Vol 2, No 3 (2011)

International Journal of Mathematical Archive-2(3), Mar. - 2011,

Page: 339-344

www.ijma.info

Available online through

A RECENT NOTE ON THE ABSOLUTE RIESZ SUMMABILITY

FACTOR OF INFINITE SERIES

W. T. Sulaiman*

Department of Computer engineering College of Engineering University of Mosul, Iraq.

E-mail: [email protected]

(Received on: 19-02-11; Accepted on: 23-02-11)

--- ABSTRACT

I

n this note we prove a new result concerning absolute summability factor of an infinite series via quasi -

f

-power increasing sequence, improving some conditions used by Bor [2] and Leindler [4] in recent results. In fact we are giving three improvements to the result of Leindler.

Key words: Infinite series, absolute summability, summability factor.

2000 (MSC): 40A05, 40D15, 40F05.

---

1. INTRODUCTION:

A positive sequence

( )

b

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

( )

c

_n and two

positive constants

A

and

B

such that

Ac

_n

≤

b

_n

≤

Bc

_n

.

A positive sequence

a

=

( )

a

_n is said to be quasi

β

−

power increasing if there exists a constant

(

,

)

≥

1

=

K

a

K

β

such that

K

n

β

a

_n

≥

m

β

a

_m (1.0)

holds for all

n

≥

m

.

If (1.0) stays with

β

=

0

then

a

is called a quasi increasing sequence . It should be noted that every almost increasing sequence is a quasi

β

−

power increasing sequence for any nonnegative

β

,

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

a

_n

=

n

−β

.

A positive sequence

α

=

(

α

_n

)

is said to be a quasi

−

f

−

power increasing sequence,

f

=

(

f

_n

)

, if there exits a constant

K

=

K

(

α

,

f

)

such that

K

f

_n

α

_n

≥

f

_m

α

_m

holds for

n

≥

m

≥

1

(see [5]). Clearly if

α

is quasi-f-power increasing sequence, then

α

f

is quasi increasing sequence.

By

t

_n we denote the nth (C, 1) mean of the sequence

(

na

_n

)

. The series

a

_n is said to be summable

,

1

,

1

,

k

≥

C

k if (see[3])

1

.

1

∞

<

∞

=

k n n

t

n

A series

a

_n with partial sums

s

_n is said to be summable

N

,

p

,

k

≥

1

,

k

n if (see [1])

₁

,

1

1

∞

<

−

₋

∞

=

−

k n n n

k

n

n

_T

_T

p

P

where

( )

p

_n is a sequence of positive numbers such that

=

→

∞

→

∞

=

n

as

p

P

n

v v n

0

and

1

.

0

=

=

n

v v v n

n

p

s

P

T

The following results are proved

Theorem: 1.1 [2]. Let

(

X

_n

)

be a quasi

β

−

power increasing sequence for some

0

<

β

<

1

,

and

( )

λ

_n be a real sequence. If the conditions

1

(

)

1

m n

m

n

P

P

n

=

Ο

=

(1.1)

λ

_n

X

_n

=

Ο

(

1

)

,

(1.2)

1

(

)

1

m k

n m

n

X

t

n

=

Ο

=

, (1.3)

(

)

1

m k

n m

n n

n

_t

_X

P

p

Ο

=

=

, (1.4)

and

(

2 ₁

)

1

2

,

+

∞

=

∆

−

∆

=

∆

∞

<

∆

_{n n n}

n

n n

X

n

λ

λ

λ

λ

(1.5)

are satisfied, then the series

a

_n

λ

_n is summable

N

,

p

,

k

≥

1

.

k

n

Theorem: 1.2[4]. If the sequence

(

X

_n

)

is quasi

β

−

power increasing for some

0

≤

β

<

1

,

( )

λ

_n satisfies the conditions

( )

m

m

n

n

ο

λ

=

=1

, (1.6)

and

( )

