# Quasimonotone and Almost Increasing Sequences and Their New Applications

Authors: Hüseyin Bor
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Recently, we have proved a main theorem dealing with the absolute Nörlund summability factors of infinite series by usingδ-quasimonotone sequences. In this paper, we prove that result under weaker conditions. A new result has also been obtained.


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## Body

## 1. Introduction

A positive sequence(bn) is said to be almost increasing if there exist a positive increasing sequence (cn) and two positive constants A and B such that Acn≤bn≤Bcn (see [1]). Obviously, every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking an example, say bn=ne(-1)n. A sequence (dn) is said to be δ-quasimonotone if dn>0 ultimately and Δdn=dn-dn+1≥-δn, where δ=(δn) is a sequence of positive numbers (see [2]). Let ∑‍an be a given infinite series with the sequence of partial sums (sn) and wn=nan. By unα and tnα, we denote the nth Cesàro means of order α, with α>-1, of the sequences (sn) and (nan), respectively, that is,
(1.1)unα=1Anα∑v=0nAn-vα-1sv,(1.2)tnα=1Anα∑v=1nAn-vα-1vav,
where
(1.3)Anα=(n+αn)=(α+1)(α+2)⋯(α+n)n!=O(nα),A-nα=0forn>0.
The series ∑‍an is said to be summable |C,α|k, k≥1, if (see [3])
(1.4)∑n=1∞‍nk-1|unα-un-1α|k=∑n=1∞‍1n|tnα|k<∞.
If we take α=1, then |C,α|k summability reduces to |C,1|k summability.Let(pn) be a sequence of constants, real or complex, and let us write
(1.5)Pn=p0+p1+p2+⋯+pn≠0,(n≥0).
The sequence-to-sequence transformation
(1.6)Vn=1Pn∑v=0n‍pn-vsv
defines the sequence (Vn) of the Nörlund mean of the sequence (sn), generated by the sequence of coefficients (pn). The series ∑‍an is said to be summable |N,pn|k, k≥1, if (see [4])
(1.7)∑n=1∞‍nk-1|Vn-Vn-1|k<∞.
In the special case when
(1.8)pn=Γ(n+α)Γ(α)Γ(n+1),α≥0,
the Nörlund mean reduces to the (C,α) mean and |N,pn|k summability becomes |C,α|k summability. For pn=1, we get the (C,1) mean and then |N,pn|k summability becomes |C,1|k summability. Also, if we take k=1, then we get |N,pn| summability. For any sequence (λn), we write Δλn=λn-λn+1. Quite recently, in [5], we have proved the following theorem dealing with the absolute Nörlund summability factors of infinite series.Theorem A.
Letp0>0, pn≥0, and (pn) be a nonincreasing sequence. Let (Xn) be an almost increasing sequence such that |ΔXn|=O(Xn/n) and (λn) is a sequence such that
(1.9)|λn|Xn=O(1)asn→∞.
Suppose also that there exists a sequence of numbers (An) such that it is δ-quasimonotone with ∑‍nδnXn<∞, ∑‍AnXn is convergent, and |Δλn|≤|An| for all n. If the sequence (wnα) defined by (see [6])
(1.10)wnα={|tnα|,α=1,max1≤v≤n|tvα|,0<α<1,
satisfies the condition
(1.11)∑n=1m‍(wnα)kn=O(Xm)asm→∞,
then the series ∑‍anPnλn(n+1)-1 is summable |N,pn|k, k≥1.


