KYUNGPOOK Math. J. 2019; 59(2): 233-240
An Application of Absolute Matrix Summability using Almost Increasing and δ-quasi-monotone Sequences
Hikmet Seyhan Özarslan
Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
e-mail : seyhan@erciyes.edu.tr and hseyhan38@gmail.com
Received: October 31, 2017; Revised: August 9, 2018; Accepted: August 13, 2018; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In the present paper, absolute matrix summability of infinite series is studied. A new theorem concerning absolute matrix summability factors, which generalizes a known theorem dealing with absolute Riesz summability factors of infinite series, is proved using almost increasing and δ-quasi-monotone sequences. Also, a result dealing with absolute Cesàro summability is given.

Keywords: summability factors, almost increasing sequences, absolute matrix summability, quasi-monotone sequences, inﬁnite series, Ho¨lder inequality, Minkowski inequality.
1. Introduction

A positive sequence (vn) is said to be almost increasing if there exists a positive increasing sequence (cn) and two positive constants K and L such that KcnvnLcn (see ). A sequence (yn) is said to be δ-quasi-monotone, if yn → 0, yn > 0 ultimately and Δyn ≥ −δn, where Δyn=yn – yn+1 and δ = (δn) is a sequence of positive numbers (see ). Let ∑an be a given infinite series with partial sums (sn). By (un) and (tn) we denote the n-th (C, 1) means of the sequences (sn) and (nan), respectively. The series ∑an is said to be |C, 1|k summable, k ≥ 1, if (see , )

$∑n=1∞nk-1∣un-un-1∣k=∑n=1∞1n∣tn∣k<∞.$

Let (pn) be a sequence of positive numbers such that

$Pn=∑v=0npv→∞ as n→∞, (P-i=p-i=0, i≥1).$

The sequence-to-sequence transformation

$zn=1Pn∑v=0npvsv$

defines the sequence (zn) of the Riesz mean of the sequence (sn), generated by the sequence of coefficients (pn) (see ). The series ∑an is said to be |N̄, pn|k summable, k ≥ 1, if (see )

$∑n=1∞(Pnpn)k-1∣Δzn-1∣k<∞,$

where

$Δzn-1=-pnPnPn-1∑v=1nPv-1av, n≥1.$

Let A = (anv) 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 = (sn) to As = (An(s)), where

$An(s)=∑i=0nanisi, n=0,1,…$

The series ∑an is said to be |A, pn|k summable, k ≥ 1, if (see )

$∑n=1∞(Pnpn)k-1∣Δ¯An(s)∣k<∞,$

where

$Δ¯An(s)=An(s)-An-1(s).$

When we take $anv=pvPn$, then |A, pn|k summability is the same as |N̄, pn|k summability. Also, when we take $anv=pvPn$ and pn = 1 for all values of n, |A, pn|k reduces to |C, 1|k summability.

Let A = (anv) be a normal matrix. Lower semimatrices Ā = (ānv) and Â = (ânv) are defined as follows:

$a¯nv=∑i=vnani, n,v=0,1,…$

and

$a^00=a¯00=a00, a^nv=a¯nv-a¯n-1,v, n=1,2,…$

Ā and Â are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we write

$An (s)=∑i=0nanisi=∑i=0na¯niai$

and

$Δ¯An (s)=∑i=0na^niai.$
2. Known Result

In [4, 5], the following theorem dealing with |N̄, pn|k summability factors of infinite series has been proved by Bor.

Theorem 2.1

Let (Xn) be an almost increasing sequence such thatXn| = O(Xn/n) and λn → 0 as n → ∞. Suppose that there exists a sequence of numbers (An) such that it is δ-quasi-monotone withnXnδn < ∞, ∑AnXn is convergent andλn| ≤ |An| for all n. If

$∑n=1m1n∣λn∣=O(1) as m→∞,$$∑n=1m1n∣tn∣k=O(Xm) as m→∞,$

and

$∑n=1mpnPn∣tn∣k=O(Xm) as m→∞,$

then the seriesanλn is |N̄, pn|k summable, k ≥ 1.

3. Main Result

The aim of this paper is to prove following more general theorem dealing with |A, pn|k summability.

Theorem 3.1

Let A = (anv) be a positive normal matrix such that

$a¯n0=1, n=0,1,…,$$an-1,v≥anv, for n≥v+1,$$ann=O (pnPn).$

If all conditions of Theorem 2.1 are satisfied, then the seriesanλn is |A, pn|k summable, k ≥ 1.

Lemma 3.2.()

Under the conditions of Theorem 3.1, we have

$∣λn∣Xn=O (1) as n→∞.$

Lemma 3.3.()

Let (Xn) be an almost increasing sequence such that nXn |= O (Xn). If (An) is a δ-quasi monotone withnXnδn < ∞, andAnXn is convergent, then

$nAnXn=O (1) as n→∞,$$∑n=1∞nXn∣ΔAn∣<∞.$
4. Proof of Theorem 3.1

Let (Mn) denotes A-transform of the series ∑anλn. Then, by (1.10) and (1.11), we have

$Δ¯Mn=∑v=0na^nvavλv=∑v=1na^nvλvvvav.$

Applying Abel’s transformation to above sum, we get

$Δ¯Mn=∑v=1n-1Δv (a^nvλvv)∑r=1vrar+a^nnλnn∑r=1nrar=n+1nannλntn+∑v=1n-1v+1vΔv (a^nv) λvtv+∑v=1n-1v+1va^n,v+1Δλvtv+∑v=1n-1a^n,v+1λv+1tvv=Mn,1+Mn,2+Mn,3+Mn,4.$

