A new application of quasi power increasing sequences. I

Hüseyin Bor

doi:10.1186/1029-242X-2013-69

Research
Open Access

A new application of quasi power increasing sequences. I

Hüseyin Bor¹Email author

Journal of Inequalities and Applications20132013:69

https://doi.org/10.1186/1029-242X-2013-69

Received: 22 December 2012
Accepted: 9 February 2013
Published: 26 February 2013

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.

Keywords

Hölder’s inequality
sequence spaces
absolute summability
increasing sequences

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} \leq b_{n} \leq B c_{n}

(see [1]). A sequence

(λ_{n})

is said to be of bounded variation, denoted by

(λ_{n}) \in BV

, if

\sum_{n = 1}^{\infty} | Δ λ_{n} | = \sum_{n = 1}^{\infty} | λ_{n} - λ_{n + 1} | < \infty

. A positive sequence

X = (X_{n})

is said to be a quasi-σ-power increasing sequence if there exists a constant

K = K (σ, X) \geq 1

such that

K n^{σ} X_{n} \geq m^{σ} X_{m}

holds for all

n \geq m \geq 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^{- σ}

for

σ > 0

(see [2]). Let

(φ_{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}^{α}

and

t_{n}^{α}

the n th Cesàro means of order α, with

α > - 1

, of the sequences

(s_{n})

and

(n a_{n})

, respectively, that is,

z_{n}^{α} = \frac{1}{A_{n}^{α}} \sum_{v = 0}^{n} A_{n - v}^{α - 1} s_{v},

(1)

t_{n}^{α} = \frac{1}{A_{n}^{α}} \sum_{v = 1}^{n} A_{n - v}^{α - 1} v a_{v},

(2)

where

A_{n}^{α} = (\binom{n + α}{n}) = \frac{(α + 1) (α + 2) \dots (α + n)}{n!} = O (n^{α}), A_{- n}^{α} = 0 for n > 0 .

(3)

The series

\sum a_{n}

is said to be summable

φ - {| C, α |}_{k}

,

k \geq 1

and

α > - 1

, if (see [3])

\sum_{n = 1}^{\infty} {| φ_{n} (z_{n}^{α} - z_{n - 1}^{α}) |}^{k} = \sum_{n = 1}^{\infty} n^{- k} {| φ_{n} t_{n}^{α} |}^{k} < \infty .

(4)

In the special case, if we take $φ_{n} = n^{1 - \frac{1}{k}}$ , then $φ - {| C, α |}_{k}$ summability is the same as ${| C, α |}_{k}$ summability (see [4]). Also, if we take $φ_{n} = n^{δ + 1 - \frac{1}{k}}$ , then $φ - {| C, α |}_{k}$ summability reduces to ${| C, α; δ |}_{k}$ summability (see [5]).

2 Known result

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

Theorem A Let

(λ_{n}) \in BV

and let

(X_{n})

be a quasi-σ-power increasing sequence for some σ (

0 < σ < 1

). Suppose also that there exist sequences

(β_{n})

and

(λ_{n})

such that

| Δ λ_{n} | \leq β_{n},

(5)

β_{n} \to 0 as n \to \infty,

(6)

\sum_{n = 1}^{\infty} n | Δ β_{n} | X_{n} < \infty,

(7)

| λ_{n} | X_{n} = O (1) as n \to \infty .

(8)

If there exists an

ϵ > 0

such that the sequence

(n^{ϵ - k} {| φ_{n} |}^{k})

is non-increasing and if the sequence

(w_{n}^{α})

defined by (see [7])

w_{n}^{α} = {\begin{matrix} | t_{n}^{α} |, & α = 1, \\ {max}_{1 \leq v \leq n} | t_{v}^{α} |, & 0 < α < 1, \end{matrix}

(9)

satisfies the condition

\sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k}} = O (X_{m}) as m \to \infty,

(10)

then the series $\sum a_{n} λ_{n}$ is summable $φ - {| C, α |}_{k}$ , $k \geq 1$ , $0 < α \leq 1$ and $k α + ϵ > 1$ .

