Applied Mathematics Letters

Applied Mathematics Letters 22 (2009) 1803–1809

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Applied Mathematics Letters

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

On a result of Flett for Cesáro matrices

Dansheng Yu

Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang 310036, China

a r t i c l e i n f o

Article history:

Received 21 April 2008

Received in revised form 17 June 2009 Accepted 17 June 2009

Keywords:

Cesáro matrix Conservative matrix Absolute summability

a b s t r a c t

Let snbe the partial sums of the seriesP∞

n=0an.We consider the sufficient conditions for a matrix T=(^tnk)^{such that}

∞

X

n=1

αn|s_n−s_n−₁|^k< ∞

implies

∞

X

n=1

βn|tn−tn−1|^s< ∞

where{αn}and{βn}are two given positive sequences and k,^s > ⁰, ^and{t_n}is the T - transformation of{_sn}.Our results extend the related results of Flett [T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc.

London Math. Soc. 7 (1957) 113–141] and Savas and Sevli [E. Savaş, H. Sevli, On extension of a result of Flett for Cesáro matrices, Appl. Math. Lett. 20 (2007) 476–478].

© 2009 Elsevier Ltd. All rights reserved.

Let s_nbe the partial sums of the series

P

∞

n=0a_n. The Cesáro means of order

α

of the series

P

∞

n=0a_nare defined by

σ

n^α

:=

¹

A^α_n

n

X

j=0

A^α−_n−j¹s_j

,

ⁿ

=

0

,

¹

, . . . ,

where

A^α_n

:=

⁰

(

ⁿ

+ α +

¹

)

0

(α +

¹

)

0

(

ⁿ

+

1

) ,

ⁿ

=

0

,

¹

, . . . .

Let

(

^C

, α)

, the Cesáro matrix of order

α

, be the lower triangular matrix A^α−_n−ν¹

/

^Aαn

.

Flett [1] introduced the concept of absolute summability of order k. A series

P

∞

n=0a_nis said to be summable

|

C

, α|

k

,

^k

≥

1

, α > −

^{1, if}

∞

X

n=1

n^k−1



 σ

n^α−1

− σ

n^α

 

k

< ∞.

Das [2] defined a matrix T

:=

t_nj

to be absolutely kth-power conservative for k

≥

1, denoted by T

∈

B

(

^Ak

)

, that is, if

{

_sn

}

_satisfies

∞

X

n=₁

n^k−1

|

s_n

−

s_n−1

|

^k

< ∞,

⁽¹⁾

E-mail address:danshengyu@yahoo.com.cn.

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

doi:10.1016/j.aml.2009.06.023

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then

∞

X

n=1

n^k−1

|

t_n

−

t_n−1

|

^k

< ∞,

⁽²⁾

where t_n

=

n

X

j=0

t_njs_j

.

Flett [1] established the following inclusion theorem for

|

C

, α|

k. If the series

P

∞

n=0a_n is summable

|

C

, α|

k, it is also summable

|

_C

, α|

rfor each r

≥

k

≥

1

, α > −

¹

, β > α +

¹_k

−

¹

r. Especially, a series

P

∞

n=0a_nwhich is

|

_C

, α|

ksummable is also

|

C

, β|

ksummable for k

≥

1

, β > α > −

¹

.

If one sets

α =

0, from the above inclusion result, we have Theorem A. Let k

≥

1, then

(

^C

, β) ∈

^B

(

^Ak

) (

^k

≥

1

)

^for

β ≥

⁰

.

Recently, Savaş and Şevli [3] claimed thatTheorem Aholds for all

β > −

¹

.

However, in their proof (see P. 477, [3]), they said that Applying Hölder’s inequality, we have

∞

X

n=1

1 n

  τ

n^β

 

k

=

∞

X

n=1

1 n

    

1 A^β_n

n

X

v=1

A^β−_n−v¹

v

^av

    

k

≤

∞

X

n=1

1 n



A^β_n



k n

X

v=1

A^β−_n−v¹

v

^k

|

a_v

|

^k

n

X

v=1

A^β−_n−v¹

!

k−1

.

