_v)\sum _{r=1}^v{\phi }_r^{k-1}{\left(\frac{p_r}{P_r}\right)}^k{|s_r|}^k+O(1)m{\beta }_m\sum _{v=1}^m{\phi }_v^{k-1}{\left(\frac{p_v}{P_v}\right)}^k{|s_v|}^k\\ =& O(1)\sum _{v=1}^{m-1}|\mathrm{\Delta }(v{\beta }_v)|X_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\sum _{v=1}^{m-1}v|\mathrm{\Delta }{\beta }_v|X_v+O(1)\sum _{v=1}^{m-1}{\beta }_{v+1}{X}_{v+1}+O(1)m{\beta }_m{X}_m\\ =& O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

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

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

This completes the proof of the theorem. If we take ${\phi }_n=\frac{P_n}{p_n}$, then we get a result concerning the ${|A,p_n|}_k$ summability factors. If we take $a_{nv}=\frac{p_v}{P_n}$, then we have another result dealing with ${|\overline{N},p_n,{\phi }_n|}_k$ summability. If we take $a_{nv}=\frac{p_v}{P_n}$ and $p_n=1$ for all values of n, then we get a result dealing with ${|C,1,{\phi }_n|}_k$ summability. If we take ${\phi }_n=n$, $a_{nv}=\frac{p_v}{P_n}$ and $p_n=1$ for all values of n, then we get a result for ${|C,1|}_k$ summability. □