On generalized quasi-beta-power increasing sequences

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Applied Mathematics Letters

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

On generalized quasi- β -power increasing sequences

Ekrem Savaş

^∗

Department of Mathematics, Istanbul Ticaret University, Üsküdar, Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 22 April 2009 Accepted 20 August 2009

Keywords:

Absolute summability Summability factors Lower triangular matrix

a b s t r a c t

The paper deals with absolute summability factors for infinite series. The main purpose of this paper is to generalize a recent paper of Savas (2009) [2].

© 2009 Elsevier Ltd. All rights reserved.

A positive sequence

{

b_n

}

is said to be almost increasing if there exists an increasing sequence

{

c_n

}

and positive constants A and B such that Ac_n

≤

b_n

≤

Bc_n.

A positive sequence

{ γ

n

}

is said to be quasi-

β

-power increasing sequence if there exists a constant K

=

K

(β, γ ) ≥

^{1 such} that Kn^β

γ

n

≥

m^β

γ

mholds for all n

≥

m

≥

1. It should be noted that every almost increasing sequence is quasi-

β

^-power increasing sequence for any non-negative

β

, but the converse need not be true as can be seen by taking the example, say

γ

n

=

n^−βfor

β >

^0.

Let A be a lower triangular matrix,

{

s_n

}

a sequence. Then A_n

:=

n

X

ν=0

a_nνs_ν

.

A series

P

a_n, with partial sums

(

^sn

)

, is 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, [1])

∞

X

n=1

n^δk+k−1

|

_An

−

_An−1

|

^k

< ∞.

⁽²⁾

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

ˆ

¯

a_nν

=

n

X

r=ν

a_nr

,

ⁿ

, ν =

⁰

,

¹

,

²

, . . . ,

and

ˆ

a_nν

= ¯

a_nν

− ¯

a_n−1,ν

,

ⁿ

=

1

,

²

,

³

, . . . .

∗Tel.: +90 2163280064.

E-mail addresses:[email protected],[email protected].

0893-9659/$ – see front matter©2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.aml.2009.08.011

A sequence

{ λ

n

}

is said to be of bounded variation

(

^b

v)

^if

P

n

|

_∆

λ

n

| < ∞

^{. Let b}

v

0

=

b

v ∩

^c0, where c₀denotes the set of all null sequences.

Quite recently, Savas [2] proved the following theorem under weaker conditions by using a quasi-

β

-power increasing sequence instead of an almost increasing sequence.

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

¯

_n0

=

1

,

ⁿ

=

0

,

¹

, . . . ,

^,

(ii) a_n−1,ν

≥

a_nνfor n

≥ ν +

^{1, and} (iii) na_nn

=

O

(

¹

)

^.

Let

(λ

n

) ∈

^b

v

0and if

{

X_n

}

is a quasi-

β

- power increasing sequence for some 0

< β <

1 such that (iv)

| λ

m

|

X_m

=

O

(

¹

)

^,

(v)

P

m

n=₁nX_n

|

_∆²

λ

n

| =

O

(

¹

)

^{, and} (vi)

P

m

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 aim of this paper is to generalizeTheorem 1for

|

A

, δ|

k

,

^k

≥

1

,

⁰

≤ δ <

¹

/

k- summability. Now we are ready to give the following main theorem.

Theorem 2. Let A be a lower triangular matrix with non-negative entries satisfying (i) a

¯

_n0

=

1

,

ⁿ

=

0

,

¹

, . . . ,

^,

(ii) a_n−1,ν

≥

a_nνfor n

≥ ν +

^{1, and} (iii) na_nn

=

O

(

¹

)

^,

(iv)

P

m+1

n=ν+1n^δk

|

_∆νa

ˆ

_nν

| =

O

ν

^δkaνν



and (v)

P

m+1

n=ν+1n^δka

ˆ

_nν+1

=

O

ν

^δk



.

