a generalization of a theorem of bor

Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179 DOI 10.1186/s13660-017-1455-3

R E S E A R C H Open Access

A generalization of a theorem of BorHikmet Seyhan Özarslan* and Bagdagül Kartal

*Correspondence:[email protected] of Mathematics, ErciyesUniversity, Kayseri, 38039, Turkey

AbstractIn this paper, a general theorem concerning absolute matrix summability isestablished by applying the concepts of almost increasing and δ-quasi-monotonesequences.

MSC: 26D15; 40D15; 40F05; 40G99

Keywords: matrix transformations; almost increasing sequences; quasi-monotonesequences; Hölder inequality; Minkowski inequality

1 IntroductionA positive sequence (yn) is said to be almost increasing if there is a positive increasingsequence (un) and two positive constants K and M such that Kun ≤ yn ≤ Mun (see []).A sequence (cn) is said to be δ-quasi-monotone, if cn → , cn > ultimately and �cn ≥ –δn,where �cn = cn – cn+ and δ = (δn) is a sequence of positive numbers (see []). Let

∑an be

a given infinite series with partial sums (sn). Let T = (tnv) be a normal matrix, i.e., a lowertriangular matrix of nonzero diagonal entries. At that time T describes the sequence-to-sequence transformation, mapping the sequence s = (sn) to Ts = (Tn(s)), where

Tn(s) =n∑

v=

tnvsv, n = , , . . . ()

Let (ϕn) be any sequence of positive real numbers. The series∑

an is said to be summableϕ – |T , pn|k , k ≥ , if (see [])

∞∑

n=

ϕk–n

∣∣�Tn(s)

∣∣k < ∞, ()

where

�Tn(s) = Tn(s) – Tn–(s).

If we take ϕn = Pnpn

, then ϕ – |T , pn|k summability reduces to |T , pn|k summability (see []).If we set ϕn = n for all n, ϕ – |T , pn|k summability is the same as |T |k summability (see []).Also, if we take ϕn = Pn

pnand tnv = pv

Pn, then we get |N , pn|k summability (see []).

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Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179 of 8

2 Known resultIn [, ], Bor has established the following theorem dealing with |N , pn|k summability fac-tors of infinite series.

Theorem . Let (Yn) be an almost increasing sequence such that |�Yn| = O(Yn/n) andλn → as n → ∞. Assume that there is a sequence of numbers (Bn) such that it is δ-quasi-monotone with

∑nYnδn < ∞,

∑BnYn is convergent and |�λn| ≤ |Bn| for all n. If

m∑

n=

n

|λn| = O() as m → ∞, ()

m∑

n=

n

|zn|k = O(Ym) as m → ∞, ()

and

m∑

n=

pn

Pn|zn|k = O(Ym) as m → ∞, ()

where (zn) is the nth (C, ) mean of the sequence (nan), then the series∑

anλn is summable|N , pn|k , k ≥ .

3 Main resultThe purpose of this paper is to generalize Theorem . to the ϕ – |T , pn|k summability.Before giving main theorem, let us introduce some well-known notations. Let T = (tnv) bea normal matrix. Lower semimatrices T = (tnv) and T = (tnv) are defined as follows:

tnv =n∑

i=v

tni, n, v = , , . . . ()

and

t = t = t, tnv = tnv – tn–,v, n = , , . . . ()

Here, T and T are the well-known matrices of series-to-sequence and series-to-seriestransformations, respectively. Then we write

Tn(s) =n∑

v=

tnvsv =n∑

v=

tnvav ()

and

�Tn(s) =n∑

v=

tnvav. ()

By taking the definition of general absolute matrix summability, we established the follow-ing theorem.


Theorem . Let T = (tnv) be a positive normal matrix such that

tn = , n = , , . . . , ()

tn–,v ≥ tnv, for n ≥ v + , ()

tnn = O(

pn

Pn

)

, ()

and ( ϕnpnPn

) be a non-increasing sequence. If all conditions of Theorem . with conditions() and () are replaced by

m∑

n=

ϕk–n

(pn

Pn

)k– n

|zn|k = O(Ym) as m → ∞ ()

and

m∑

n=

ϕk–n

(pn

Pn

)k

|zn|k = O(Ym) as m → ∞, ()

then the series∑

anλn is ϕ – |T , pn|k summable, k ≥ .