It should be remarked that we have added the condition ‘ $(λ_{n}) \in BV$ ’ 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 < σ < 1

). If there exists an

ϵ > 0

such that the sequence

(n^{ϵ - k} {| φ_{n} |}^{k})

is non-increasing and if the conditions from (5) to (8) are satisfied and if the condition

\sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k} X_{n}^{k - 1}} = O (X_{m}) as m \to \infty

(11)

is satisfied, then the series $\sum a_{n} λ_{n}$ is summable $φ - {| C, α |}_{k}$ , $k \geq 1$ , $0 < α \leq 1$ , and $k (α - 1) + ϵ > 1$ .

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{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k} X_{n}^{k - 1}} = O (\frac{1}{X_{1}^{k - 1}}) \sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k}} = O (X_{m}) .

If (11) is satisfied, then for

k > 1

we obtain that

\sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k}} = \sum_{n = 1}^{m} X_{n}^{k - 1} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k} X_{n}^{k - 1}} = O (X_{m}^{k - 1}) \sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k} X_{n}^{k - 1}} = O (X_{m}^{k}) \neq O (X_{m}) .

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

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

Lemma 1 [9]

If

0 < α \leq 1

and

1 \leq v \leq n

, then

| \sum_{p = 0}^{v} A_{n - p}^{α - 1} a_{p} | \leq max_{1 \leq m \leq v} | \sum_{p = 0}^{m} A_{m - p}^{α - 1} a_{p} | .

(12)

Lemma 2 [2]

Under the conditions on

(X_{n})

,

(β_{n})

and

(λ_{n})

, as expressed in the statement of the theorem, we have the following:

n β_{n} X_{n} = O (1) as n \to \infty,

(13)

\sum_{n = 1}^{\infty} β_{n} X_{n} < \infty .

(14)

4 Proof of the theorem

Let

(T_{n}^{α})

be the n th

(C, α)

, with

0 < α \leq 1

, mean of the sequence

(n a_{n} λ_{n})

. Then, by (2), we have

T_{n}^{α} = \frac{1}{A_{n}^{α}} \sum_{v = 1}^{n} A_{n - v}^{α - 1} v a_{v} λ_{v} .

(15)

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

\begin{matrix} T_{n}^{α} = \frac{1}{A_{n}^{α}} \sum_{v = 1}^{n - 1} Δ λ_{v} \sum_{p = 1}^{v} A_{n - p}^{α - 1} p a_{p} + \frac{λ_{n}}{A_{n}^{α}} \sum_{v = 1}^{n} A_{n - v}^{α - 1} v a_{v}, \\ \begin{aligned} | T_{n}^{α} | & \leq \frac{1}{A_{n}^{α}} \sum_{v = 1}^{n - 1} | Δ λ_{v} | | \sum_{p = 1}^{v} A_{n - p}^{α - 1} p a_{p} | + \frac{| λ_{n} |}{A_{n}^{α}} | \sum_{v = 1}^{n} A_{n - v}^{α - 1} v a_{v} | \\ \leq \frac{1}{A_{n}^{α}} \sum_{v = 1}^{n - 1} A_{v}^{α} w_{v}^{α} | Δ λ_{v} | + | λ_{n} | w_{n}^{α} \\ = T_{n, 1}^{α} + T_{n, 2}^{α} . \end{aligned} \end{matrix}

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

\sum_{n = 1}^{\infty} n^{- k} {| φ_{n} T_{n, r}^{α} |}^{k} < \infty for r = 1, 2 .