⁽³⁾

Unfortunately,(3)is not true when

−

1

< β <

0. In fact, take a₁

=

1

,

^a2

=

a₃

= · · · =

0, then the middle term of(3)equals

∞

X

n=1

1 n

    

A^β−_n−1¹ A^β_n

    

k

>

^Aβ−

1 0

A^β₁

!

k

=

¹

(β +

¹

)

^k

,

while the right-hand side of(3)is

∞

X

n=1

A^β−_n−1¹ n



A^β_n



k A^β_n

−

A^β−_n ¹

k−1

<

^Aβ−

1 0



A^β₁



k



A^β₁

−

A^β−₁ ¹



k−1

=

¹

(β +

¹

)

^k

,

where we used the following facts:

A^β−_n−1¹

<

⁰

,

^Aβ−n ¹

<

^{0 and Aβ}n

>

⁰

,

ⁿ

≥

2

, −

¹

< β <

⁰

.

Since(3)plays an essential role, their proof of the result is not true. The correctness ofTheorem Afor

−

1

< β <

^{0 need} to be further verified. In spite of this, Savaş and Şevli [3] have given a simple proof ofTheorem Aby adopting some new ideas.

Motivated byTheorem A, a natural problem is that what are the sufficient conditions for T such that

∞

X

n=1

α

n

|

s_n

−

s_n−1

|

^k

< ∞

⁽⁴⁾

implies that

∞

X

n=1

β

n

|

t_n

−

t_n−1

|

^s

< ∞

⁽⁵⁾

where

{ α

n

}

and

{ β

n

}

are two given positive sequences and k

,

^s

>

^0?

The object of this note is to answer the above problem by establishing the followingTheorem 1. We also prove few corollaries(1)–(4)on absolute Cesáro summability andCorollary 5on absolute Riesz summability.

Theorem 1. Let

{ α

n

}

and

{ β

n

}

be positive sequences and k

>

⁰

,

^s

≥

1. Assume that T

:=

t_nj

is a lower triangular matrix satisfying

P

n

j=0t_nj

=

1 for any n

≥

0. If

∞

X

n=i

β

n n

X

v=1

 e

t_nv



 v

!

s−1

 e

t_ni



 =

O i^1−k

α

i

,

ⁱ

=

₁

,

²

, . . . ,

⁽⁶⁾

and

(

ⁱ

|

s_i

−

s_i−1

| )

^s−k

=

O

(

¹

) ,

⁽⁷⁾

where

e

t_ni

:=



 

 

n

X

j=i

t_nj

−

n−1

X

j=i

t_n−1,j

,

⁰

≤

i

≤

n

−

1

,

t_nn

,

ⁱ

=

n

,

then(4)implies(5).

Proof. Since (set s−1

:=

0)

t_n

=

n

X

j=0

t_njs_j

=

n

X

j=0

t_nj

j

X

i=0

(

^si

−

s_i−1

)

!

=

n

X

i=0

(

^si

−

s_i−1

) X

ⁿ

j=i

t_nj

! ,

then

t_n

−

t_n−1

=

n

X

i=0

(

^si

−

s_i−1

)

n

X

j=i

t_nj

!

−

n−1

X

i=0

(

^si

−

s_i−1

)

n−1

X

j=i

t_n−1,j

!

=

n

X

i=0

e

t_ni

(

^si

−

s_i−1

) =

n

X

i=1

e

t_ni

(

^si

−

s_i−1

) ,

where in the last inequality, we used the fact that

e

t_n0

=

0, which follows from

P

n

j=0t_nj

=

1 and the definition of

e

t_n0

.

By Hölder’s inequality,(6)and(7), we have

∞

X

n=₁

β

n

|

t_n

−

t_n−1

|

^s

=

∞

X

n=₁

β

n

    

n

X

i=₁

e

t_ni

(

^si

−

s_i−1

)     

s

≤

∞

X

n=1

β

n n

X

i=1

 e

t_ni





i

!

s−1 n

X

i=1

 e

t_ni





i i^s

|

s_i

−

s_i−1

|

^s

!

=

O

(

¹

)

∞

X

n=1

β

n n

X

i=1

 e

t_ni





i

!

s−1 n

X

i=1

 e

t_ni





i^k−1

|

_si

−

s_i−1

|

^k

!

=

O

(

¹

)

∞

X

i=1

i^k−1

|

s_i

−

s_i−1

|

^k





∞

X

n=i

β

n n

X

v=¹

 e

t_nv



 v

!

s−1

 e

t_ni









=

O

(

¹

)

∞

X

i=1

α

i

|

s_i

−

s_i−1

|

^k

.

Theorem 1is proved. 

A non-negative sequence

{

_an

}

is said to be almost decreasing, if there is a positive constant K such that a_n

≥

Ka_m

holds for all n

≤

m, and it is said to be quasi-

β

-power decreasing with some real number

β

^{, if}



n^βa_n

is almost decreasing.

