A new application of quasi power increasing sequences. I

Abstract

In (Rocky Mt. J. Math. 38:801-807, 2008), we proved a theorem dealing with an application of quasi-σ-power increasing sequences. In the present paper, we prove that theorem under less and more weaker conditions. This theorem also includes some new and known results.

MSC: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 $A{c}_n\le b_n\le B{c}_n$ (see [1]). A sequence $({\lambda }_n)$ is said to be of bounded variation, denoted by $({\lambda }_n)\in \mathcal{BV}$, if ${\sum }_{n=1}^{\mathrm{\infty }}|\mathrm{\Delta }{\lambda }_n|={\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_n-{\lambda }_{n+1}|<\mathrm{\infty }$. A positive sequence $X=(X_n)$ is said to be a quasi-σ-power increasing sequence if there exists a constant $K=K(\sigma ,X)\ge 1$ such that $K{n}^{\sigma }{X}_n\ge m^{\sigma }{X}_m$ holds for all $n\ge m\ge 1$. 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 $X_n=n^{-\sigma }$ for $\sigma >0$ (see [2]). Let $({\phi }_n)$ be a sequence of complex numbers and let $\sum a_n$ be a given infinite series with partial sums $(s_n)$. We denote by $z_n^{\alpha }$ and $t_n^{\alpha }$ the n th Cesàro means of order α, with $\alpha >-1$, of the sequences $(s_n)$ and $(n{a}_n)$, respectively, that is,

$$z_n^{\alpha }=\frac{1}{A_n^{\alpha }}\sum _{v=0}^n{A}_{n-v}^{\alpha -1}{s}_v,$$

(1)

$$t_n^{\alpha }=\frac{1}{A_n^{\alpha }}\sum _{v=1}^n{A}_{n-v}^{\alpha -1}v{a}_v,$$

(2)

where

$$A_n^{\alpha }=\left(\genfrac{}{}{0ex}{}{n+\alpha }{n}\right)=\frac{(\alpha +1)(\alpha +2)\cdots (\alpha +n)}{n!}=O\left(n^{\alpha }\right),A_{-n}^{\alpha }=0\text{for }n>0.$$

(3)

The series $\sum a_n$ is said to be summable $\phi -{|C,\alpha |}_k$, $k\ge 1$ and $\alpha >-1$, if (see [3])

$$\sum _{n=1}^{\mathrm{\infty }}{\left|{\phi }_n(z_n^{\alpha }-z_{n-1}^{\alpha })\right|}^k=\sum _{n=1}^{\mathrm{\infty }}{n}^{-k}{\left|{\phi }_n{t}_n^{\alpha }\right|}^k<\mathrm{\infty }.$$

(4)

In the special case, if we take ${\phi }_n=n^{1-\frac{1}{k}}$, then $\phi -{|C,\alpha |}_k$ summability is the same as ${|C,\alpha |}_k$ summability (see [4]). Also, if we take ${\phi }_n=n^{\delta +1-\frac{1}{k}}$, then $\phi -{|C,\alpha |}_k$ summability reduces to ${|C,\alpha ;\delta |}_k$ summability (see [5]).

2 Known result

In [6], we have proved the following theorem.

Theorem A Let $({\lambda }_n)\in \mathcal{BV}$ and let $(X_n)$ be a quasi-σ-power increasing sequence for some σ ($0<\sigma <1$). Suppose also that there exist sequences $({\beta }_n)$ and $({\lambda }_n)$ such that

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

(5)

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

(6)

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

(7)

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

(8)

If there exists an $ϵ>0$ such that the sequence $(n^{ϵ-k}{|{\phi }_n|}^k)$ is non-increasing and if the sequence $(w_n^{\alpha })$ defined by (see [7])

$$w_n^{\alpha }=\{\begin{array}{cc}|t_n^{\alpha }|,\hfill & \alpha =1,\hfill \\ {max}_{1\le v\le n}|t_v^{\alpha }|,\hfill & 0<\alpha <1,\hfill \end{array}$$

(9)

satisfies the condition

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k}=O(X_m)\mathit{\text{as }}m\to \mathrm{\infty },$$

(10)

then the series $\sum a_n{\lambda }_n$ is summable $\phi -{|C,\alpha |}_k$, $k\ge 1$, $0<\alpha \le 1$ and $k\alpha +ϵ>1$.

