On absolute summability factors of infinite series

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www.elsevier.com/locate/camwa

On absolute summability factors of infinite series

Ekrem Savas¸

Department of Mathematics, Istanbul Ticaret University, ¨Usk¨udar, Istanbul, Turkey Received 27 September 2007; accepted 17 October 2007

Abstract

In this paper a general theorem on | A, δ|k-summability methods has been proved. This theorem includes, as a special case, a known result in [E. Savas, Factors for | A|_kSummability of infinite series, Comput. Math. Appl. 53 (2007) 1045–1049].

c

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

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

n

X

ν=0

a_n_νs_ν.

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

∞

X

n=1

n^k−1|A_n−A_n−1|^k < ∞ (1)

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

∞

X

n=1

n^δk+k−1|An−An−1|^k< ∞. (2)

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

a¯nν =

n

X

r =ν

anr, n, ν = 0, 1, 2, . . . , and

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

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

E-mail address:ekremsavas@yahoo.com.

doi:10.1016/j.camwa.2007.10.031

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We may write T_n=

n

X

ν=0

a_n_ν Xν i =0

a_iλi =

n

X

i =0

a_iλi n

X

ν=i

a_n_ν =

n

X

i =0

a¯_nia_iλi.

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 −si −1)

=

n

X

i =1

aˆ_niλis_i −

n

X

i =1

aˆ_niλis_{i −1}

=

n−1

X

i =1

aˆniλisi + ˆannλnsn−

n

X

i =1

aˆniλisi −1

=

n−1

X

i =1

aˆ_niλis_i +a_nnλns_n−

n−1

X

i =0

aˆ_n_,i+1λi +1s_i

=

n

X

i =1

(ˆaniλi− ˆan,i+1λi +1)si+annλnsn.

We may write

(âniλi − ân,i+1λi +1) = âniλi− ân,i+1λi +1− ân,i+1λi+ ân,i+1λi

=(âni− â_n,i+1)λi+ â_n,i+1(λi−λi +1)

=λi∆_iaˆ_ni+ ˆa_n_,i+1∆λi. Therefore

T_n−T_n−1=

n−1

X

i =0

∆_iaˆ_niλis_i +

n−1

X

i =1

aˆ_n_,i+1∆λis_i +a_nnλns_n

=T_n1+T_n2+T_n3, say.

We shall prove the following theorem.

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

(ii) a_n−1,ν ≥a_nνfor n ≥ν + 1, (iii) nann =O(1),

(iv) Pm+1

n=ν+1n^δk|∆_νaˆ_n_ν| =O ν^δka_νν, and (v) Pm+1

n=ν+1n^δkaˆ_nν+1=O ν^δk.

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

(vi)P∞

n=1X_n|4λn|< ∞,

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(vii) P∞

n=1n^δk−1(|λn|X_n)^k< ∞, and (viii) P∞

n=1n^δkX_n|4λn|< ∞,

then the seriesP anλnis summable | A, δ|k, k ≥ 1, 0 ≤ δ < 1/k.

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

∞

X

n=1

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

Using H´older’s inequality,

I₁:=

m+1

X

n=1

n^δk+k−1|T_n1|^k≤

m+1

X

n=1

n^δk+k−1

n−1

X

i =0

|∆_iaˆ_ni||λi||s_i|

!k

≤

m+1

X

n=1

n^δk+k−1

n−1

X

i =0

|∆_iaˆ_ni||λi|^k(Xi)^k

! _n−1 X

i =0

|∆_iaˆ_ni|

!k−1

.

From (ii)

∆_iaˆnν = ˆani− ˆan,i+1

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

=ani−an−1,i ≤0.

Thus, using (i),

n−1

X

i =0

|∆_iaˆni| =

n−1

X

i =0

(an−1,i−ani) = 1 − 1 + ann =ann.

