A summability factor theorem for absolute summability involving delta-quasi-monotone and almost increasing sequences

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Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

A summability factor theorem for absolute summability involving

δ -quasi-monotone and almost increasing sequences

Ekrem Savaş

Department of Mathematics, Istanbul Commerce University, Uskudar-Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 16 July 2006 Accepted 15 May 2008

Keywords:

Summability factor Almost increasing Quasi monotone

a b s t r a c t

Quite recently Bor [H. Bor, An application of almost increasing andδ-quasi-monotone sequences, JIPAM. J. Inequal. Pure Appl. Math. 1 (2) (2000). Article 18, 6 pp.] has proved a theorem on|Np_n|_k-summability factors of an infinite series. The present paper deals with a further generalization of his result.

Bor [2] obtained sufficient conditions for

P

a_n

λ

nto be summable

|

N

,

^pn

|

_k

,

^k

≥

1. We generalize this result by replacing the weighted mean matrix with a triangular matrix, and by using the correct definition of absolute summability, [3].

A sequence

(

^bn

)

of positive numbers is said to be

δ

-quasi-monotone, if b_n

>

0 ultimately and∆^bn

≥ − δ

n, where

(δ

n

)

^is a sequence of positive numbers [2].

Let A be a lower triangular matrix,

{

s_n

}

a sequence. Then A_n

:=

n

X

ν=⁰ a_n_νs_ν

.

A series

P

a_nis said to be summable

|

A

|

_k

,

^k

≥

1 if

∞

X

n=1

n^k⁻¹

|

A_n

−

A_n−1

|

^k

< ∞.

⁽¹⁾

We may associate with A two lower triangular matrices A andA defined as follows:

ˆ

a

¯

_n_ν

=

n

X

r=ν

a_nr

,

ⁿ

, ν =

⁰

,

¹

,

²

, . . . ,

and

a

ˆ

_n_ν

= ¯

a_n_ν

− ¯

a_n−1,ν

,

ⁿ

=

1

,

²

,

³

, . . . .

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

A positive sequence

{

d_n

}

is said to be almost increasing if there exist a positive increasing sequence

{

c_n

}

and two positive constants A and B such that Ac_n

≤

d_n

≤

Bc_nfor each n.

Theorem 1. Let A be a lower triangular matrix with non-negative entries satisfying

E-mail addresses:esavas@iticu.edu.tr,ekremsavas@yahoo.com.

doi:10.1016/j.mcm.2008.05.049

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(i) _a

¯

_n0

=

1

,

ⁿ

=

0

,

¹

, . . . ,

(ii) a_n−1,ν

≥

a_n_νfor n

≥ ν +

^{1, and} (iii) na_nn

=

O

(

¹

)

^.

If

{

X_n

}

is an almost increasing sequence such that

|

_∆X_n

| =

O

(

^X_nⁿ

)

^and

λ

n

→

0 as n

→ ∞

. Suppose that there exists a sequence of numbers

(

^An

)

such that it is

δ

-quasi-monotone with

P

nX_n

δ

n

< ∞, P

^AnX_nis convergent and

|4 λ

n

| ≤ |

A_n

|

for all n. If

(iv)

P

∞ n=1

|λn|

n

< ∞

^{, and} (v)

P

∞

n=1 1

n

|

t_n

|

^k

=

O

(

^Xm

)

^{, where t}n

:=

¹

n+1

P

n k=1ka_k

,

then the series

P

a_n

λ

nis summable

|

A

|

_k

,

^k

≥

1.

The following lemmas are pertinent for the proof ofTheorem 1.

Lemma 1 ([2]). Under the conditions of the theorem, we have that (1)

| λ

n

|

X_n

=

O

(

¹

)

^.

Lemma 2 ([1]). Let

{

X_n

}

be an almost increasing sequence such that

|

_∆X_n

| =

O

(

^X_nⁿ

)

^{. If}

(

^An

)

^is

δ

P

nX_n

δ

n

< ∞, P

^AnX_nis convergent, then

(2)

P

∞

n=1nX_n

|

_∆A_n

| < ∞

^{, and} (3) nA_nX_n

=

O

(

¹

)

^.