It should be remarked that we have added the condition ‘$({\lambda }_n)\in \mathcal{BV}$’ in the statement of Theorem A because it is necessary.

3 The main result

The aim of this paper is to prove Theorem A under less and weaker conditions. Now, we will prove the following theorem.

Theorem Let $(X_n)$ be a quasi-σ-power increasing sequence for some σ ($0<\sigma <1$). If there exists an $ϵ>0$ such that the sequence $(n^{ϵ-k}{|{\phi }_n|}^k)$ is non-increasing and if the conditions from (5) to (8) are satisfied and if the condition

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O(X_m)\mathit{\text{as }}m\to \mathrm{\infty }$$

(11)

is satisfied, then the series $\sum a_n{\lambda }_n$ is summable $\phi -{|C,\alpha |}_k$, $k\ge 1$, $0<\alpha \le 1$, and $k(\alpha -1)+ϵ>1$.

Remark 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 [8] we can show that if (10) is satisfied, then we get that

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O\left(\frac{1}{X_1^{k-1}}\right)\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k}=O(X_m).$$

If (11) is satisfied, then for $k>1$ we obtain that

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k}=\sum _{n=1}^m{X}_n^{k-1}\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O\left(X_m^{k-1}\right)\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O\left(X_m^k\right)\ne O(X_m).$$

Also, it should be noted that the condition ‘$({\lambda }_n)\in \mathcal{BV}$’ has been removed.

We need the following lemmas for the proof of our theorem.

Lemma 1 [9]

If $0<\alpha \le 1$ and $1\le v\le n$, then

$$\left|\sum _{p=0}^v{A}_{n-p}^{\alpha -1}{a}_p\right|\le \underset{1\le m\le v}{max}\left|\sum _{p=0}^m{A}_{m-p}^{\alpha -1}{a}_p\right|.$$

(12)

Lemma 2 [2]

Under the conditions on $(X_n)$, $({\beta }_n)$ and $({\lambda }_n)$, as expressed in the statement of the theorem, we have the following:

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

(13)

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

(14)

4 Proof of the theorem

Let $(T_n^{\alpha })$ be the n th $(C,\alpha )$, with $0<\alpha \le 1$, mean of the sequence $(n{a}_n{\lambda }_n)$. Then, by (2), we have

$$T_n^{\alpha }=\frac{1}{A_n^{\alpha }}\sum _{v=1}^n{A}_{n-v}^{\alpha -1}v{a}_v{\lambda }_v.$$

(15)

First, applying Abel’s transformation and then using Lemma 1, we get that

$$\begin{array}{c}{T}_n^{\alpha }=\frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}\mathrm{\Delta }{\lambda }_v\sum _{p=1}^v{A}_{n-p}^{\alpha -1}p{a}_p+\frac{{\lambda }_n}{A_n^{\alpha }}\sum _{v=1}^n{A}_{n-v}^{\alpha -1}v{a}_v,\hfill \\ \begin{array}{rl}\left|T_n^{\alpha }\right|& \le \frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}|\mathrm{\Delta }{\lambda }_v|\left|\sum _{p=1}^v{A}_{n-p}^{\alpha -1}p{a}_p\right|+\frac{|{\lambda }_n|}{A_n^{\alpha }}\left|\sum _{v=1}^n{A}_{n-v}^{\alpha -1}v{a}_v\right|\\ \le \frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}{A}_v^{\alpha }{w}_v^{\alpha }|\mathrm{\Delta }{\lambda }_v|+|{\lambda }_n|w_n^{\alpha }\\ =T_{n,1}^{\alpha }+T_{n,2}^{\alpha }.\end{array}\hfill \end{array}$$

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

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-k}{\left|{\phi }_n{T}_{n,r}^{\alpha }\right|}^k<\mathrm{\infty }\text{for }r=1,2.$$

