A summability factor theorem involving an almost increasing sequence for generalized absolute summability

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A summability factor theorem involving an almost increasing sequence for generalized absolute summability

Ekrem Savas¸

Istanbul Ticaret University, Department of Mathematics, Uskudar, Istanbul, Turkey Received 17 July 2006; received in revised form 28 October 2006; accepted 12 December 2006

Abstract

In this paper we prove a general theorem on | A;δ|k-summability factors of infinite series under suitable conditions by using an almost increasing sequence, where A is a lower triangular matrix with non-negative entries. Also, we deduce a similar result for the weighted mean method.

c

2007 Elsevier Ltd. All rights reserved.

Keywords:Absolute summability; Almost increasing summability factors

Let A be a lower triangular matrix, {sn}a sequence. Then A_n:=

n

X

ν=0

a_nνs_ν.

A seriesP a_n, with partial sums {s_n}, is said to be summable | A|_k, k ≥ 1 if

∞

X

n=1

n^k−1|An−An−1|^k< ∞, (1)

and it is said to be summable | A, δ|k, k ≥ 1 and δ ≥ 0 if [2]

∞

X

n=1

n^δk+k−1|A_n−An−1|^k < ∞. (2)

We may associate with A two lower triangular matrices A and ˆAdefined as follows:

a¯_nν =

n

X

r =ν

a_nr, ν = 0, 1, 2, . . . , n and n = 0, 1, 2, . . . ,

E-mail address:ekremsavas@yahoo.com.

0898-1221/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.camwa.2006.12.082

and

aˆnν = ¯anν− ¯an−1,ν, n = 1, 2, 3, . . . .

A positive sequence {b_n}is said to be almost increasing if there exists an increasing sequence {c_n}and two positive constants A and B such that Ac_n≤b_n≤Bc_nfor each n.

Given any sequence {x_n}, the notation x_n O(1) means xn= O(1) and 1/xn=O(1). For any matrix entry anν,

∆_νanν :=anν−anν+1.

Quite recently, Rhoades and Savas [5] proved the following theorem for | A;δ|k-summability factors of infinite series.

Theorem 1. Let {X_n}be an almost increasing sequence and let {βn}and {λn}be sequences such that (i) |∆λn| ≤βn,

(ii) limβn=0, (iii) P∞

n=1n|∆βn|X_n< ∞, and (iv) |λn|X_n=O(1).

Let A be a lower triangular matrix with non-negative entries satisfying (v) na_nnO(1),

(vi) a_n−1,ν≥anνfor n ≥ν + 1, (vii) ¯an0=1 for all n,

(viii) Pn−1

ν=1a_ννaˆ_nν+1=O(ann), (ix) Pm+1

n=ν+1n^δk|∆_νaˆ_nν| =O ν^δka_νν
and (x) Pm+1

n=ν+1n^δkaˆ_nν+1=O ν^δk
.

If (xi)Pm

n=1n^δk−1|t_n|^k =O(Xm), where tn:= ¹

n+1

Pn k=1ka_k, then the seriesP a_nλnis summable | A, δ|k, k ≥ 1, 0 ≤ δ < 1/k.

In this paper, we shall prove the above theorem in a more general form as follows. Now we have

Theorem 2. Let {X_n} be an almost increasing sequence and {βn} and {λn} be sequences such that the conditions(i)–(iv) ofTheorem1and the following condition are satisfied,

(viii) P∞ n=1

|λn| n < ∞.

Let A be a triangle satisfying conditions(v)–(vii) and (ix)–(xi) of Theorem1, then the seriesP anλnis summable

|A, δ|k, k ≥ 1, 0 ≤ δ < 1/k.

The following lemma is pertinent for the proof ofTheorem 2.

Lemma 1 ([3]). Under the conditions(ii) and (iii) on the sequences {Xn}, {βn}and {λn}as stated inTheorem2, the following conditions hold,

(1) nβnX_n=O(1), and (2) P∞

n=1βnX_n< ∞.

