Factors for | A |k summability of infinite series

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Factors for | A| _k summability of infinite series

Ekrem Savas¸

Istanbul Ticaret University, Department of Mathematics, ¨Usk¨udar, ´Ystanbul, Turkey Received 27 September 2005; received in revised form 16 May 2006; accepted 19 May 2006

Abstract

In this paper we generalize Bor’s result by using the correct definition of absolute summability.

c

2007 Elsevier Ltd. All rights reserved.

Keywords:Absolute summability; Weighted mean matrix; Ces´aro matrix; Summability factor

Recently Bor [1] generalized a result of Sulaiman [3]. Unfortunately he used an incorrect definition of absolute summability. In this paper we obtain the corresponding result for a lower triangular matrix using the correct definition (see, e.g. [2]). We obtain the correct form of [1] as a corollary.

Let T be a lower triangular matrix, {s_n}a sequence. Then we put T_n:=

n

X

ν=0

t_nνs_ν.

A seriesP a_nis said to be summable |T |_k, k ≥ 1, if

∞

X

n=1

n^k−1|T_n−T_n−1|^k < ∞. (1)

We may associate with T two lower triangular matrices T and ˆT defined as follows:

t¯_nν =

n

X

r =ν

t_nr, n, ν = 0, 1, 2, . . . , and

tˆ_nν = ¯t_nν− ¯t_n−1,ν, n = 1, 2, 3, . . . . We may write

Tn=

n

X

ν=0

anν

Xν i =0

aiλi =

n

X

i =0

aiλi n

X

ν=i

a_ν =

n

X

i =0

a¯niaiλi.

E-mail address:ekremsavas@yahoo.com.

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

doi:10.1016/j.camwa.2006.05.019

Thus

Tn−Tn−1=

n

X

i =0

a¯niaiλi−

n−1

X

i =0

a¯n−1,iaiλi

=

n

X

i =0

a¯niaiλi−

n

X

i =0

a¯n−1,iaiλi

=

n

X

i =0

(¯ani − ¯an−1,i)aiλi

=

n

X

i =0

aˆ_nia_iλi =

n

X

i =1

aˆ_niλi(si −s_{i −1})

=

n

X

i =1

aˆ_niλis_i −

n

X

i =1

aˆ_niλis_{i −1}

=

n−1

X

i =1

aˆ_niλis_i + ˆa_nnλns_n−

n

X

i =1

aˆ_niλisi −1

=

n−1

X

i =1

aˆniλisi +annλnsn−

n−1

X

i =0

aˆn,i+1λi +1si

=

n

X

i =1

(ˆaniλi− ˆan,i+1λi +1)si+annλnsn.

We may write

(ˆaniλi − ˆa_n,i+1λi +1) = ˆaniλi− ˆa_n,i+1λi +1− ˆa_n,i+1λi+ ˆa_n,i+1λi

=(ˆani− ˆa_n,i+1)λi+ ˆa_n,i+1(λi−λi +1)

=λi1iaˆni + ˆan,i+11λi. Therefore

T_n−Tn−1=

n−1

X

i =0

1iaˆ_niλis_i+

n−1

X

i =1

aˆ_n,i+11λis_i+a_nnλns_n

=Tn1+Tn2+Tn3, say.

A triangle is a lower triangular matrix with all nonzero main diagonal entries.

Theorem 1. Let A be a lower triangular matrix with nonnegative entries satisfying (i) ¯a_n0=1, n = 0, 1, . . . ,

(ii) an−1,ν ≥a_nνfor n ≥ν + 1, and (iii) nann =O(1).

Let {X_n}be given sequence of positive numbers and let s_n = O(Xn) as n → ∞. If (λn)n≥0 is a sequence of complex numbers such that

(iv) P∞

n=1a_nn(|λn|X_n)^k< ∞, and (v) P∞

n=1Xn|4λn|< ∞,

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

Proof. To prove the theorem it will be sufficient to show that

∞

X

n=1

n^k−1|T_nr|^k< ∞, for r = 1, 2, 3.

Using H´older’s inequality, (iii), and (iv),

I1:=

m+1

X

n=1

n^k−1|Tn1|^k≤

m+1

X

n=1

n^k−1

n−1

X

i =0

|1iaˆni||λi||si|

!k

≤

m+1

X

n=1

n^k−1

n−1

X

i =0

|1iaˆni||λi|^k(Xi)^k

! _n−1 X

i =0

|1iaˆni|

!k−1

.

