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 : and
Received: October 31, 2017; Revised: August 9, 2018; Accepted: August 13, 2018; Published online: June 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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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 [1]). 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 [2]). 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 [6], [8])


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

Pn=v=0npv         as         n,         (P-i=p-i=0,         i1).

The sequence-to-sequence transformation


defines the sequence (zn) of the Riesz mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [7]). The series ∑an is said to be |N̄, pn|k summable, k ≥ 1, if (see [3])



Δzn-1=-pnPnPn-1v=1nPv-1av,         n1.

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 [9])




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  = (ânv) are defined as follows:

a¯nv=i=vnani,         n,v=0,1,


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

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



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=1m1nλn=O(1)         as         m,n=1m1ntnk=O(Xm)         as         m,


n=1mpnPntnk=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,vanv,         for         nv+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.([4])

Under the conditions of Theorem 3.1, we have

λnXn=O(1)         as         n.

Lemma 3.3.([5])

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=1nXnΔAn<.
4. Proof of Theorem 3.1

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


Applying Abel’s transformation to above sum, we get


To prove Theorem 3.1, we will show that

n=1(Pnpn)k-1Mn,rk<,         for         r=1,2,3,4.

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

n=1m(Pnpn)k-1Mn,1k=n=1m(Pnpn)k-1|n+1nannλntn|k=O(1)n=1m(Pnpn)k-1annkλnktnk=O(1)n=1mpnPnλnk-1λntnk=O(1)n=1mpnPnλntnk=O(1)n=1m-1Δλnr=1nprPrtrk+O(1)λmn=1mpnPntnk=O(1)n=1m-1ΔλnXn+O(1)λmXm=O(1)n=1m-1AnXn+O(1)λmXm=O(1)         as         m.

Now, as in Mn,1, we have




by (1.8) and (1.9), we have


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


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



n=2m+1(Pnpn)k-1Mn,2k=O(1)v=1mλvtvkavv=O(1)v=1mλvtvkpvPv=O(1)         as         m,

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


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


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


then we get

n=2m+1(Pnpn)k-1Mn,3k=O(1)v=1mAvtvk=O(1)v=1mvAv1vtvk=O(1)v=1m-1Δ(vAv)r=1v1rtrk+O(1)mAmv=1m1vtvk=O(1)v=1m-1vΔAvXv+O(1)v=1m-1AvXv+O(1)mAmXm=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-1Mn,4kn=2m+1(Pnpn)k-1(v=1n-1a^n,v+1λv+1tvv)kn=2m+1(Pnpn)k-1(v=1n-1a^n,v+1λv+1tvkv)×(v=1n-1a^n,v+1λv+1v)k-1=O(1)v=1mλv+1vtvkn=v+1m+1a^n,v+1=O(1)v=1mλv+1vtvk=O(1)v=1m-1Δλv+1r=1v1rtrk+O(1)λm+1v=1m1vtvk=O(1)v=1m-1Δλv+1Xv+1+O(1)λm+1Xm+1=O(1)v=1m-1Av+1Xv+1+O(1)λm+1Xm+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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