Let

(λ

n

) ∈

^b

v

0and if

{

X_n

}

is a quasi-

β

- power increasing sequence for some 0

< β <

1 such that (vi)

| λ

m

|

X_m

=

O

(

¹

)

^,

(vii)

P

m

n=1nX_n

|

_∆²

λ

n

| =

O

(

¹

)

^{, and} (viii)

P

m

n=1n^δk−1

|

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

, δ|

k

,

^k

≥

1

,

⁰

≤ δ <

¹

/

^k.

It should be note that if we take

δ =

0 in the above theorem then we can getTheorem 1and also if we take

{

X_n

}

is an almost increasing sequence instead of a quasi-

β

-power increasing sequence then ourTheorem 2reduces to Theorem 1 in [3].

We shall need the following lemma for the proof ofTheorem 2.

Lemma 1 ([2]). Let

(λ

n

) ∈

^b

v

0and if

{

X_n

}

is a quasi-

β

- power increasing sequence for some 0

< β <

1, then under the conditions

(

^vi

)

^and

(

^vii

)

^{of theTheorem 2we have}

(1)

P

∞

n=1X_n

|

_∆

λ

n

| < ∞

^{, and} (2) nX_n

|

_∆

λ

n

| =

O

(

¹

)

^.

We shall now proveTheorem 2.

Proof. Let y_nbe 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

¯

_nν

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

_nν

λ

νa_ν

.

⁽³⁾

Using(3)we may write

Y_n

=

n

X

ν=1



_a

ˆ

_nν

λ

ν

ν ν

^aν

=

n

X

ν=1



a

ˆ

_nν

λ

ν

ν

h X

^ν

r=1

ra_r

−

ν−1

X

r=1

ra_r

i

=

n1

X

ν=1

∆ν



a

ˆ

_nν

λ

ν

ν

 X

^ν

r=1

ra_r

+

a

ˆ

_nn

λ

n

n

n

X

ν=1

ν

^aν

=

n−1

X

ν=1

(

∆ν_a

ˆ

_nν

)λ

ν

ν +

¹

ν

^tν

+

n−1

X

ν=1

ˆ

a_n,ν+1

(

∆

λ

ν

) ν +

¹

ν

^tν

+

n−1

X

ν=1

a

ˆ

_n,ν+1

λ

ν+1

1

ν

^tν

+ (

ⁿ

+

₁

)

^ann

λ

nt_n n

=

T_n1

+

T_n2

+

T_n3

+

T_n4

,

^say

.

In order to prove our theorem 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

.

Using Hölder’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

∆νa

ˆ

_nν

λ

ν

ν +

¹

ν

^tν

    

k

=

O

(

¹

)

m+1

X

n=1

n^δk+k−1

 X

ⁿ⁻¹

ν=1

|

_∆νa

ˆ

_nν

k λ

ν

k

t_ν

|



k

=

O

(

¹

)

^m

+1

X

n=1

n^δk+k−1

 X

ⁿ⁻¹

ν=1

|

_∆νa

ˆ

_nν

k λ

ν

|

^k

|

t_ν

|

^k

 X

ⁿ⁻¹

ν=1

|

_∆νa

ˆ

_nν

| 

k−1

.

Using the fact that, from (vi),

{ λ

n

}

is bounded, (iii), (viii) and condition (1) ofLemma 1, I₁

=

O

(

¹

)

^m

+1

X

n=1

n^δk

(

^nann

)

^k−1n

−1

X

ν=1

| λ

ν

|

^k

|

t_ν

|

^k

|

_∆ν

ˆ

a_nν

|

=

O

(

¹

)

m+1

X

n=1

n^δk

(

^nann

)

^k−1

 X

ⁿ⁻¹

ν=1

| λ

ν

|

^k−1

| λ

ν

k

_∆νa

ˆ

_nν

k

t_ν

|

^k



=

O

(

¹

)

m

X

ν=1

| λ

ν

k

t_ν

|

^k

m+1

X

n=ν+1

n^δk

(

^nann

)

