IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 2 (Sep. - Oct. 2013), PP 90-94 www.iosrjournals.org

On An Extension Of Absolute Cesaro Summability Factor Aditya Kumar Raghuvanshi, B.K. Singh & Ripendra Kumar Department of Mathematics IFTM University Moradabad (U.P.) India-244001

Abstract: In this paper we have generalised the theorem of Sulaiman which gives some unknown results and known result.s.. Keywords: Absolute summability, increasing sequence, Hölder inequality Minkowski inequality and infinite series.

I. A positive sequence

Introduction

(bn ) is called an almost increasing sequence if there exist a positive increasing

(cn ) and two positive constants A and B such that (Bari [2]) Acn  bn  Bcn (1.1) A sequence (n ) is said to be of bounded variation denoted by (n )  BV if

sequence

n 1

n 1

 | n | | n  n1 |  A positive sequence

(1.2)

x  ( xn ) is said to be a quasi-  -power increasing seqeuence if there exist a constant

  K  K ( , X )  1 such that Kn X n  m X n holds for all n  m  1 (Leindler [5]).

Let

( n ) be a sequence of complex numbers and let an be a given infinite series with partial sums ( sn )

. We denote by

zn and tn the nth cesaro means of order  with   1 of the sequence ( sn ) and

(nan ) respectively, that is zn  tn 

n

1 An 1 An

 Anv1sv

v 0

(1.3)

n

 Anv1vav

v 0

(1.4)

 n    (  1)(  2)...(  n)  O(n ), A n  0 for n  0  n!  n 

where An  

an is said to be summable   | C,  |k , k  1 and   1 if (Balci [1])

The series 

 | n ( zn  zn1) |   nk | ntn |k   

k

n 1

If

(1.5)

n 1

1

n  n

1 k

then

  | C,  |k -summability is the same as | C ,  |k -summability (Flett [4]). II.

Known theorem

Sulaiman [6] has proved the following theorem. Theorem 2.1 Let ( n ) be a sequence of positive real numbers. Let 

( X n ) be a quasi- f -increasing sequence

f  ( f n ), f n  n (log n) , 0    1,   0 . Let (n ) and ( n ) be sequences of numbers such that ( n ) is positive non-decreasing sequences. If www.iosrjournals.org

90 | Page

On An Extension Of Absolute Cesaro Summability Factor m

 nk 1

  k 1   O  vk   v 

 nk 1

n v

(2.1)

 n 1 (log n) X n | 2n |  n 1

n  0

as

n 

(2.2)

(2.3)

 1  n1  (log n) X n n     O(1) as n   (2.4)  n  m  nk 1 | tn |k  O(m (log m) X m m ) an m    nk (n (log  k 1 n) X n ) n2

| n |  n 1 n

(2.5)

m

(2.6)

and

 1   | n |    O   n   n | n1 |  are satisfied then the series an n n is summable   (C,1)k , k  1 .

n  2 

III.

Main theorem

In this paper we have proved the following theorem. Theorem 3.1 Let ( n ) be a sequence of complex numbers. Let

( X n ) be a quasi- f -power increasing

f  ( f n ), f n  n (log n) ,0    1,   0 . Let (m ) and ( n ) be sequences of the numbers

sequence, such that

(2.7)

( n ) is positive non-decreasing sequence if (2.1), (2.2), (2.3) (2.4), (2.6) (2.7) and

 nk | Wn |k

m

 nk (n (log n) X

n2

n)

k 1

 O((m (log m) X m )k m )

  | tn |,

 1

where Wn  

  max1  v  n | t v |, 0    1 a Are satisfied then  n n is summable   | C,  |k , k  1 . n

IV.

Lemmas

We have need the following lemmas for the the proof of our theorem. Lemma 4.1 (Sulaiman [6]) Let ( X n ) be a positive non decreasing sequence and, let numbers if

(n ) be a sequence of

m  O(1) , m   | n | X n  O(1) , n   m

 n | 2n | X n  O(1)

m

n 1

are satisfied then 

 | n | , nor    n  | n |  n 1 n 1 Lemma 4.2 (Sulaiman [6]) Let ( X n ) be a quasi- f -power increasing sequence

neither

www.iosrjournals.org

91 | Page

On An Extension Of Absolute Cesaro Summability Factor

f  ( f n ), f n  n (log n) , 0    1,   0. if n  0 as n   

 n 1 (log n) X n | 2n |  n 1

m 1 (log m) X n | m | O(1) as m  

Then

 n (log n) X n | n | O(1) n 1

and

n (log n) X n | n | O(1) as n   . V.

