A New Application of Generalized Almost Increasing Sequence

A New Application of Generalized Almost Increasing Sequence

Aditya Kumar Raghuvanshi, B.K. Singh, and Ripendra Kumar

Department of Mathematics

IFTM University, Moradabad, (U.P.) India, 244001

Abstract

A new result concerning absolute summability of infinite series us- ing almost increasing sequence is obtained. An application gives some generaliztion of Sulaiman [3].

Keywords: Absolute summability, almost increasing sequence and se- quence of bounded variation.

1. n

   n

   n

   n

   Let an be an infinite series with sequence of partial sums (sn). By u, t

   we denote the nth Cesaro mean of order > 1 of the sequence (sn), (nan)

   respectively, that is

   n

   n

   u = 1 A1s

   A

   A

   (1.1)

   n

   n v=0 n

   nv v

   n

   n

   n

   n

   nv

   nv

   t = A A1vav (1.2)

   v=0

   v=0

   v=0

   v=0

   The series an is summable |C, |k, k 1 if

   | n |t | < . (1.3)

   | n |t | < . (1.3)

   n

   n

   n1

   n1

   n

   n

   nk1|u u k 1 k

   n=1

   n=1

   n=1

   n=1

   n=1

   n=1

   n=1

   n=1

   For = 1, |C, |k summability reduces to |C, 1|k summability. Let (pn) be a sequence of constants such that

   Pn = p0 + p1 + …. + pn as n (1.4)

   The positive sequence (bn) is said to be almost increasing sequence if there exists a positive increasing sequence (cn) and two positive constants M and N

   such that, Mcn bn Ncn. Every increasing sequence is almost increasing sequence.

   A sequence (n) is said to be of bounded variation, denoted by (n) BV

   if

   |n| = |n n+1| < .

   n=1 n=1

Here we generalized the Sulaiman theorem [3].

Theorem 2.1. Let p > 0, pn 0 and (pn) be a non increasing sequence (Sulaiman [3]) (Xn) be almost increasing sequence if the following conditions (Bor [1]), (Mazhar [2]) and (Verma [4]). Where n BV

n|2n|Xn < (2.1)

n=1

|n|Xn = O(1) as n (2.2)

nXn|n| = 0(1) as n (2.3)

Xn|n| < (2.4)

n=1

v = O(1) as v (2.5)

vv = O(1) as v (2.6)

and

vk1 k

v

v

n

n

v=1

Xk1 |tv |

= O(Xn) as n (2.7)

are satisfied, then the series annn is summable |C, 1, |k, k 1, 0.

Proof. Let Tn be the n-th (C, 1) means of the sequence (nannn).

),

),

Therefore

1 n

T = va .

n

Abels transformation gives

n + 1

v=1

v v v

/

/

1 n1

Tn = n + 1

v=1

(vv )

v

v

r=1

r ar + nn

n

n

v=1

v av \

/

/

1 n1

=

n + 1

v=1

(v + 1)tv vv +

n1

v=1

(v + 1)tvv+1v \

+ tnnn.

= Tn,1 + Tn,2 + Tn,3.

In order to complete the proof, by Minkowaskis inequality, it is sufficient to show that

nk1|Tn,j|k < , j = 1, 2, 3

n=1

Applying H¨older inequality, we have

m+1

n=2

nk1|Tn,1|k =

m+1

n=2

1 n1

nk1

nk1

n + 1 v=1

m+1 n1

m+1 n1

= O(1)

= O(1)

vk|tv|k|v|k|v|k ·

vk|tv|k|v|k|v|k ·

nk1

nk1

nk1

n=2

nk

v=1

n=2

nk

v=1

k

(v + 1)tv vv

/ n1

v=1

v=1

\k1

1

1

n=2

nk

v=1

v=1

n=2

nk

v=1

v=1

= O(1)

m+1 n1

nk

vk|tv|k|v|k|v|k (n)k1

n=2

m+1

v=1

n1

= O(1) nk2 vk|tv|k|v|k|v|k

n=2 m

v=1

m+1

= O(1) vk|tv|k|v|k|v|k

v=1

m

m

n =v+1

v

v

nk2

= O(1)

v=1 m

vk|tv|k|v|k|v|k

v

v

xk2dx

= O(1) vk|tv|k|v|k|v|k (v)k1

v=1 m

= O(1) vk1|tv|k|vv|k|v|k

v=1 m

= O(1) vk1|tv|k|v|k

v=1

m

m

vk1|t |k| |

v

v

= O(1)

= O(1)

v=1

m

m

v=1

m1

v

Xk1

v

|v|

r=1

v

v

v

|tr|k rk1 Xk1

+ O(1)|m|

m

m

v=1

v

v

|tv|kvk1

Xk1

= O(1) Xv|v| + O(1)Xm|m| = O(1)

v=1

m+1

n=2

nk1|Tn,2|k =

m+1

n=2

nk1 1

k

(v + 1)tvv+1v

n + 1

n + 1

nk1

nk1

= O(1)

m+1 n1

nk

vk|tv|k|v+1|k|v|k

n=2 v=1

m+1 n1

m+1 n1

= O(1)

= O(1)

v

v

v+1

v+1

nk1 vk|t |k|

|k| |

/ n1

·

·

v

v

\k1

X | |

X | |

n=2 m

nk

v=1

k k

Xk1

v

v

m+1

v v

v=1

= O(1) v |tv| |v |

Xk1

nkk1

v=1 v

·

·

m k k

n=v+1

= O(1) v |tv| |v | vkk

Xk1

= O(1)

v=1

m

m

v=1

v

|tv|kvk1

v

v

Xk1

|vv|

v

v

= O(1)

m

m

v=1

m1

|(v|v|)|

v

v

r=1

|tr|krk1

r

r

Xk1

m1

+ O(1)|m

m

m

v=1

|tv|k vk1

Xk1

= O(1) v|2V |Xv + O(1) Xv|v| + O(1)m|m|Xm

= O(1)

v=1

v=1

And

m m

nk1|Tn,3|k = nk1|tnnn|k

n=1

n=1

= O(1)

m

m

n=1

|tn

n

n

|k · nk1 Xk1

|n|

= O(1) as in the case of Tn,1

This completes the proof of theorem.

1. Bor, H.; On a new application of almost increasing sequences, to be published in Mathem and Computer Modelling, 35 (2011), 230-233.

2. Mazhar, S.M.; Absolute summablity factor of infinite series Kyungpook Math. J. 39 (1999).

3. Sulaiman, W.T.; On a application of almost increasing sequences, Bul. of Math. Analysis and applications Vol. 4 (2012), 29-33.

4. Verma, R.S.; On the absolute N¨orlund summability factors Riv. Mat. Univ. Parma (4), 3 (1977), 27-33.