^k−1

|

_∆νa

ˆ

_nν

|

=

O

(

¹

)

m

X

ν=1

| λ

ν

k

t_ν

|

^k

m+1

X

n=ν+1

n^δk

|

_∆νa

ˆ

_nν

|

=

O

(

¹

)

m

X

ν=1

ν

^δk

| λ

ν

|

a_νν

|

t_ν

|

^k

=

O

(

¹

) X

^m

ν=1

| λ

ν

| h X

^ν

r=1

a_rr

|

t_r

|

^kr^δk

−

ν−1

X

r=1

a_rr

|

t_r

|

^kr^δk

i

=

O

(

¹

)

m−1

X

ν=1

∆

(|λ

ν

| ) X

^ν

r=1

|

t_r

|

^kr^δk−1

+ | λ

m

|

m

X

r=1

|

t_r

|

^kr^δk−1

=

O

(

¹

)

m−1

X

ν=1

|

_∆

λ

ν

|

X_ν

+

O

(

¹

)|λ

m

|

X_m

=

O

(

¹

).

Using Hölder’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

a

ˆ

_n,ν+1

(

∆

λ

ν

) ν +

¹

ν

^tν

    

k

≤

m+1

X

n=2

n^δk+k−1

h X

ⁿ⁻¹

ν=1

|ˆ

a_n,ν+1

||

_∆

λ

ν

| ν +

¹

ν |

t_ν

|

i

k

=

O

(

¹

)

m+1

X

n=2

n^δk+k−1

h X

ⁿ⁻¹

ν=¹

|ˆ

a_n,ν+1

||

_∆

λ

ν

k

t_ν

| i

k

=

O

(

¹

)

m+1

X

n=2

n^δk+k−1

h X

ⁿ⁻¹

ν=1

|ˆ

a_n,ν+1

||

_∆

λ

ν

k

t_ν

|

^k

ih X

ⁿ⁻¹

ν=1

|ˆ

a_n,ν+1

k

_∆

λ

ν

| i

k−1

.

It is easy to see that

n−1

X

ν=1

|ˆ

a_n,ν+1

k

_∆

λ

ν

| ≤

a_nn

n−1

X

ν=1

|

_∆

λ

ν

| ≤

Ma_nn

.

Using (iii)

I₂

:=

O

(

¹

)

m+1

X

n=2

n^δk

(

^nann

)

^k−1

n−1

X

ν=1

a

ˆ

_n,ν+1

|

_∆

λ

ν

k

t_ν

|

^k

=

O

(

¹

)

m

X

ν=1

|

_∆

λ

ν

k

t_ν

|

^k

m+1

X

n=ν+1

n^δk

(

^nann

)

^k−1a

ˆ

_n,ν+1

=

O

(

¹

)

m

X

ν=1

|

_∆

λ

ν

k

t_ν

|

^k

m+1

X

n=ν+1

n^δka

ˆ

_n,ν+1

.

Therefore

I₂

:=

O

(

¹

) X

^m

ν=1

ν

^δk

|

_∆

λ

ν

k

t_ν

|

^k

.

We may write

I₂

:=

O

(

¹

) X

^m

ν=1

ν

^δk

ν|

∆

λ

ν

| |

t_ν

|

^k

ν .

Using summation by parts and (viii)

I₂

:=

O

(

¹

)

m

X

ν=1

∆

(ν|

∆

λ

ν

| ) X

^ν

r=1

1

r

|

t_r

|

^kr^δk

+

O

(

¹

)

^m

|

_∆

λ

m

|

m

X

ν=1

1 r

|

t_r

|

^kr^δk

=

O

(

¹

)

m

X

ν=1

|

_∆

(ν

∆

λ

ν

)|

^Xν

+

O

(

¹

)

^m

|

_∆

λ

m

|

X_m

.

Using (vii), the properties ofLemma 1, and the fact that

{

X_n

}

is quasi-

β

-power increasing, I₂

:=

O

(

¹

)

m

X

ν=1

ν|

∆²

λ

ν

|

X_ν

+

O

(

¹

)

m−1

X

ν=1

|

_∆

λ

ν+1

|

X_ν+1

+

O

(

¹

)

^m

|

_∆

λ

m

|

X_m

=

O

(

¹

).