We need the following lemmas for the proof of Theorem ..

Lemma . ([]) Let (Yn) be an almost increasing sequence and λn → as n → ∞. If (Bn)is δ-quasi-monotone with

∑BnYn is convergent and |�λn| ≤ |Bn| for all n, then we have

|λn|Yn = O() as n → ∞. ()

Lemma . ([]) Let (Yn) be an almost increasing sequence such that n|�Yn| = O(Yn). If(Bn) is δ-quasi monotone with

∑nYnδn < ∞, and

∑BnYn is convergent, then

nBnYn = O() as n → ∞, ()∞∑

n=

nYn|�Bn| < ∞. ()

4 Proof of Theorem 3.1Let (In) indicate the T-transform of the series

∑anλn. Then we obtain

�In =n∑

v=

tnvavλv =n∑

v=

tnvλv

vvav ()

by means of () and ().


Using Abel’s formula for (), we obtain

�In =n–∑

v=

�v

(tnvλv

v

) v∑

r=

rar +tnnλn

n

n∑

r=

rar

=n–∑

v=

v + v

�v(tnv)λvzv +n–∑

v=

v + v

tn,v+�λvzv

+n–∑

v=

tn,v+λv+zv

v+

n + n

tnnλnzn

= In, + In, + In, + In,.

For the proof of Theorem ., it suffices to prove that

∞∑

n=

ϕk–n |In,r|k < ∞

for r = , , , .By Hölder’s inequality, we have

m+∑

n=

ϕk–n |In,|k = O()

m+∑

n=

ϕk–n

( n–∑

v=

∣∣�v(tnv)

∣∣|λv||zv|

)k

= O()m+∑

n=

ϕk–n

( n–∑

v=

∣∣�v(tnv)

∣∣|λv|k|zv|k

)( n–∑

v=

∣∣�v(tnv)

∣∣

)k–

.

By () and (), we have

�v(tnv) = tnv – tn,v+

= tnv – tn–,v – tn,v+ + tn–,v+

= tnv – tn–,v. ()

Thus using (), () and ()

n–∑

v=

∣∣�v(tnv)

∣∣ =

n–∑

v=

(tn–,v – tnv) ≤ tnn.

Hence, we get

m+∑

n=


m+∑

n=

ϕk–n tk–

nn

( n–∑

v=

∣∣�v(tnv)

∣∣|λv|k|zv|k

)


by using ()

m+∑

n=


m+∑

n=

(ϕnpn

Pn

)k–( n–∑

v=

∣∣�v(tnv)

∣∣|λv|k|zv|k

)

= O()m∑

v=

|λv|k|zv|km+∑

n=v+

(ϕnpn

Pn

)k–∣∣�v(tnv)

∣∣

= O()m∑

v=

(ϕvpv

Pv

)k–

|λv|k|zv|km+∑

n=v+

∣∣�v(tnv)

∣∣.

Now, using () and (), we obtain

m+∑

n=v+

∣∣�v(tnv)

∣∣ =

m+∑

n=v+

(tn–,v – tnv) ≤ tvv.

Thus, by using Abel’s formula, we obtain

m+∑

n=


m∑

v=

(ϕvpv

Pv

)k–

|λv|k–|λv||zv|ktvv

= O()m∑

v=

ϕk–v

(pv

Pv

)k

|λv||zv|k

= O()m–∑

v=

�|λv|v∑

r=

ϕk–r

(pr

Pr

)k

|zr|k + O()|λm|m∑

v=

ϕk–v

(pv

Pv

)k

|zv|k

= O()m–∑

v=

|�λv|Yv + O()|λm|Ym

= O()m–∑

v=

|Bv|Yv + O()|λm|Ym

= O() as m → ∞,

in view of () and ().Again, using Hölder’s inequality, we have

m+∑

n=


m+∑

n=

ϕk–n

( n–∑

v=

|tn,v+||�λv||zv|)k

= O()m+∑

n=

ϕk–n

( n–∑

v=

|tn,v+||Bv||zv|k)

×( n–∑

v=

|tn,v+||Bv|)k–

.