From (ii)

1iaˆnν = ˆani− ˆan,i+1

= ¯a_ni− ¯a_n−1,i− ¯a_n,i+1+ ¯a_n−1,i+1

=a_ni−a_n−1,i ≤0. Thus, using (i),

n−1

X

i =0

|1iaˆni| =

n−1

X

i =0

(an−1,i−ani) = 1 − 1 + ann =ann.

Using (iv), I1:= O(1)

m+1

X

n=1

(nann)^k−1

n−1

X

i =0

|1iaˆ_ni||λi|^k(Xi)^k

= O(1)X^m

i =0

(Xi|λi|)^{k m+1}X

n=i +1

(nann)^k−1|1iaˆni|

= O(1)X^m

i =0

ai i(|λi|Xi)^k

= O(1).

By H´older’s inequality, (iii) and (v),

I₂:=

m+1

X

n=1

n^k−1|T_n2|^k≤

m+1

X

n=1

n^k−1     

n−1

X

i =0

aˆ_n,i+1s_i1λi

k

≤

m+1

X

n=1

n^k−1

n−1

X

i =0

| ˆa_n,i+1||1λi||s_i|

!k

≤

m+1

X

n=1

n^k−1

n−1

X

i =0

| ˆa_n,i+1||1λi|X_i

! _n−1 X

i =0

| ˆa_n,i+1||1λi|X_i

!k−1

.

From the definition of ˆAand ¯A, and using (i) and (ii), aˆn,i+1 = ¯an,i+1− ¯an−1,i+1

=

n

X

ν=i+1

a_nν−

n−1

X

ν=i+1

a_n−1,ν

=1 −

i

X

ν=0

a_nν−1 +

i

X

ν=0

a_n−1,ν

=

i

X

ν=0

a_n−1,ν−a_n,ν
≥ 0. (2)

Using (i) aˆn,i+1 =

i

X

ν=0

an−1,ν−an,ν

≤

n−1

X

ν=0

a_n−1,ν−a_n,ν

=1 − 1 + a_nn. (3)

Therefore I2 := O(1)

m+1

X

n=1

(nann)^k−1

n−1

X

i =0

aˆn,i+1|1λi|Xi

= O(1)X^m

i =1

|1λi|X_i

m+1

X

n=i +1

(nann)^k−1aˆ_n,i+1,

= O(1)

m

X

i =1

|1λi|X_i

m+1

X

n=i +1

aˆ_n,i+1. From(2)

m+1

X

n=i +1 i

X

ν=0

(an−1,ν−a_nν)

!

=

i

X

ν=0 m+1

X

n=i +1

(an−1,ν−a_nν)

=

i

X

ν=0

(ai,ν−am+1,ν)

≤

i

X

ν=0

a_i,ν =1. (4)

Hence

I₂ := O(1)

m

X

i =1

|1λi|X_i,

= O(1).

Using (iii) and (iv),

m+1

X

n=1

n^k−1|T_n3|^k ≤

m+1

X

n=1

n^k−1|a_nnλns_n|^k

= O(1)

m

X

n=1

(nann)^k−1a_nn(|λn|X_n)^k

= O(1)

m

X

n=1

a_nn(Xn|λn|)^k,

= O(1). 

Corollary 1. Let { p_n}be a positive sequence such that Pn:=Pn

k=0pk → ∞, and satisfies (i) np_n=O(Pn).

Let {X_n}be given sequence of positive numbers and let s_n = O(Xn) as n → ∞. If (λn)n≥0is a sequence of complex numbers such that

(ii) P∞ n=1

pn

P_n(|λn|Xn)^k< ∞, and

(iii) P∞

n=1X_n|4λn|< ∞,

are satisfied, then the seriesP a_nλnis summable |N, pn|_k, k ≥ 1.

Proof. Condition (iii) of Corollary 1 is condition (v) of Theorem 1. Conditions (i) and (ii) of Theorem 1 are automatically satisfied for any weighted mean method. Conditions (iii) and (iv) ofTheorem 1become, respectively, conditions (i) and (ii) ofCorollary 1. 

It should be noted that, in [1], an incorrect definition of absolute summability was used. Corollary 1 gives the correct version of Bor’s theorem.

Acknowledgement

This research was completed while the author was a Fulbright scholar at Indiana University, Bloomington, IN, USA during the spring semester of 2004.

References

[1] H. Bor, Factors for |N, pn|_k, summability of infinite series, Ann. Acad. Sci. Fenn. Math. Ser. A 16 (1991) 151–154.

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

[3] W.T. Sulaiman, Multipliers for |C, 1| summability of Jacobi seies, Indian J. Pure Appl. Math. 18 (1987) 1121–1130.