## 2. The Main Results

The aim of this paper is to prove Theorem A under weaker conditions. We will prove the following theorems.Theorem 2.1.
If the sequences(Xn), (An), and (λn) are as in Theorem A and if conditions (1.9) and
(2.1)∑n=1m‍(wnα)knXnk-1=O(Xm)asm→∞
are satisfied, then the series ∑‍anλn is summable |C,α|k, 0<α≤1 and k≥1.Theorem 2.2.
Let(pn) be as in Theorem A. If the sequences (Xn), (An), and (λn) are as in Theorem A and if conditions (1.9) and (2.1) are satisfied, then the series ∑‍anPnλn(n+1)-1 is summable |N,pn|k, k≥1.Remark 2.3.
The following sequences satisfy the conditions of the theorems:(2.2)δn=1n3,An=1n2,λn=1n,Xn=nϵ,0<ϵ<1.Remark 2.4.
It should be noted that condition (2.1) is the same as condition (1.11) when k=1.  When k>1, condition (2.1) is weaker than condition (1.11), but the converse is not true. In fact, if (1.11) is satisfied, then we get that
(2.3)∑n=1m‍(wnα)knXnk-1=O(1X1k-1)∑n=1m‍(wnα)kn=O(Xm).
To show that the converse is false when k>1, the following example is sufficient. We can take Xn=nϵ, 0<ϵ<1, and then construct a sequence (an) such that
(2.4)(wnα)knXnk-1=Xn-Xn-1,
whence
(2.5)∑n=1m‍(wnα)knXnk-1=Xm=mϵ,
and so
(2.6)∑n=1m‍(wnα)kn=∑n=1m‍(Xn-Xn-1)Xnk-1=∑n=1m‍(nϵ-(n-1)ϵ)nϵ(k-1)≥ϵ∑n=1m(n-1)ϵ-1nϵ(k-1)=ϵ∑n=1m(n-1)ϵk-1~mϵkkasm→∞.
This is because vϵ-1≥nϵ-1 for n-1≤v≤n.
This shows that, whenk>1, (1.11) implies (2.1) but not conversely.
We need the following lemmas for the proof of our theorem.Lemma 2.5 (see [7]).
If0<α≤1 and 1≤v≤n, then
(2.7)|∑p=0vAn-pα-1ap|≤max1≤m≤v|∑p=0mAm-pα-1ap|.Lemma 2.6 (see [8]).
If-1<α≤β, k>1 and the series ∑‍an is summable |C,α|k, then it is also summable |C,β|k.Lemma 2.7 (see [9]).
Let(Xn) be an almost increasing sequence such that n|ΔXn|=O(Xn).
If(An) is a δ-quasimonotone with ∑‍nδnXn<∞, ∑‍AnXn is convergent, then
(2.8)nAnXn=O(1)asn→∞,∑n=1∞nXn|ΔAn|<∞.Lemma 2.8 (see [10]).
Letp0>0, pn≥0, and (pn) be a nonincreasing sequence. If the series ∑‍an is summable |C,1|k, then the series ∑‍anPn(n+1)-1 is summable |N,pn|k, k≥1.


## 3.  Proof of Theorem2.1

Let(Tnα) be the nth (C,α), with 0<α≤1, mean of the sequence (nanλn). Then, by (1.2), we have
(3.1)Tnα=1Anα∑v=1nAn-vα-1vavλv.
First applying Abel's transformation and then using Lemma 2.5, we have that
(3.2)Tnα=1Anα∑v=1n-1Δλv∑p=1vAn-pα-1pap+λnAnα∑v=1nAn-vα-1vav,(3.3)|Tnα|≤1Anα∑v=1n-1|Δλv||∑p=1vAn-pα-1pap|+|λn|Anα|∑v=1nAn-vα-1vav|≤1Anα∑v=1n-1Avαwvα|Δλv|+|λn|wnα=Tn,1α+Tn,2α.
To complete the proof of Theorem 2.1, by Minkowski's inequality, it is sufficient to show that
(3.4)∑n=1∞n-1|Tn,rα|k<∞forr=1,2.
Whenever k>1, we can apply Hölder's inequality with indices k and k', where (1/k)+(1/k′)=1, we get that
(3.5)∑n=2m+1n-1|Tn,1α|k≤∑n=2m+1n-1(Anα)-k{∑v=1mAvαwvα|Δλv|}k≤∑n=2m+1n-1-αk{∑v=1n-1vαk(wvα)k|Av|k}×{∑v=1n-11}k-1=O(1)∑v=1mvαk(wvα)k|Av||Av|k-1∑n=v+1m+11n2+(α-1)k=O(1)∑v=1mvαk(wvα)k|Av|1(vXv)k-1∫v∞dxx2+(α-1)k=O(1)∑v=1mv|Av|(wvα)kvXvk-1=O(1)∑v=1m-1Δ(v|Av|)∑r=1v(wrα)krXrk-1+O(1)m|Am|∑v=1m(wvα)kvXvk-1=O(1)∑v=1m-1|Δ(v|Av|)|Xv+O(1)m|Am|Xm=O(1)∑v=1m-1v|ΔAv|Xv+O(1)∑v=1m|Av|Xv+O(1)m|Am|Xm=O(1)asm→∞,
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.7. Again, we have that
(3.6)∑n=1m‍n-1|Tn,2α|k=O(1)∑n=1m|λn||λn|k-1(wnα)kn=O(1)∑n=1m|λn|(wnα)knXnk-1=O(1)∑n=1m-1Δ|λn|∑v=1n(wvα)kvXvk-1+O(1)|λm|∑n=1m(wnα)knXnk-1=O(1)∑n=1m-1|Δλn|Xn+O(1)|λm|Xm=O(1)∑n=1m-1|An|Xn+O(1)|λm|Xm=O(1)asm→∞,
by virtue of the hypotheses of Theorem 2.1. This completes the proof of Theorem 2.1. If we take α=1, then we get a new result dealing with |C,1|k summability factors.Proof of Theorem2.2.
In order to prove Theorem2.2, we need to consider only the special case in which (N,pn) is (C,α). Therefore, Theorem 2.2 will then follow by means of Theorem 2.1, Lemma 2.6 (for β=1), and Lemma 2.8. If we take α=1, then we get a new result for the absolute Nörlund summability factors of infinite series.



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*Source: 793548-2012-10-17.xml*