Proof. Let

(

^yn

)

be the nth term of the A-transform of

P

n

i=0

λ

ia_i.Then, y_n

:=

n

X

i=0

a_nis_i

=

n

X

i=0

a_ni

i

X

ν=0

λ

νa_ν

=

n

X

ν=0

λ

νa_ν

n

X

i=ν a_ni

=

n

X

ν=0

a

¯

_n_ν

λ

νa_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

a

ˆ

_n_ν

λ

νa_ν

.

⁽²⁾

We may write

Y_n

=

n

X

ν=1

ˆ

a_n_ν

λ

ν

^aν

=

n

X

ν=1

ˆ

a_n_ν

λ

ν

"

_ν

X

r=₁ ra_r

−

ν−1

X

r=₁ ra_r

#

=

n−1

X

ν=1

∆ν

ˆ

a_n_ν

λ

ν

X

r=1

ra_r

+

_a

ˆ

_nn

λ

n

X

ν=1

ν

^aν

=

n−1

X

ν=1

(

∆νa

ˆ

_n_ν

)λ

ν

ν +

¹

ν

^t^ν

+

n−1

X

ν=1

ˆ

a_n_,ν+₁

(

∆

λ

ν

) ν +

¹

ν

^t^ν

+

n−1

X

ν=1

a

ˆ

_n_,ν+₁

λ

ν+1

1

ν

^t^ν

+ (

ⁿ

+

1

)

^ann

λ

nt_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^k⁻¹

|

T_nr

|

^k

< ∞,

^{for r}

=

1

,

²

,

³

,

⁴

.

Using Hölder’s inequality and (iii),

I₁

:=

m

X

n=1

n^k⁻¹

|

T_n1

|

^k

=

m

X

n=1

n^k⁻¹

n−1

X

ν=1

∆ν_a

ˆ

_n_ν

λ

ν

ν +

¹

ν

^t^ν

k

=

O

(

¹

)

^m

+1

X

n=1

n^k⁻¹

n−1

X

ν=1

|

_∆_νa

ˆ

_n_ν

|| λ

ν

||

t_ν

|

!

k

(3)

=

O

(

¹

)

^m

+1

X

n=1

n^k⁻¹

n−1

X

ν=1

|

_∆_ν

ˆ

_a_n_ν

|| λ

ν

|

^k

|

t_ν

|

^k

!

_n₋₁

X

ν=1

|

_∆_ν_a

ˆ

_n_ν

|

!

k−1

.

Also,

∆νa

ˆ

_n_ν

= ˆ

a_n_ν

− ˆ

a_n_,ν+₁

= ¯

a_n_ν

− ¯

a_n−1,ν

− ¯

a_n_,ν+₁

+ ¯

a_n−1,ν+1

=

a_n_ν

−

a_n−1,ν

≤

0

.

Thus, using (ii),

n−1

X

ν=0

|

_∆_νa

ˆ

_n_ν

| =

n−1

X

ν=0

(

^an−1,ν

−

a_n_ν

) =

¹

−

1

+

a_nn

=

a_nn

.

From condition (1) ofLemma 1,

{ λ

n

}

is bounded, and (v)

I₁

=

O

(

¹

)

m+1

X

n=1

(

^nann

)

^k⁻¹

n−1

X

ν=1

| λ

ν

|

^k

|

t_ν

|

^k

|

_∆_ν

ˆ

a_n_ν

|

=

O

(

¹

)

^m

+1

X

n=1

(

^nann

)

^k⁻¹ ⁿ

−1

X

ν=1

| λ

ν

|

^k⁻¹

| λ

ν

||

_∆_ν

ˆ

a_n_ν

||

t_ν

|

^k

!

=

O

(

¹

) X

^m

ν=1

| λ

ν

||

t_ν

|

^k

m+₁

X

n=ν+1

(

^nann

)

^k⁻¹

|

_∆_νa

ˆ

_n_ν

|

=

O

(

¹

) X

^m

ν=1

| λ

ν

||

t_ν

|

^k

m+1

X

n=ν+1

|

_∆_νa

ˆ

_n_ν

|

=

O

(

¹

) X

^m

ν=1

| λ

ν

|

a_νν

|

t_ν

|

^k

=

O

(

¹

)

m−1

X

ν=1

|

_∆

λ

ν

||

X_ν

| +

O

(

¹

)|λ

m

||

X_m

|

=

O

(

¹

)

m−1

X

ν=1

|

A_ν

||

X_ν

| +

O

(

¹

)|λ

m

||

X_m

|

=

O

(

¹

),

again using the hypothesis of the theorem andLemma 1.