Using (iii), (iv) and (vii),

I1:= O(1)

m+1

X

n=1

n^δk(nann)^k−1

n−1

X

i =0

|∆_iaˆni||λi|^k(Xi)^k

= O(1)

m

X

i =0

(Xi|λi|)^k

m+1

X

n=i +1

n^δk(nann)^k−1|∆_iaˆ_ni|

= O(1)

m

X

i =0

(Xi|λi|)^k

m+1

X

n=i +1

n^δk|∆_iaˆ_ni|

= O(1)

m

X

i =0

i^δk−1(|λi|Xi)^k

= O(1).

From H´older’s inequality, and (vi),

I₂:=

m+1

X

n=1

n^δk+k−1|T_n2|^k≤

m+1

X

n=1

n^δk+k−1

n−1

X

i =0

aˆ_n,i+1s_i∆λi

k

≤

m+1

X

n=1

n^δk+k−1

n−1

X

i =0

| ˆa_n,i+1||∆λi||s_i|

!k

≤

m+1

X

n=1

n^δk+k−1

n−1

X

i =0

| ˆa_n,i+1||∆λi|X_i

! _n−1 X

i =0

| ˆa_n,i+1||∆λi|X_i

!k−1

.

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From the definition of ˆAand ¯A, and using (i) and (ii);

aˆ_n_,i+1 = ¯a_n_,i+1− ¯a_n−1_,i+1

=

n

X

ν=i+1

anν−

n−1

X

ν=i+1

an−1,ν

=1 −

i

X

ν=0

a_n_ν−1 +

i

X

ν=0

a_n−1_,ν

=

i

X

ν=0

an−1,ν−an,ν ≥ 0. (3)

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. (4)

Therefore, using (iii), (v) and (viii)

I2 := O(1)^m+1X

n=1

n^δk(nann)^k−1Xⁿ⁻¹

i =0

aˆn,i+1|∆λi|Xi

= O(1)

m

X

i =1

|∆λi|X_i

m+1

X

n=i +1

n^δk(nann)^k−1aˆ_n_,i+1

= O(1)

m

X

i =1

|∆λi|Xi m+1

X

n=i +1

n^δkaˆn,i+1

= O(1)X^m

i =1

i^δk|∆λi|Xi.

Using (iii) and (vii),

m+1

X

n=1

n^δk+k−1|Tn3|^k ≤

m+1

X

n=1

n^δk+k−1|annλnsn|^k

= O(1)

m

X

n=1

n^δk(nann)^k−1a_nn(|λn|X_n)^k

= O(1)

m

X

n=1

n^δk−1(Xn|λn|)^k

= O(1).

Settingδ = 0 in the theorem yields the following corollary:

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

(ii) a_n−1_,ν ≥a_n_νfor n ≥ν + 1, and (iii) nann =O(1).

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

(iv) P∞ n=11

n(|λn|X_n)^k < ∞, and (v) P∞

n=1X_n|4λn|< ∞,

then the seriesP anλnis summable | A|k, k ≥ 1.

Corollary 2. Let { p_n}be a positive sequence such that P_n:=Pn

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

(ii) Pm+1

n=ν+1n^δk| ^pⁿ

PnPn−1| =O_ν_δk

P_ν

.

Let {X_n}be a given sequence of positive numbers and let s_n = O(Xn) as n → ∞. If (λn)n≥0is a sequence of complex numbers, satisfying conditions (vi)–(viii) ofTheorem1, then the seriesP a_nλnis summable | ¯N, p, δ|k, k ≥ 1 for0 ≤δ < 1/k.

Proof. Conditions (vi)–(viii) ofCorollary 2are, respectively, conditions (vi)–(viii) ofTheorem 1.

Conditions (i) and (ii) ofTheorem 1are automatically satisfied for any weighted mean method. Condition (iii) of Theorem 1becomes condition (i) ofCorollary 2, and conditions (iv) and (v) ofTheorem 1become condition (ii) of Corollary 2.

References

[1] E. Savas, Factors for | A|_kSummability of infinite series, Comput. Math. Appl. 53 (2007) 1045–1049.

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