Proof of the theorem

 na   (Tn ) be the nth (C,  ) with 0    1 , mean of the sequ ence  n n  , then  n  va  1 n Tn    Anv1 v v v An v 1

Let

Applying Abel’s transformation and using lemma (4.1), we get that

Tn 

1 An

| Tn | 

n 1

 v  v  1        An p pa p   An  v 1  v  p 1 n n

1 An

n 1

     v  v 1  v

n 1

1 An

 AvWv

1 An

n 1

v

v

v 1

v

 Anp1 pa p  p 1

n

 Anv1vav v 1

n 1 n An

n

 Anv1vav v 1

n  W n n

  1   n   v     Wn  v         v v  1 n        

Wv Av     v 1

 Tn,1  Tn,2  Tn,3 To complete the proof of the theorem by minkowskiys inequality, it is sufficient to show that 

 nk | nTn,r |k  

for r  1, 2,3

k 1

Now, when m1

n

n2

k

k  1 applying Hölder’s inequality with indices k and k  , where m1

|  nTn,1 |   n ( An )  k

 k

k

n2

 n1  1   |  n |  AvWv | v |   v   v1 

|  n |k  k n1  k  k  1    k (n)  | Wv | v    | v |k n2 n v 1  v 

 nk

m

n2

m

n

 O(1) v v 1

k (1 )

k

k

k

m

 O(1) 

1 1   1 we get that k k 1

n 1

 n1 1       v 1 v 

k 1

 1  k  | v |  v

 v k | Wv |k    v 1

 1  | Wv |    | v |k  v   k

m

 nk

 nk (1 )

n v

www.iosrjournals.org

92 | Page

On An Extension Of Absolute Cesaro Summability Factor

 1  v k | Wv |k k   k 1  O(1) k  k 1    | v |  v (| v | v (log v) X v |  k 1  (v (log v) X v ) v 1 v  v  m

 1  v | Wv |k k   | v |  v k   k 1   v 1 v (v (log v) X v )  v m

 O(1)

 vk | Wv |k

 1   |  |    v k   k 1 v 1 v (v (log v) X v )  v  m  1   k n1  O(1)  k (1v )  v k | Wv |k    | v |k n2 n v 1  v  m  v  1   rk | Wr |k  O(1)   k   ( v |  |     v  k 1  v 1  r 1 r (r (log r ) X r )  v   m

 O(1)

 m   1   vk | Wvk |k O(1)   k  m(m )   r k 1    m   v1 v (v (log v) X v )  m   1  1 1   O(1) v  (log v) X v v  | v |     (v  1) | v |   (v  1) | v 1 |  2   v v  v 1  v    1  O(1)m 1 (log m) X m m |  m |     m  m1 | v |  O(1)  v  (log v) X v | v | v 1 v v 1 m

 O(1) m1

O(1)    (log v) X v | v | O(1) v 1

 O(1) m

n2

|  nTn,2 |k nk

m



n2

 nk 1

nk An

n 1

 Av Wv 

v 1

v

k

v1

 v k | Wv |k v  n1   O(1)  k  k   v (log v) X v v   k 1 k   v1  v1 n2 n v 1 (v (log v) X v )  m

 nk

n 1

k 1

v k | Wv |k v m  nk  O(1)    k 1 vk1 nv nk  k v 1 (v (log v) X v ) m

v | Wv |k v m  nk    k 1 vk1 nv nk 1 v 1 (v (log v) X v ) m

 O(1)

| Wv |k  nk v | v | k   k 1 v v 1 v (v (log v) X v ) m

 O(1)

m1  v   v | v |  | W  |k  rk  O(1)    k  r    k 1   v 1  r 1 r ( r (log r ) X r )   v   m  m | m | | W  |k  k    k  v  v k 1   v 1 v (v (logv) X v )  m

www.iosrjournals.org

93 | Page

On An Extension Of Absolute Cesaro Summability Factor m1   1  1  O(1)  v  (log v) X v v     v | v |  | v | (v  1) |  2v |    v 1 v 1   v  O(1)mX m | m |

m1

m1

v 1 m1

v 1



   

 O(1)  v  (log v) X v | v | O(1)  v  (log v) X v | v | O(1)  v  1 (log v) X v |  2v |  O(1) v 1

 O(1) m

n 1

|  nTn,3 |k nk

 k W  O(1) kn n n n n 1 n m

 nk | Wn |k

m

 O(1)

k

k

n 1 n (n (log n) X n )

 nk | Wn |k

m

 O(1)

n 1 n

k

(n (log n) X n )

k 1

k 1

| n |

nk

(n  (log n) X n | n |)k 1

| n |

n

m1  n   v | n |  | W  |k  vk  O(1)    k  v   k 1    n n 1  v 1 v (v (log v) X v )     m  | m |  k | W  |k    k  n n  k 1    n1 n (n (log n) X n )  m m1   1  | n |   O(1)  n  (log n) X n n     | n |    O(1) X m | m |   n1  n 1   n m1 m1 | n |  O(1)   O(1)  n  (log n) X n | n |  O(1) n 1 n n 1

 O(1)

This completes the proof of theorem.

References

[1]

Balci, M.; Absolute

-summability factors, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 29, 1980.

[2] [3] [4] [5] [6]

Bari, N.K.; Best approximation and differential properties of two conjugate functions, T. Mat. Obs. 5, 1956. Bor, H.; A new application of quasi power-increasing sequences II, Bor fixed point theory and application, 2013. Flett, T.M.; On an extension of absoulte summability and some theorems of littlewood and paley, Proc. London Math. Soc. 7, 1957. Leindler, L.; A new application of quasi power increasing sequences, Pub. Math. (Debar.) 58, 2001. Sulaiman, W.T.; On some generalization of absolute Cesaro summability factors, J. of classical Analysis Vol. 1, 2012.

www.iosrjournals.org

94 | Page