Proof. Let(yn) be the nth term of the A-transform of the partial sums of Pⁿ_{i =0}λia_i. Then,

y_n :=

n

X

i =0

a_nis_i =

n

X

i =0

a_ni

i

X

ν=0

λνa_ν

=

n

X

ν=0

λνa_ν

n

X

i =ν

a_ni =

n

X

ν=0

a¯_nνλνa_ν

and

Y_n :=y_n−y_n−1=

n

X

ν=0

(¯anν− ¯a_n−1,ν)λ_νa_ν=

n

X

ν=0

aˆ_nνλ_νa_ν. (3)

We may write (note that (vii) implies that ˆa_n0=0):

Yn =

n

X

ν=1

 ˆa_nνλ_ν ν

 νa_ν

=

n

X

ν=1

 ˆanνλν

ν

" _ν X

r =1

r a_r −

ν−1X

r =1

r a_r

#

=

n−1

X

ν=1

∆_ν ˆa_nνλ_ν ν

 _ν X

r =1

r a_r +aˆ_nnλn

n

n

X

r =1

r a_r

=

n−1

X

ν=1

(∆_νaˆnν)λ_νν + 1 ν t_ν+

n−1

X

ν=1

aˆn,ν+1(∆λ_ν)ν + 1 ν t_ν+

n−1

X

ν=1

aˆn,ν+1λ_ν+11

νt_ν+(n + 1)annλnt_n n

=T_n1+T_n2+T_n3+T_n4, say.

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

∞

X

n=1

n^δk+k−1|T_nr|^k< ∞, for r = 1, 2, 3, 4.

From the definition of ˆAand using (vi) and (vii), aˆn,ν+1 = ¯an,ν+1− ¯an−1,ν+1

=

n

X

i =ν+1

a_ni−

n−1

X

i =ν+1

a_n−1,i

=1 − Xν i =0

a_ni−1 + Xν i =0

a_n−1,i

= Xν i =0

a_n−1,i−a_n,i
≥ 0. (4)

Using H¨older’s inequality,

I₁ :=

m

X

n=1

n^δk+k−1|T_n1|^k =

m

X

n=1

n^δk+k−1     

n−1

X

ν=1

∆_νaˆ_nνλ_νν + 1 ν t_ν

k

= O(1)

m+1

X

n=1

n^δk+k−1

n−1

X

ν=1

|∆_νaˆ_nν||λ_ν||t_ν|

!k

= O(1)

m+1

X

n=1

n^δk+k−1

n−1

X

ν=1

|∆_νaˆ_nν||λ_ν|^k|t_ν|^k

! _n−1 X

ν=1

|∆_νaˆ_nν|

!k−1

.

Also,

∆_νaˆnν = ˆanν − ˆan,ν+1

= ¯a_nν − ¯a_n−1,ν− ¯a_n,ν+1+ ¯a_n−1,ν+1

=anν −an−1,ν≤0.

Thus, using (vii),

n−1

X

ν=0

|∆_νaˆnν| =

n−1

X

ν=0

(an−1,ν−anν) = 1 − 1 + ann=ann.

Since {X_n}is an almost increasing sequence, condition (iv) implies that {λn}is bounded. Then, using (v), (ix),(xi), and (i) and condition (2) ofLemma 1,

I1= O(1)^m+1X

n=1

n^δk(nann)^k−1n−1X

ν=1

|λ_ν|^k|t_ν|^k|∆_νaˆnν|

= O(1)

m+1

X

n=1

n^δk

n−1

X

ν=1

|λ_ν|^k−1|λ_ν||∆_νaˆ_nν||t_ν|^k

!