,

1

m

m

n

n

ο

λ

=

∆

=

(1.7)

further the conditions

( )

,

1

∞

<

∆

∆

∞

=

n n

n

X

n

β

λ

(1.8)

(1.3) and (1.4) holds, where

X

_n

( )

β

=

max

(

n

β

X

_n

,

log

n

)

,

∆

λ

_n

=

λ

_n

−

λ

_n+1

, then the series

a

_n

λ

_nis

summable

N

,

p

,

k

≥

1

.

k n

2. LEMMAS:

Page: 339-344

λ

_n

→

0

as

n

→

∞

,

(2.1)

(

)

,

1

∞

<

∆

∆

∞

=

n n

n

X

n

β

λ

(2.2)

are satisfied. Then

_n

+1

_X

_∆

₌

_Ο

( )

1

,

_as

_n

_→

_∞

,

n

n

λ

β

(2.3)

,

1

∞

<

∆

∞

=

n n

n

X

n

β

λ

(2.4)

and

n

β

X

_n

λ

_n

=

Ο

(

1

),

as

n

→

∞

,

(2.5) where

X

_n

(

β

)

=

n

β

X

_n

.

Proof: As

∆

λ

_n

→

0

,

and

n

β

X

_n is non-decreasing, we have

( )

v v

n v

n v

v n

n n

X

v

X

n

X

n

λ

λ

λ

β β β

∆

∆

Ο

=

∆

∆

=

∆

∞

= + ∞

= + +

1 1 1

1

=

Ο

( )

1

.

This proves (2.3). To prove (2.4), we observe that

∞

=

∞

= ∞

=

∆

∆

=

∆

1

1 n v n

v n

n

n

n

n

X

X

n

β

λ

β

λ

∞

=

∞

=

∆

∆

Ο

=

1

)

1

(

n v n

v n

X

n

β

λ

= ∞

=

∆

∆

Ο

=

v

n

n v

v

n

X

1 1

)

1

(

λ

β

∞

=

+

_∆

_∆

Ο

=

1 1

)

1

(

v

v v

X

v

β

λ

=

Ο

(

1

)

.

Finally

∞

= ∞

=

∆

≤

∆

=

n v

v v n v

v n n

n

X

v

X

n

X

n

λ

λ

λ

β β β

=

Ο

(

1

),

by (2.4).

Lemma: 2.2[4]. The conditions (1.6) and (1.7) implies (2.1).

3. MAIN RESULT:

Theorem: 3.1. If the sequences:

(

X

_n

)

is quasi

−

β

−

power increasing,

0

<

β

<

1

,

( )

λ

_n is a sequence of constants both satisfyingconditions (1.1),(2.1), (2.2) and

(

)

,

)

(

1

1

1 m

k n m

n

k n

X

m

t

X

n

n

β

β

=

Ο

=

− (3.1)

_n k

(

_m

)

m

n

k n n

n

_t

_m

_X

X

n

P

p

_β

β

=

Ο

=1 −

1

)

(

1

. (3.2)

Then the series

a

_n

λ

_n is summable

N

,

p

,

k

≥

1

.

k

n .

Lemma: 3.2 Conditions (3.1) and (3.2) are weaker than conditions (1.3) and (1.4) respectively.

Proof: If (1.3) holds, then we have

_nk

(

_m

)

m

n k m

n

k n k n

X

s

n

X

X

n

n

s

Ο

=

Ο

=

= − =

−

1 1 1 1

1

1

1

)

(

β ,

while if (3.1) is satisfied then,

1

1

1 1

)

(

)

(

1

1

−

= −

=

=

k

n m

n

k n k n m

n

k

n

s

n

X

X

n

n

s

n

β β

1

1

1 1

1

1 1

1

(

)

)

(

)

(

)

(

− =

− −

=

− =

−

∆

+

=

k

m m

n

k n k n m

n

k n n

v

k v k v

X

m

X

n

n

s

X

n

X

v

v

s

_β

β β

β

( )