By means of (), () and (), we have

tn,v+ = tn,v+ – tn–,v+

=n∑

i=v+

tni –n–∑

i=v+

tn–,i

= tnn +n–∑

i=v+

(tni – tn–,i)

≤ tnn.

In this way, we have

m+∑

n=


m+∑

n=

ϕk–n tk–

nn

( n–∑

v=

|tn,v+||Bv||zv|k)

= O()m+∑

n=

(ϕnpn

Pn

)k–( n–∑

v=

|tn,v+||Bv||zv|k)

= O()m∑

v=

|Bv||zv|km+∑

n=v+

(ϕnpn

Pn

)k–

|tn,v+|

= O()m∑

v=

(ϕvpv

Pv

)k–

|Bv||zv|km+∑

n=v+

|tn,v+|.

By (), (), () and (), we obtain

|tn,v+| =v∑

i=

(tn–,i – tni).

Thus, using () and (), we have

m+∑

n=v+

|tn,v+| =m+∑

n=v+

v∑

i=

(tn–,i – tni) ≤ ,

then we get

m+∑

n=


m∑

v=

ϕk–v

(pv

Pv

)k–

v|Bv| v|zv|k

= O()m–∑

v=

�(v|Bv|

) v∑

r=

ϕk–r

(pr

Pr

)k– r|zr|k

+ O()m|Bm|m∑

v=

ϕk–v

(pv

Pv

)k– v|zv|k

= O()m–∑

v=

v|�Bv|Yv + O()m–∑

v=

|Bv|Yv + O()m|Bm|Ym

= O() as m → ∞,

in view of (), () and ().


Also, we have

m+∑

n=

ϕk–n |In,|k ≤

m+∑

n=

ϕk–n

( n–∑

v=

|tn,v+||λv+| |zv|v

)k

≤m+∑

n=

ϕk–n

( n–∑

v=

|tn,v+||λv+| |zv|kv

)( n–∑

v=

|tn,v+| |λv+|v

)k–

≤m+∑

n=

ϕk–n tk–

nn

( n–∑

v=


)( n–∑

v=

|λv+|v

)k–

= O()m+∑

n=

ϕk–n

(pn

Pn

)k–( n–∑

v=


)

= O()m+∑

n=

(ϕnpn

Pn

)k–( n–∑

v=


)

= O()m∑

v=

|λv+| |zv|v

k m+∑

n=v+

(ϕnpn

Pn

)k–

|tn,v+|

= O()m∑

v=

(ϕvpv

Pv

)k–

|λv+| |zv|v

k m+∑

n=v+

|tn,v+|

= O()m∑

v=

ϕk–v

(pv

Pv

)k–

|λv+| |zv|v

k

= O()m–∑

v=

|�λv+|v∑

r=

ϕk–r

(pr

Pr

)k– r|zr|k

+ O()|λm+|m∑

v=

ϕk–v

(pv

Pv

)k– v|zv|k

= O()m–∑

v=

|Bv+|Yv+ + O()|λm+|Ym+

= O() as m → ∞,

in view of (), (), () and ().Finally, as in In,, we have

m∑

n=


m∑

n=

ϕk–n tk

nn|λn|k|zn|k

= O()m∑

n=

ϕk–n

(pn

Pn

)k

|λn|k–|λn||zn|k

= O()m∑

n=

ϕk–n

(pn

Pn

)k

|λn||zn|k = O() as m → ∞,

in view of (), () and (). Finally, the proof of Theorem . is completed.


5 CorollaryIf we take ϕn = Pn

pnand tnv = pv

Pnin Theorem ., then we get Theorem .. In this case,

conditions () and () reduce to conditions () and (), respectively. Also, the condition‘( ϕnpn

Pn) is a non-increasing sequence’ and the conditions ()-() are clearly satisfied.

6 ConclusionsIn this study, we have generalized a well-known theorem dealing with an absolute summa-bility method to a ϕ – |T , pn|k summability method of an infinite series by using almostincreasing sequences and δ-quasi-monotone sequences.

AcknowledgementsThis work was supported by Research Fund of the Erciyes University, Project Number: FBA-2014-3846.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally to the manuscript and read and approved the final manuscript.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 3 May 2017 Accepted: 18 July 2017

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a generalization of a theorem of bor

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