Now, when

k > 1

, applying Hölder’s inequality with indices k and

k^{'}

, where

\frac{1}{k} + \frac{1}{k^{'}} = 1

, we get that

\begin{array}{rcl} \sum_{n = 2}^{m + 1} n^{- k} {| φ_{n} T_{n, 1}^{α} |}^{k} & \leq & \sum_{n = 2}^{m + 1} n^{- k} {(A_{n}^{α})}^{- k} {| φ_{n} |}^{k} {\sum_{v = 1}^{n - 1} A_{v}^{α} w_{v}^{α} | Δ λ_{v} |}^{k} \\ \leq & \sum_{n = 2}^{m + 1} n^{- k} n^{- α k} {| φ_{n} |}^{k} \sum_{v = 1}^{n - 1} v^{α k} {(w_{v}^{α})}^{k} {| Δ λ_{v} |}^{k} \times {\sum_{v = 1}^{n - 1} 1}^{k - 1} \\ = & O (1) \sum_{v = 1}^{m} v^{α k} {(w_{v}^{α})}^{k} {(β_{v})}^{k} \sum_{n = v + 1}^{m + 1} \frac{n^{ϵ - k} {| φ_{n} |}^{k}}{n^{k (α - 1) + ϵ + 1}} \\ = & O (1) \sum_{v = 1}^{m} v^{α k} {(w_{v}^{α})}^{k} β_{v} {(β_{v})}^{k - 1} v^{ϵ - k} {| φ_{v} |}^{k} \sum_{n = v + 1}^{m + 1} \frac{1}{n^{k (α - 1) + ϵ + 1}} \\ = & O (1) \sum_{v = 1}^{m} v^{α k} {(w_{v}^{α})}^{k} β_{v} {(β_{v})}^{k - 1} v^{ϵ - k} {| φ_{v} |}^{k} \int_{v}^{\infty} \frac{d x}{x^{k (α - 1) + ϵ + 1}} \\ = & O (1) \sum_{v = 1}^{m} β_{v} {(β_{v})}^{k - 1} {(w_{v}^{α} | φ_{v} |)}^{k} \\ = & O (1) \sum_{v = 1}^{m} β_{v} {(\frac{1}{v X_{v}})}^{k - 1} {(w_{v}^{α} | φ_{v} |)}^{k} \\ = & O (1) \sum_{v = 1}^{m - 1} Δ (v β_{v}) \sum_{r = 1}^{v} \frac{{(| φ_{r} | w_{r}^{α})}^{k}}{r^{k} X_{r}^{k - 1}} + O (1) m β_{m} \sum_{v = 1}^{m} \frac{{(| φ_{v} | w_{v}^{α})}^{k}}{v^{k} X_{v}^{k - 1}} \\ = & O (1) \sum_{v = 1}^{m - 1} | Δ (v β_{v}) | X_{v} + O (1) m β_{m} X_{m} \\ = & O (1) \sum_{v = 1}^{m - 1} | (v + 1) Δ β_{v} - β_{v} | X_{v} + O (1) m β_{m} X_{m} \\ = & O (1) \sum_{v = 1}^{m - 1} v | Δ β_{v} | X_{v} + O (1) \sum_{v = 1}^{m - 1} β_{v} X_{v} + O (1) m β_{m} X_{m} \\ = & O (1) as m \to \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} {| φ_{n} T_{n, 2}^{α} |}^{k} & = & \sum_{n = 1}^{m} | λ_{n} | {| λ_{n} |}^{k - 1} n^{- k} {(w_{n}^{α} | φ_{n} |)}^{k} \\ = & O (1) \sum_{n = 1}^{m} | λ_{n} | {(\frac{1}{X_{n}})}^{k - 1} n^{- k} {(w_{n}^{α} | φ_{n} |)}^{k} \\ = & O (1) \sum_{n = 1}^{m - 1} Δ | λ_{n} | \sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k} X_{n}^{k - 1}} + O (1) | λ_{m} | \sum_{n = 1}^{m} \frac{{(| φ_{n} | w_{n}^{α})}^{k}}{n^{k} X_{n}^{k - 1}} \\ = & O (1) \sum_{n = 1}^{m - 1} | Δ λ_{n} | X_{n} + O (1) | λ_{m} | X_{m} \\ = & O (1) \sum_{n = 1}^{m - 1} β_{n} X_{n} + O (1) | λ_{m} | X_{m} = O (1) as m \to \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 $φ_{n} = n^{1 - \frac{1}{k}}$ (resp. $ϵ = 1$ , $α = 1$ and $φ_{n} = n^{1 - \frac{1}{k}}$ ), then we get a new result dealing with ${| C, α |}_{k}$ (resp. ${| C, 1 |}_{k}$ ) summability factors. Also, if we set $ϵ = 1$ and $φ_{n} = n^{δ + 1 - \frac{1}{k}}$ , then we get another new result concerning the ${| C, α; δ |}_{k}$ 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]).

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)

P.O. Box 121, Bahçelievler, Ankara, TR-06502, Turkey

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