= O(1)

m

X

ν=1

|λν||t_ν|^k

m+1

X

n=ν+1

n^δk|∆_νaˆ_nν|

= O(1)

m

X

ν=1

ν^δk|λν|a_νν|t_ν|^k

= O(1)X^m

ν=1

|λ_ν|

" _ν X

r =1

arr|tr|^kr^δk−

ν−1X

r =1

arr|tr|^kr^δk

#

= O(1)

" _m X

ν=1

|λ_ν| Xν r =1

a_rr|t_r|^kr^δk−

m−1

X

ν=0

|λ_ν+1| Xν r =1

a_rr|t_r|^kr^δk

#

= O(1)

"_m−1 X

ν=1

∆(|λν|)X^ν

r =1

arr|tr|^kr^δk+ |λm|

m

X

r =1

arr|tr|^kr^δk

#

= O(1)

"_m−1 X

ν=1

∆(|λ_ν|)X^ν

r =1

r^δk−1|t_r|^k+ |λm|

m

X

r =1

r^δk−1|t_r|^k

#

= O(1)

m−1

X

ν=1

|∆λ_ν|X_ν+O(1)|λm|X_m

= O(1)

m

X

ν=1

β_νX_ν+O(1)|λm|X_m

= O(1).

Using (i) and H¨older’s inequality

I₂:=

m+1

X

n=2

n^δk+k−1|T_n2|^k=

m+1

X

n=2

n^δk+k−1     

n−1

X

ν=1

aˆ_n,ν+1(∆λ_ν)ν + 1 ν t_ν

k

= O(1)

m+1

X

n=2

n^δk+k−1

"_n−1 X

ν=1

aˆ_n,ν+1β_ν|t_ν|

#k

= O(1)

m+1

X

n=2

n^δk+k−1

"_n−1 X

ν=1

β_ν|t_ν|^kaˆ_n,ν+1

# "_n−1 X

ν=1

aˆ_n,ν+1β_ν

#k−1

.

It is easy to see that

n−1

X

ν=1

aˆ_n,ν+1β_ν ≤ Ma_nn.

We have, using (v) and (x),

I₂ = O(1)

m+1

X

n=2

n^δk(nann)^k−1

n−1

X

ν=1

aˆ_n,ν+1β_ν|t_ν|^k

= O(1)

m

X

ν=1

β_ν|t_ν|^k

m+1

X

n=ν+1

n^δkaˆ_n,ν+1.

Therefore,

I2 = O(1)X^m

ν=1

ν^δkβ_ν|t_ν|^k

= O(1)X^m

ν=1

νβ_ν|t_ν|^k ν ν^δk.

Using summation by parts, (iii), (xi) and condition (2) ofLemma 1,

I2 := O(1)

m

X

ν=1

νβν

" _ν X

r =1

r^δk−1|tr|^k− Xν−1 r =1

r^δk−1|tr|^k

#

= O(1)

" _m X

ν=1

νβ_νX^ν

r =1

r^δk−1|t_r|^k−

m−1

X

ν=1

(ν + 1β_ν+1)X^ν

r =1

r^δk−1|t_r|^k

#

= O(1)

m−1

X

ν=1

∆(νβ_ν)X^ν

r =1

r^δk−1|t_r|^k+O(1)mβm m

X

r =1

r^δk−1|t_r|^k

= O(1)^m−1X

ν=1

|∆(νβ_ν)|X_ν +O(1)mβmXm

= O(1)

m−1

X

ν=1

ν|∆(β_ν)|X_ν+O(1)

m−1

X

ν=1

β_ν+1X_ν+O(1)

= O(1).

Using (v), (x), (xi) and H¨older’s inequality, summation by parts, condition (2) ofLemma 1, and (viii), we have

m+1

X

n=2

n^δk+k−1|Tn3|^k =

m+1

X

n=2

n^δk+k−1     

n−1

X

ν=1

aˆn,ν+1λν+11 νt_ν

k

≤

m+1

X

n=2

n^δk+k−1

"_n−1 X

ν=1

|λ_ν+1|

ν aˆn,ν+1|t_ν|

#k

= O(1)

m+1

X

n=2

n^δk+k−1

"_n−1 X

ν=1

|λ_ν+1|

ν |t_ν|^kaˆn,ν+1

#

×

"_n−1 X

ν=1

aˆn,ν+1

|λ_ν+1| ν

#k−1

= O(1)