To prove Theorem 3.1, we will show that

$∑n=1∞(Pnpn)k-1∣Mn,r∣k<∞, for r=1,2,3,4.$

First, by using (2.3), (3.3) and (3.4), we have

$∑n=1m(Pnpn)k-1∣Mn,1∣k=∑n=1m(Pnpn)k-1|n+1nannλntn|k=O(1)∑n=1m(Pnpn)k-1annk∣λn∣k∣tn∣k=O(1)∑n=1mpnPn∣λn∣k-1∣λn∣∣tn∣k=O(1)∑n=1mpnPn∣λn∣∣tn∣k=O(1)∑n=1m-1Δ∣λn∣∑r=1nprPr∣tr∣k+O(1)∣λm∣∑n=1mpnPn∣tn∣k=O(1)∑n=1m-1∣Δλn∣Xn+O(1)∣λm∣Xm=O(1)∑n=1m-1∣An∣Xn+O(1)∣λm∣Xm=O(1) as m→∞.$

Now, as in Mn,1, we have

$∑n=2m+1(Pnpn)k-1∣Mn,2∣k=O(1)∑n=2m+1(Pnpn)k-1(∑v=1n-1∣Δv(a^nv)∣∣λv∣∣tv∣)k=O(1)∑n=2m+1(Pnpn)k-1(∑v=1n-1∣Δv(a^nv)∣ ∣λv∣k ∣tv∣k)×(∑v=1n-1∣Δv(a^nv)∣)k-1.$

Since

$Δv(a^nv)=a^nv-a^n,v+1=a¯nv-a¯n-1,v-a¯n,v+1+a¯n-1,v+1=anv-an-1,v$

by (1.8) and (1.9), we have

$∑v=1n-1∣Δv(a^nv)∣=∑v=1n-1(an-1,v-anv)≤ann$

by using (1.8), (3.1) and (3.2). Hence, we get

$∑n=2m+1(Pnpn)k-1∣Mn,2∣k=O(1)∑v=1m∣λv∣k-1∣λv∣∣tv∣k∑n=v+1m+1(Pnpn)k-1annk-1∣Δv(a^nv)∣=O(1)∑v=1m∣λv∣ ∣tv∣k∑n=v+1m+1∣Δv(a^nv)∣.$

Now, using (3.2) and (4.1), we obtain

$∑n=v+1m+1∣Δv(a^nv)∣=∑n=v+1m+1(an-1,v-anv)≤avv,$

then

$∑n=2m+1(Pnpn)k-1∣Mn,2∣k=O(1)∑v=1m∣λv∣ ∣tv∣kavv=O(1)∑v=1m∣λv∣ ∣tv∣kpvPv=O(1) as m→∞,$

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. Also, we have

$∑n=2m+1(Pnpn)k-1∣Mn,3∣k=∑n=2m+1(Pnpn)k-1|∑v=1n-1v+1va^n,v+1Δλvtv|k=O(1)∑n=2m+1(Pnpn)k-1(∑v=1n-1∣a^n,v+1∣∣Δλv∣∣tv∣)k=O(1)∑n=2m+1(Pnpn)k-1(∑v=1n-1∣a^n,v+1∣∣Av∣∣tv∣)k=O(1)∑n=2m+1(Pnpn)k-1(∑v=1n-1∣a^n,v+1∣∣Av∣ ∣tv∣k)×(∑v=1n-1∣a^n,v+1∣∣Av∣)k-1=O(1)∑v=1m∣Av∣ ∣tv∣k∑n=v+1m+1∣a^n,v+1∣.$

By (1.8), (1.9), (3.1) and (3.2), we obtain

$∣a^n,v+1∣=∑i=0v(an-1,i-ani).$

Thus, using (1.8) and (3.1), we have

$∑n=v+1m+1∣a^n,v+1∣=∑n=v+1m+1∑i=0v(an-1,i-ani)≤1,$

then we get

$∑n=2m+1(Pnpn)k-1∣Mn,3∣k=O(1)∑v=1m∣Av∣∣tv∣k=O(1)∑v=1mv∣Av∣1v∣tv∣k=O(1)∑v=1m-1Δ(v∣Av∣)∑r=1v1r∣tr∣k+O(1)m∣Am∣∑v=1m1v∣tv∣k=O(1)∑v=1m-1v∣ΔAv∣Xv+O(1)∑v=1m-1∣Av∣Xv+O(1)m∣Am∣Xm=O(1) as m→∞,$

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.3. Again, operating Hölder’s inequality, we have

$∑n=2m+1(Pnpn)k-1∣Mn,4∣k≤∑n=2m+1(Pnpn)k-1(∑v=1n-1∣a^n,v+1∣∣λv+1∣∣tv∣v)k≤∑n=2m+1(Pnpn)k-1(∑v=1n-1∣a^n,v+1∣∣λv+1∣∣tv∣kv)×(∑v=1n-1∣a^n,v+1∣∣λv+1∣v)k-1=O(1)∑v=1m∣λv+1∣v∣tv∣k∑n=v+1m+1∣a^n,v+1∣=O(1)∑v=1m∣λv+1∣v∣tv∣k=O(1)∑v=1m-1Δ∣λv+1∣∑r=1v1r∣tr∣k+O(1)∣λm+1∣∑v=1m1v∣tv∣k=O(1)∑v=1m-1∣Δλv+1∣Xv+1+O(1)∣λm+1∣Xm+1=O(1)∑v=1m-1∣Av+1∣Xv+1+O(1)∣λm+1∣Xm+1=O(1) as m→∞,$

by (2.1), (2.2), (3.3) and (3.4). This completes the proof of Theorem 3.1.

If we take $anv=pvPn$ in this theorem, then we get Theorem 2.1. If we take $anv=pvPn$ and pn = 1 for all values of n, then we get a result for |C, 1|k summability.

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