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

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{n}^{-k}{\left|{\phi }_n{T}_{n,1}^{\alpha }\right|}^k& \le & \sum _{n=2}^{m+1}{n}^{-k}{\left(A_n^{\alpha }\right)}^{-k}{|{\phi }_n|}^k{\{\sum _{v=1}^{n-1}{A}_v^{\alpha }{w}_v^{\alpha }|\mathrm{\Delta }{\lambda }_v|\}}^k\\ \le & \sum _{n=2}^{m+1}{n}^{-k}{n}^{-\alpha k}{|{\phi }_n|}^k\sum _{v=1}^{n-1}{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{|\mathrm{\Delta }{\lambda }_v|}^k\times {\left\{\sum _{v=1}^{n-1}1\right\}}^{k-1}\\ =& O(1)\sum _{v=1}^m{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{({\beta }_v)}^k\sum _{n=v+1}^{m+1}\frac{n^{ϵ-k}{|{\phi }_n|}^k}{n^{k(\alpha -1)+ϵ+1}}\\ =& O(1)\sum _{v=1}^m{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{\beta }_v{({\beta }_v)}^{k-1}{v}^{ϵ-k}{|{\phi }_v|}^k\sum _{n=v+1}^{m+1}\frac{1}{n^{k(\alpha -1)+ϵ+1}}\\ =& O(1)\sum _{v=1}^m{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{\beta }_v{({\beta }_v)}^{k-1}{v}^{ϵ-k}{|{\phi }_v|}^k{\int }_v^{\mathrm{\infty }}\frac{dx}{x^{k(\alpha -1)+ϵ+1}}\\ =& O(1)\sum _{v=1}^m{\beta }_v{({\beta }_v)}^{k-1}{(w_v^{\alpha }|{\phi }_v|)}^k\\ =& O(1)\sum _{v=1}^m{\beta }_v{\left(\frac{1}{v{X}_v}\right)}^{k-1}{(w_v^{\alpha }|{\phi }_v|)}^k\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }(v{\beta }_v)\sum _{r=1}^v\frac{{(|{\phi }_r|w_r^{\alpha })}^k}{r^k{X}_r^{k-1}}+O(1)m{\beta }_m\sum _{v=1}^m\frac{{(|{\phi }_v|w_v^{\alpha })}^k}{v^k{X}_v^{k-1}}\\ =& O(1)\sum _{v=1}^{m-1}|\mathrm{\Delta }(v{\beta }_v)|X_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\sum _{v=1}^{m-1}|(v+1)\mathrm{\Delta }{\beta }_v-{\beta }_v|X_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\sum _{v=1}^{m-1}v|\mathrm{\Delta }{\beta }_v|X_v+O(1)\sum _{v=1}^{m-1}{\beta }_v{X}_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\text{as }m\to \mathrm{\infty }\end{array}$$

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

$$\begin{array}{rcl}\sum _{n=1}^m{n}^{-k}{\left|{\phi }_n{T}_{n,2}^{\alpha }\right|}^k& =& \sum _{n=1}^m|{\lambda }_n|{|{\lambda }_n|}^{k-1}{n}^{-k}{(w_n^{\alpha }|{\phi }_n|)}^k\\ =& O(1)\sum _{n=1}^m|{\lambda }_n|{\left(\frac{1}{X_n}\right)}^{k-1}{n}^{-k}{(w_n^{\alpha }|{\phi }_n|)}^k\\ =& O(1)\sum _{n=1}^{m-1}\mathrm{\Delta }|{\lambda }_n|\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}+O(1)|{\lambda }_m|\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}\\ =& O(1)\sum _{n=1}^{m-1}|\mathrm{\Delta }{\lambda }_n|X_n+O(1)|{\lambda }_m|X_m\\ =& O(1)\sum _{n=1}^{m-1}{\beta }_n{X}_n+O(1)|{\lambda }_m|X_m=O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take $ϵ=1$ and ${\phi }_n=n^{1-\frac{1}{k}}$ (resp. $ϵ=1$, $\alpha =1$ and ${\phi }_n=n^{1-\frac{1}{k}}$), then we get a new result dealing with ${|C,\alpha |}_k$ (resp. ${|C,1|}_k$) summability factors. Also, if we set $ϵ=1$ and ${\phi }_n=n^{\delta +1-\frac{1}{k}}$, then we get another new result concerning the ${|C,\alpha ;\delta |}_k$ summability factors. Finally, if we take $(X_n)$ as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see [10]).