It should be noted that every decreasing sequence is an almost decreasing sequence, and every almost decreasing sequence is a quasi-

β

-power decreasing sequence for any non-positive index

β

, but the converse is not true.

The following properties of A^α_nare well known (see [4], P.77):

Lemma 1. (i) A^α_nis positive for

α > −

1, increasing (as a function of n) for

α >

0 and decreasing for

−

1

< α <

⁰

;

and A⁰_n

=

1 for all n

.

(ii) A^α_n

=

_0(α+^nα

1) 1

+

O ¹_n

.

We give some applications ofTheorem 1now.

Corollary 1. Let T be the Cesáro matrix of order

α ≥

^{0 and k}

>

⁰

,

^s

≥

1. Assume that

{ β

n

}

is a positive sequence such that



n^1−s

β

n

is quasi-

ε

-power decreasing for some

ε > −

¹

.

^{If (7)}holds, then(4)implies(5)with

α

n

=

n^k−s

β

n

.

Proof. Let

t_nj

:=

^A

α−¹ n−j

A^α_n

,

^j

=

0

,

¹

, . . . ,

ⁿ

.

It is clear that

n

X

j=0

t_nj

=

¹ A^α_n

n

X

j=0

A^α−_n−j¹

=

1

.

For 0

≤

i

≤

n

−

1, a direct calculation yields

e

t_ni

=

¹ A^α_n

n

X

j=i

A^α−_n−j¹

−

¹ A^α_n−1

n

X

j=i

A^α−_n−1¹_−j

=

¹ A^α_n

n−i

X

j=0

A^α−_j ¹

−

¹ A^α_n−1

n−1−i

X

j=0

A^α−_j ¹

=

^A

αn−i

A^α_n

−

^A

αn−1−i

A^α_n−1

=

ⁱ n

A^α−_n−i¹

A^α_n

,

⁽⁸⁾

and

e

t_nn

=

^A

α−1 0

A^α_n

=

¹

A^α_n

.

⁽⁹⁾

By(8)and(9)andLemma 1, we have

n

X

v=¹

 e

t_nv



 v =

¹

nA^α_n

n

X

v=¹

A^α−_n−v¹

=

¹ nA^α_n

n

X

v=⁰

A^α−_n−v¹

−

^A

α−1 n

nA^α_n

=

O



1 n



.

⁽¹⁰⁾

Since



n^1−s

β

n

is quasi-

ε

-power decreasing for some

ε > −

^{1, by}(10), we have

∞

X

n=i

β

n n

X

v=¹

 e

t_nv



 v

!

s−1

 e

t_ni



 =

O

(

¹

)

∞

X

n=i

n^1−s

β

n

 e

t_ni





=

O

(

¹

)

∞

X

n=i

n^1−s+ε

β

n

 e

t_ni





n^ε

=

O i^2−s+ε

β

i

∞

X

n=i

A^α−_n−i¹ n^1+εA^α_n

.

Therefore, byLemma 1and the condition

ε > −

^{1, we have}

∞

X

n=i

β

n n

X

v=1

 e

t_nv



 v

!

s−1

 e

t_ni



 =

O i^2−s+ε

β

i

1

i^1+εA^α_i

2i

X

n=i

A^α−_n−i¹

+

∞

X

n=2i+1

A^α−_n−i¹ n^1+εA^α_n

!

=

O i^2−s+ε

β

i

1

i^1+εA^α_i

i

X

n=0

A^α−_n ¹

+

∞

X

n=2i+1

(

ⁿ

−

i

)

^α−1 n^1+ε+α

!

=

O i^2−s+ε

β

i

i^−1−ε

+

∞

X

n=2i+1

n^−2−ε

!

=

O i^1−s

β

i

=

_{O i}^1−k

α

i

.

Therefore, we obtainCorollary 1by applyingTheorem 1. 

Let

β

n

=

n^δk+k−1log^γn

,

ⁿ

≥

1

,

^k

>

⁰

,

^s

≥

1. If

δ <

¹⁺_k^s−k

,

then there is an

ε

^{such that}

−

1

< ε < −δ

^k

−

k

+

s. Thus, by noting that n^ε n^1−s

β

n

=

n^{ε−(−δk−k+s)}log^γn, we see that



n^1−s

β

n

is quasi-

ε

-power decreasing for

γ ∈

^R

.