m+1

X

n=2

n^δk(nann)^k−1

"_n−1 X

ν=1

|λ_ν+1|

ν |t_ν|^kaˆn,ν+1

#

×

"_n−1 X

ν=1

|λ_ν+1| ν

#k−1

= O(1)

m+1

X

n=2

n^δk(nann)^k−1

n−1

X

ν=1

|λ_ν+1|

ν |t_ν|^kaˆn,ν+1

= O(1)

m

X

ν=1

|λν+1| ν |t_ν|^k

m+1

X

n=ν+1

n^δk(nann)^k−1aˆ_n,ν+1

= O(1)

m

X

ν=1

|λ_ν+1| ν |t_ν|^k

m+1

X

n=ν+1

n^δkaˆn,ν+1

= O(1)X^m

ν=1

|λ_ν+1| ν ν^δk|t_ν|^k

= O(1)

m−1

X

ν=1

(|∆λ_ν+1|)X^ν

r =1

r^δk−1|t_r|^k+O(1)|λm+1|

m

X

r =1

r^δk−1|t_r|^k

= O(1)

m−1

X

ν=1

(|∆λν+1|)Xν+O(1)|λm+1|X_m

= O(1)

m−1

X

ν=1

β_ν+1X_ν+O(1)|λm+1|Xm

= O(1).

Finally, using (iv) and (v), we have

m

X

n=1

n^δk+k−1|Tn4|^k =

m

X

n=1

n^δk+k−1    

(n + 1)annλnt_n n

k

= O(1)X^m

n=1

n^δk+k−1|ann|^k|λn|^k|tn|^k

= O(1)

m

X

n=1

n^δk(nann)^k−1ann|λn|^k−1|λn||tn|^k

= O(1)

m

X

n=1

n^δka_nn|λn||t_n|^k

= O(1), as in the proof of I1. 

Settingδ = 0 in the theorem yields the following corollary.

Corollary 1. Let {X_n} be an almost increasing sequence and {βn} and {λn} sequences satisfying condi- tions(i)–(iv) and (viii) ofTheorem2. Let A be a triangle satisfying conditions(v)–(vii) of Theorem2. If

(ix)Pm n=11

n|tn|^k =O(Xm), where tn:= ¹

n+1

Pn k=1kak, then the seriesP a_nλnis summable | A|_k, k ≥ 1.

Corollary 2. Let {X_n} be an almost increasing sequence, and {βn} and {λn} be sequences satisfying conditions(i)–(iv) and (viii) of Theorem2. Also, let { p_n}be a positive sequence such that P_n :=Pn

k=0p_k → ∞, and satisfies

(v) np_nO(Pn), (vi)Pm+1

n=ν+1n^δk| ^pn

PnP_n−1| =O

ν^δk P_ν

 . If

(vii)Pm

n=1n^δk−1|t_n|^k =O(Xm), where tn:= ¹

n+1

Pn k=1ka_k,

then the seriesP anλnis summable | ¯N, p, δ|k, k ≥ 1 for 0 ≤ δ < 1/k.

Proof. Conditions (i)–(iv), (vii) and (viii) of Corollary 2 are, respectively, conditions (i)–(iv), (xi) and (viii) of Theorem 2.

Conditions (vi) and (vii) ofTheorem 2are automatically satisfied for any weighted mean method. Condition (v) of Theorem 2becomes condition (v) ofCorollary 2, and conditions (ix) and (x) ofTheorem 2become condition (vi) of Corollary 2. 

It should be noted that, in [1], an incorrect definition of absolute summability was used (see, [4]).Corollary 2gives the correct version of Bor and ¨Ozarslan’s theorem.

Acknowledgements

The author wish to thank the referees for their constructive comments and suggestions.

References

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[2] T.M. Fleet, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 3 (7) (1957) 113–141.

[3] S.M. Mazhar, A note on absolute summability, Bull. Inst. Math. Acad. Sinica 25 (1997) 233–242.

[4] B.E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math. Anal. Appl. 238 (1999) 82–90.

[5] B. Rhoades, E. Savas¸, A summability factor theorem for generalized absolute summability, Real Anal. Exchange 31 (2) (2006) 355–364.