Using Hölder’s inequality,

I₂

:=

m+1

X

n=2

n^k⁻¹

|

T_n2

|

^k

=

m+1

X

n=2

n^k⁻¹

n−1

X

ν=¹

a

ˆ

_n_,ν+₁

(

∆

λ

ν

) ν +

¹

ν

^t^ν

k

≤

m+1

X

n=2

n^k⁻¹

"

_n₋₁

X

ν=1

|ˆ

a_n_,ν+₁

||

_∆

λ

ν

| ν +

¹

ν |

t_ν

|

#

k

=

O

(

¹

)

^m

+1

X

n=2

n^k⁻¹

"

_n₋₁

X

ν=1

a

ˆ

_n_,ν+₁

|

A_ν

||

t_ν

|

#

k

=

O

(

¹

)

^m

+1

X

n=2

n^k⁻¹

"

_n₋₁

X

ν=1

|

A_ν

||

t_ν

|

^ka

ˆ

_n_,ν+₁

# "

_n₋₁

X

ν=1

|

A_ν

|ˆ

a_n_,ν+₁

#

k−1

.

Thus we have,

I₂

:=

O

(

¹

)

m+1

X

n=2

(

^nann

)

^k⁻¹

n−1

X

ν=1

|

A_ν

||

t_ν

|

^k

ˆ

a_n_,ν+₁

.

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Therefore, using (iii),

I₂

=

O

(

¹

)

m

X

ν=¹

|

A_ν

||

t_ν

|

^k

m+1

X

n=ν+¹

(

^nann

)

^k⁻¹a

ˆ

_n_,ν+₁

=

O

(

¹

)

m

X

ν=¹

|

A_ν

||

t_ν

|

^k

m+1

X

n=ν+¹ a

ˆ

_n_,ν+₁

.

Therefore

I₂

=

O

(

¹

) X

^m

ν=1

|

A_ν

||

t_ν

|

^k

.

We may write

I₂

=

O

(

¹

)

m

X

ν=1

ν|

^Aν

| |

t_ν

|

^k

ν .

Using summation by parts and (v),

I₂

:=

O

(

¹

)

m−1

X

ν=¹

∆

(ν|

^Aν

| ) X

^ν

r=1

1

r

|

t_r

|

^k

+

O

(

¹

)

^m

|

A_m

|

m

X

r=1

1 r

|

t_r

|

^k

=

O

(

¹

)

m−1

X

ν=1

|

_∆

(ν

^Aν

)|

^Xν

+

O

(

¹

)

^m

|

A_m

|

X_m

.

Using (v) and properties (2) and (3) ofLemma 2, and the fact that

{

X_n

}

is almost increasing

I₂

=

O

(

¹

)

m−1

X

ν=¹

ν|

∆^Aν

|

X_ν

+

O

(

¹

)

m−1

X

ν=¹

|

A_ν+₁

|

X_ν+₁

+

O

(

¹

)

^m

|

A_m

|

X_m

=

O

(

¹

).

Using the hypothesis ofTheorem 1, Hölder’s inequality, summation by parts andLemma 1,

m+1

X

n=2

n^k⁻¹

|

T_n3

|

^k

=

m+1

X

n=2

n^k⁻¹

n−1

X

ν=1

a

ˆ

_n_,ν+₁

λ

ν+1

1

ν

^t^ν

k

≤

m+1

X

n=2

n^k⁻¹

"

_n₋₁

X

ν=¹

| λ

ν+1

|

ν

a

ˆ

_n_,ν+₁

|

t_ν

|

#

k

=

O

(

¹

)

m+1

X

n=2

n^k⁻¹

"

_n₋₁

X

ν=1

| λ

ν+1

|

ν |

t_ν

|

^k

ˆ

_a_n_,ν+₁

#

×

"