Now, applying Corollary 1, we have

Corollary 2. Let T be the Cesáro matrix of order

α ≥

0, and let k

>

⁰

,

^s

≥

1 and

γ ∈

^{R. If}

δ <

^1+s_k^−kand(7)holds, then

∞

X

n=1

n^{δk+2k−s−1}log^γn

|

s_n

−

s_n−1

|

^k

< ∞

implies

∞

X

n=1

n^δk+k−1log^γn



 σ

n^α

− σ

n^α−1

 

s

< ∞, α ≥

⁰

.

In particular (by taking s

=

k), if

δ <

¹_k^{, then}

∞

X

n=1

n^δk+k−1log^γn

|

s_n

−

s_n−1

|

^k

< ∞

implies

∞

X

n=1

n^δk+k−1log^γn



 σ

n^α

− σ

n^α−1

 

k

< ∞, α ≥

⁰

.

Remark 1. By taking s

=

k

, δ = γ =

^{0 in}Corollary 2, we obtainTheorem A.

Corollary 3. Let T be the Cesáro matrix of order

−

1

< α <

^{0 and k}

>

⁰

,

^s

≥

1. Assume that

{ β

n

}

is a positive sequence such that



n^{(1+α)(1−s)}

β

n

is quasi-

ε

-power decreasing for some

ε > −

¹

.

^{If (7)}holds, then(4)implies(5)with

α

n

=

n^k−(1+α)s

β

n

.

Proof. When

−

1

< α <

0, it is clear that A^α−_n ¹

<

^{0 for n}

=

1

,

²

, . . . ,

^{and Aα−}0 ¹

=

1. ByLemma 1,(8)and(9), we get

n

X

v=1

 e

t_nv





v =

¹

nA^α_n

    

n−1

X

v=1

A^α−_n−v¹

    

+

^A

α−1 0

nA^α_n

=

¹ nA^α_n

    

n

X

v=0

A^α−_n−v¹

−

A^α−_n ¹

−

A^α−₀ ¹

    

+

¹ nA^α_n

=

¹ nA^α_n





A^α_n

−

A^α−_n ¹

−

A^α−₀ ¹

  +

1 nA^α_n

=

O



1 n^1+α



,

⁽¹¹⁾

2i

X

n=i+1

 

A^α−_n−i¹



 =     

2i

X

n=i+1

A^α−_n−i¹

    

=     

i

X

n=0

A^α−_n ¹

−

A^α−₀ ¹

    

= 



A^α_i

−

A^α−₀ ¹



 =

O

(

¹

) ,

⁽¹²⁾

and

∞

X

n=2i+1

 

A^α−_n−i¹





n^1+εA^α_n

=

O

(

¹

)

∞

X

n=2i+1

(

ⁿ

−

i

)

^α−1 n^1+ε+α

=

O

(

¹

)

∞

X

n=2i+1

n^−2−ε

=

O i^−1−ε

(13)

for

ε > −

1. Combining(11)–(13)and noting that



n^{(1+α)(1−s)}

β

n

is quasi-

ε

-power decreasing with

ε > −

^{1, we have}

∞

X

n=i

β

n n

X

v=1

 e

^tnv



 v

!

s−1

 e

^tni

  =

O

(

¹

)

∞

X

n=i

n^{(1+α)(1−s)}

β

n

 e

^tni

 

=

O

(

ⁱ

)

∞

X

n=i

n^{(1+α)(1−s)}

β

n

    

A^α−_n−i¹ nA^α_n

    

=

O



i^{1+(1+α)(1−s)}

β

i

1 iA^α_i

 +

O

(

ⁱ

)

∞

X

n=i+1

n^{(1+α)(1−s)}

β

n

    

A^α−_n−i¹ nA^α_n

    

=

O



i^{1+(1+α)(1−s)}

β

i

1 iA^α_i

 +

O

(

ⁱ

)

∞

X

n=i+1

n^{(1+α)(1−s)+}

β

n

    

A^α−_n−i¹ n^1+A^α_n

    

=

O i^1−(1+α)s

β

i

+

O i^{1+(1+α)(1−s)+ε}

β

i

∞

X

n=i+1

 

A^α−_n−i¹





n^1+εA^α_n

=

O i^1−(1+α)s

β

i

+

O i^{1+(1+α)(1−s)+}

β

i

2i

X

n=i+1

 

A^α−_n−i¹





n^1++α

+

O i^{1+(1+α)(1−s)+ε}

β

i

∞

X

n=2i+1

 

A^α−_n−i¹





n^1+εA^α_n (byLemma 1again)

=

O i^1−(1+α)s

β

i

+

_{O i}^{(1+α)(1−s)−α}

β

i

2i

X

n=i+1

 

A^α−_n−i¹



 +

O i^{1+(1+α)(1−s)+ε}

β

i

∞

X

n=2i+1

 

A^α−_n−i¹





n^1+εA^α_n

=

O i^1−(1+α)s

β

i

+

O i^{(1+α)(1−s)}

β

i

=

O i^1−(1+α)s

β

i

=

O i^1−k

α

i

.