1

(

)

1

1 1

)

(

)

(

1

− −

−

=

Ο

+

∆

Ο

=

k

m m

k n m

n

n

n

X

m

X

m

X

X

n

β β β β

(

) (

)

_m k m

n

k n k

n

m

n

X

n

X

m

X

X

m

1

)

((

1

)

)

(

)

(

)

(

1 1

1 1

1 1

β β

β β

Ο

+

−

+

−

Ο

=

−

=

− −

+ −

=

Ο

(

m

β

X

_m

)(

(

m

β

X

_m

)

k−1

)

+

Ο

(

m

β

X

_m

)

k

=

Ο

(

m

β

X

_m

)

k

.

Therefore (1.3) implies (1.4) but not conversely.

The proof of the other part is similar.

Remark: Althoughthe condition (1.1) has been added to the statement of theorem 3.1 butit may be mentioned that

Theorem: 3.1 give three improvements in comparing with Theorem 1.2 in the following sense:

1. (i) Conditions (3.1) and (3.2) are weaker than conditions (1.3) and (1.4) respectively (see lemma 3.2) .

(ii) The more advantage of our conditions is to obtain the desired result without any loss of powers through estimations. As an example the proof via conditions (1.3) and (1.4) impose to deal with

( )

_n

,

n

k n k n k

n

as

λ

λ

λ

λ

λ

=

=

Ο

loosing

λ

_n k−1 as considered to be

Ο

(

1

)

.

We have no such case via our conditions.

2. Condition (1.8) has been replaced by condition (2.2) which is weaker

Page: 339-344

Proof of Theorem: 3.1. Let

( )

T

_n denote the

(

N

,

p

_n

)

mean of the series

a

_n

λ

_n. Then

1

1

(

)

.

0 1 0 0 v v n v v n n v r r r n v v n

n

P

P

a

P

a

p

P

T

λ

λ

= − = =

−

=

=

Therefore, for

n

≥

1

,

we have

−

=

=

− = − = − −

− v v

n v v n n n n v v v v n n n n n

P

v

va

P

P

p

a

P

P

P

p

T

T

λ

₁

λ

1 1 1 1 1 1

1

,

and via Abel’s transformation,

− = − − = − −

+

∆

+

−

+

=

−

1 1 1 1 1 1 1

1

1

n v v v v n n n n v v v v n n n n n n n n

n

P

t

P

P

p

v

v

t

p

P

P

p

t

P

p

n

n

T

T

λ

λ

λ

− = −

+

1 1 1

1

n v v v v n n n

v

t

P

P

P

p

λ

=

T

_n1

+

T

_n2

+

T

_n3

+

T

_n4

.

To prove the theorem, by Minkowski’s inequality, it is enough to show that

,

1

,

2

,

3

,

4

.

1 1

=

∞

<

− ∞ =

j

T

p

P

k nj k n n n

Applying Holder’s inequality, we have, in view of (2.4), (2.5) and (3.2),

( )

_n k _n k

m n n n m n k n k n n

_t

P

p

T

p

P

λ

= = −

Ο

=

1 1 1 1

1

( )

_k

(

_{n n}

)

k _n n

k n m

n n

n

_n

_X

X

n

t

P

p

λ

λ

β β 1 1

1

(

)

1

₋ −

=

Ο

=

( )

_{k n} n k n m n n n

X

n

t

P

p

λ

β 1

1

(

)

1

₋

=

Ο

=

( )

( )

_{k m} n k n m n n n n m n n v k v k v v v

X

n

t

P

p

X

v

t

P

p

λ

λ

_β

β

∆

+

Ο

Ο

=

₋ = − = = − 1 1 1 1 1 1

)

(

1

)

(

1

( )

_n

( )

_{m m} m

n

n

m

X

X

n

β

λ

₁

β

λ

1

1 1

Ο

+

∆

Ο

=

− =

=

Ο

( )