_n₋₁

X

ν=1

a

ˆ

_n_,ν+₁

| λ

ν+1

| ν

#

k−1

=

_O

(

¹

)

^m

+1

X

n=2

(

^nann

)

^k⁻¹

"

_n₋₁

X

ν=1

| λ

ν+¹

|

ν |

_t_ν

|

^k_a

ˆ

_n_,ν+₁

#

×

"

_n₋₁

X

ν=1

| λ

ν+¹

| ν

#

k−1

=

O

(

¹

)

^m

+1

X

n=2

(

^nann

)

^k⁻¹ⁿ

−1

X

ν=1

| λ

ν+1

|

ν |

t_ν

|

^ka

ˆ

_n_,ν+₁

=

O

(

¹

)

m

X

ν=1

| λ

ν+1

| ν |

t_ν

|

^k

m+1

X

n=ν+1

(

^nann

)

^k⁻¹_a

ˆ

_n_,ν+₁

=

O

(

¹

)

m

X

ν=1

| λ

ν+1

| ν |

t_ν

|

^k

m+1

X

n=ν+1

a

ˆ

_n_,ν+₁

=

O

(

¹

)

m

X

ν=¹

| λ

ν+1

| ν |

t_ν

|

^k

=

O

(

¹

).

(5)

Finally, using (iii) and the hypothesis ofTheorem 1, we have

m

X

n=1

n^k⁻¹

|

T_n4

|

^k

=

m

X

n=1

n^k⁻¹

(

ⁿ

+

1

)

^ann

λ

nt_n n

k

=

O

(

¹

)

m

X

n=1

n^k⁻¹

|

a_nn

|

^k

| λ

n

|

^k

|

t_n

|

^k

=

O

(

¹

)

m

X

n=1

(

^nann

)

^k⁻¹^ann

| λ

n

|

^k⁻¹

| λ

n

||

t_n

|

^k

=

O

(

¹

) X

^m

n=1

a_nn

| λ

n

||

t_n

|

^k

=

O

(

¹

),

as in the proof of I₁.

Corollary 1. Let

{

p_n

}

be a positive sequence such that P_n

:= P

n

k=0p_k

→ ∞

, and satisfies (i) np_n

=

O

(

^Pn

)

^.

If

{

X_n

}

is an almost increasing sequence such that

|

_∆X_n

| =

O

(

^X_nⁿ

)

^and

λ

n

→

0 as n

→ ∞ .

Suppose that there exists a sequence of numbers

(

^An

)

such that it is

δ

P

nX_n

δ

n

< ∞, P

^AnX_nis convergent and

|4 λ

n

| ≤ |

A_n

|

for all n. If

(ii)

P

∞ n=1

|λn|

n

< ∞

^{, and} (iv)

P

m

n=1 1

n

|

t_n

|

^k

=

O

(

^Xm

)

^, then the series

P

a_n

λ

nis summable

|

N

,

^pn

|

_k

,

^k

≥

1.

Proof. Conditions (ii) and (iv) ofCorollary 1are, respectively, conditions (iv), and (v) ofTheorem 1.

Conditions (i) and (ii) ofTheorem 1 are automatically satisfied for any weighted mean method. Condition (iii) of Theorem 1becomes condition (i) ofCorollary 1.

References

[1] H. Bor, Corrigendum on the paper: ‘‘An application of almost increasing andδ-quasi-monotone sequences’’, JIPAM. J. Inequal. Pure Appl. Math. 3 (1) (2002). Article 16, 2 pp.

[2] H. Bor, An application of almost increasing andδ-quasi-monotone sequences, JIPAM. J. Inequal. Pure Appl. Math. 1 (2) (2000). Article 18, 6 pp.

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

Further reading

[1] R.P. Boas Jr., Quasi-positive sequence and trigonometric series, Proc. Lond. Math. Soc. 14 (A) (1965) 38–46.

[2] B.E. Rhoades, E. Savas, On|A|_ksummability Factors, Acta Math. Hung. 112 (1–2) (2006) 15–23.

[3] E. Savas, On a recent result on absolute summability factors, Appl. Math. Lett. 18 (2005) 1273–1280.

[4] E. Savas, A summability factor theorem for absolute summability involving quasi-monotone sequences (Preprint).