Therefore, we obtainCorollary 3byTheorem 1. 

Similar to the deduction ofCorollary 2, we have the following corollary ofCorollary 3:

Corollary 4. Let T be the Cesáro matrix of order

−

1

< α <

0 and let k

>

⁰

,

^s

≥

1 and

γ ∈

^R

.

^If

δ <

^2+(1+α)(_k^{s−1)−kand(7)} holds, then

∞

X

n=1

n^{δk+2k−(1+α)s−1}log^γn

|

_sn

−

s_n−1

|

^k

< ∞

implies

∞

X

n=1

n^δk+k−1log^γn



 σ

n^α

− σ

n^α−1

 

s

< ∞, α ≥

⁰

.

In particular (by taking s

=

k), if

δ <

^αk−_k^α+1

,

^then

∞

X

n=1

n^{δk+k−αk−1}log^γn

|

s_n

−

s_n−1

|

^k

< ∞

implies

∞

X

n=1

n^δk+k−1log^γn



 σ

n^α

− σ

n^α−1

 

k

< ∞, α ≥

⁰

.

Corollary 5. Let s

=

k

≥

1

, δ ∈

^R

, {

^pn

}

be a positive sequence, such that P_n

= P

n

n=0p_i

→ ∞

as n

→ ∞ ,

^and

n

X

i=1

P_i

i

=

O

(

^Pn

) ,

⁽¹⁴⁾

∞

X

n=i



P_n p_n



_δk+k−s

p_n P_nP_n−1

=

O



P_i p_i



_δk+k−s

1 P_i−1

!

.

⁽¹⁵⁾

If (7)is satisfied, then(4)implies(5)with

α

n

=

n^k−1



Pn pn



δk+k−s

, β

n

=



Pn pn



δk+k−1

. Proof. Let

t_nj

:=

^pj

P_n

,

^j

=

0

,

¹

, . . . ,

ⁿ

,

then

P

n

j=0t_nj

=

1 for any n

≥

0. Direct calculations yield

e

t_ni

=

n

X

j=i

t_nj

−

n−1

X

j=i

t_n−1,j

=

^pn P_n

+



1 P_n

−

¹

P_n−1



n−1

X

j=i

p_j

=

^pnPi−1 P_nP_n−1

,

¹

≤

i

≤

n

−

1

,

⁽¹⁶⁾

and

e

t_n0

=

0

, e

t_nn

=

^pn

P_n

.

⁽¹⁷⁾

Therefore, by(14), we have

n

X

v=1

 e

t_nv



 v =

^pn

P_nP_n−1 n

X

v=1

P_v−1

v =

O



p_n P_n

 .

Now, by(15), we have

∞

X

n=i

β

n n

X

v=1

 e

t_nv



 v

!

s−1

 e

t_ni



 =

O

∞

X

n=i



P_n p_n



δk+k−1



p_n P_n



s−1

p_nP_i−1

P_nP_n−1

!

=

O

(

^Pi−1

)

∞

X

n=i



P_n p_n



_δk+k−s

p_n P_nP_n−1

=

O



P_i p_i



δk+k−s

! ,

which completes the proof ofCorollary 5. 

Remark 2. If

δ =

⁰

,

^s

=

k

,

⁽¹⁵⁾can be deduced from the facts that P_n

→ ∞

as n

→ ∞

, and _P^pn

nPn−1

=

¹

Pn−1

−

¹

Pn

.

Remark 3. If

{

p_n

}

, in addition, satisfies np_n

≈

P_n, then

α

ninCorollary 5becomes n^δk+k−1when s

=

k

.

Acknowledgements

The author is grateful to the referees for their careful reading and kind suggestions which improved the presentation of the paper. The research is supported by Research Project of Hangzhou Normal University.

References

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

[2] G. Das, A tauberian theorem for absolute summability, Proc. Cambridge Philos. Soc. 67 (1970) 321–326.

[3] E. Savaş, H. Sevli, On extension of a result of Flett for Cesáro matrices, Appl. Math. Lett. 20 (2007) 476–478.

[4] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1977.