1

,

( )

1 1 1 1 1 1 1 2 1 1 2 2 1

1

− − = − − = + = − + = −

Ο

=

k n v n v k n v v k v v m

n n n n m n k n k n n

P

p

t

p

P

P

p

T

p

P

λ

( )

+ + = − =

Ο

=

1 1 1 1

1

m v n n n

n k v m v k v v

P

P

p

t

p

λ

( )

_v k _v k m v v v

_t

P

p

λ

=

Ο

=

1

1

=

Ο

( )

1

,

as in the case of

T

_n1

.

( )

1 1 1 1 1 1 1 2 1 3 1 2 1

)

(

1

− − = − = − + = − + = −

∆

∆

Ο

=

k v n v v v n v k v k v k v m n k n n n k n m n k n

n

_v

_X

X

v

t

P

P

P

p

T

p

P

λ

λ

β β

( )

+ + = − − =

∆

Ο

=

1 1 1 1

1

(

)

1

m v n k n n n v k v k v m v k v

P

P

p

X

v

t

P

_β

λ

( )

(

_v

)

m v k v k v

v

X

v

t

v

β

∆

λ

Ο

=

= − 1 1

)

(

1

1

( )

(

)

( )

_m m v k v k v v m v v r k r k r

m

X

v

t

v

v

X

r

t

r

β

∆

∆

λ

+

Ο

β

∆

λ

Ο

=

= − − = = − 1 1 1 1 1

1

₍

₎

1

1

)

(

1

1

(

_{v v}

)

_{m m} m

v

v

v

mX

X

v

β

λ

λ

λ

∆

Ο

+

∆

∆

+

+

∆

−

Ο

=

− =

)

1

(

)

1

(

)

1

(

1 1

( )

( )

( )

_{m m} m v v v v m v

v

v

X

m

X

X

v

β

λ

β

λ

β

λ

∆

Ο

+

∆

∆

Ο

+

∆

Ο

=

+ = + = 1 1 1 1

1

1

1

=

Ο

( )

1

.

In view of (2.5) and (3.1),

( )

1 ` 1 1 1 1 1 2 1 4 1 1 2

1

− − = − = + = − − + =

Ο

=

k n v v k v k v n v v m n k n n n k n k m n n n

v

P

t

v

P

P

P

p

T

p

P

λ

( )

+ + = − =

Ο

=

1 1 1 1

1

m v n n n

n k v k v m v v

P

P

p

t

v

P

λ

( )

+ + = − =

Ο

=

1 1 1 1

1

m v n n n

n k v k v m v v

P

P

p

t

v

P

λ

=

Ο

( )

1

_k

(

_{v v}

)

k _v v k v m v

X

v

X

v

t

v

λ

λ

β β

1 1

1

(

)

1

−

− =

( )

_{k v} v

k v m

v

v

X

t

v

β 1

λ

1

(

)

1

1

₋

=

Ο

=

( )

( )

_{k m} v k v m v v m v v r k r k r

X

v

t

v

X

r

t

r

β

∆

λ

+

Ο

β

λ

Ο

=

₋ = − = = − 1 1 1 1 1 1

)

(

1

1

)

(

1

1

( )

_v

( )

_{m m} m

v

v

m

X

X

v

β

λ

β

λ

1

1

1

Ο

+

∆

Ο

=

=

=

Ο

( )

1

.

This completes the proof of the theorem.

REFERENCES:

[1] H. Bor, A note on two summability methods, Proc. Amer. Math. Soc. 98 (1986), 81-84.

[2] H. Bor and L. Debnath, Quasi

β

−

power increasing sequences, Internal. J. And Math. Sci., 44 (2004), 371-2376.

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

[4] L. Leindler, A recent note on the absolute Riesz summability factors, J. Ineq. Pure Appl. Math. Volume 7, Issue 2, Article 44, 2006.