Using Hölder’s inequality, and condition (1) ofLemma 1.

I₃

:=

m+1

X

n=2

n^δk+k−1

|

T_n3

|

^k

=

m+1

X

n=2

n^δk+k−1

    

n−1

X

ν=1

ˆ

a_n,ν+1

λ

ν+1

1

ν

^tν

    

k

≤

m+1

X

n=2

n^δk+k−1

h X

ⁿ⁻¹

ν=1

| λ

ν+1

|

ν

a

ˆ

_n,ν+1

|

t_ν

| i

k

=

O

(

¹

)

m+1

X

n=2

n^δk+k−1

h X

ⁿ⁻¹

ν=1

| λ

ν+1

|

ν

^k

|

t_ν

|

^k_a

ˆ

_n,ν+1

i

× h X

ⁿ⁻¹

ν=1

a

ˆ

_n,ν+1

| λ

ν+1

| i

k−1

=

O

(

¹

)

m+1

X

n=2

n^δk

(

^nann

)

^k−1

h X

ⁿ⁻¹

ν=¹ 1

ν

^k

| λ

ν+1

k

t_ν

|

^ka

ˆ

_n,ν+1

i.

From (vi),

{ λ

n

}

is bounded. Therefore, using (iii), (v) and (viii) I₃

:=

O

(

¹

)

^m

+1

X

n=2

n^δk

(

^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

n^δk

(

^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

n^δk_a

ˆ

_n,ν+1

=

O

(

¹

)

m

X

ν=¹

| λ

ν+1

| ν |

t_ν

|

^k

ν

^δk

=

O

(

¹

)

m−1

X

ν=1

(|

∆

λ

ν+1

| )

^Xν+1

+

O

(

¹

)|λ

m+1

|

X_m+1

=

O

(

¹

).

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

m

X

n=1

n^δk+k−1

|

T_n4

|

^k

=

m

X

n=1

n^δk+k−1

   

(

ⁿ

+

1

)

^ann

λ

nt_n n

   

k

=

O

(

¹

)

m

X

n=1

n^δk+k−1

|

a_nn

|

^k

| λ

n

|

^k

|

t_n

|

^k

=

O

(

¹

) X

^m

n=1

n^δk

(

^nann

)

^k−1ann

| λ

n

|

^k−1

| λ

n

k

t_n

|

^k

=

O

(

¹

)

m

X

n=1

n^δka_nn

| λ

n

k

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

)

^.

(ii)

P

m+1

n=ν+1n^δk

|

^pn

PnPn−1

| =

O



ν^δk Pν



.

Let

(λ

n

) ∈

^b

v

0and if

{

X_n

}

is a quasi-

β

-power increasing sequence for some 0

< β <

1 such that (iii)

| λ

m

|

X_m

=

O

(

¹

)

^,

(iv)

P

m

n=1nX_n

|

_∆²

λ

n

| =

O

(

¹

)

^{, and} (v)

P

m

n=1n^δk−1

|

t_n

|

^k

=

O

(

^Xm

)

^. then the series

P

a_n

λ

nis summable

| ¯

N

,

^p

, δ|

k

,

^k

≥

1 for 0

≤ δ <

¹

/

^k.

Proof. Conditions (iii), (iv) and (v) ofCorollary 1are, respectively, conditions (vi), (vii))and (viii) ofTheorem 1.

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



References

[1] T.M. Fleet, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 37 (1957) 113–141.

[2] E. Savas, On quasiβ-power increasing sequences, Nonlinear Anal. 70 (2009) 1459–1464.

[3] E. Savas, On generalized|A|_k-summability factors, Comput. Math. Appl. (2009) (preprint).

Further reading

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

[2] B. Rhoades, E. Savas, A summability factor theorem for generalized absolute summability, Real Anal. Exchange 31 (2) (2005/06) 355–363.