C0751720
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This document presents a theorem on generalized kC |,,,,| -summability factor. It begins with introducing some relevant concepts such as almost increasing sequences, sequences of bounded variation, and quasi- -power increasing sequences. It then states an existing theorem on kC |,,,,| summability by Tuncer. The main result of the document is a theorem that generalizes Tuncer's theorem to kC |,,,,| summability
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 5 (Jul. - Aug. 2013), PP 17-20
www.iosrjournals.org
www.iosrjournals.org 17 | Page
On Generalized kC |,,,,| -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 established a theorem on kC |,,,,| -summability factor, which
gives some new results.
I. Introduction
A positive sequence )( nb is said to be almost increasing if there exist a positive sequence )( nc and
two positive constants A and B such that nnn BcbAc (Bari [2]).
A sequence )( n is said to be of bounded variation, denoted by BVn )( if
|<||=| 11=1= nnnnn
.
A positive sequence )(= nXX is said to be quasi- -power increasing sequence if there exist a
constant 1),(= Xkk such that 1, mnXmXkn mn
(Leindler [7]).
Let n be a sequence of complex numbers. Let na be a given infinite series with partial sum )( ns
. We denote by
,
nz and
,
nt the n
th
Cesaro means of order ),( with 1> of the sequences
)( ns and )( nna respectively (Borwein [5]).
vvvn
n
vn
n sAA
A
z
0=
, 1
=
vvvn
n
vn
n vaAA
A
t
1=
, 1
=
where 0=1,=1,>),(= 0
nn AAnOA for 0>n .
The series na is said to be summable 1,|,,| kC k and 1> (Das [6]) if
<
1
= ,
1=
,
1
,1
1=
k
n
n
k
nn
k
n
t
n
zzn
The series na is said to be summable kC |,,,,| , 1k 1> , 0 and is a
real number (Bor [4]) if
<= ,1)(
1=
,
1
,1(
1=
k
n
kkk
n
k
nn
kk
n
tnzzn
The series na is said to be summable 1>1,,|,| kC k if (Balci [1])
<||=|)(|
1=
1
1=
k
n
k
n
k
nnn
n
tnzz
And the series na is said to be summable | , , , , |kC if
<=)( ,1)(
1=
,
1
,1)(
1=
k
nn
kkk
n
k
nnn
kk
n
tnzzn ](https://image.slidesharecdn.com/c0751720-150428031117-conversion-gate01/85/C0751720-1-320.jpg)
![On Generalized kC |,,,,| -Summability Factor
www.iosrjournals.org 18 | Page
II. Known theorem
Tuncer has proved the following theorem
Theorem 2.1 Let 1<01, k and be a real number such that 1>1))(1( k
and let the sequences )( nB and )( n such that )( nB is -quasi-monotone with
| | | |, 0asn n nB n (1)
=1
log < and logn n
n
n n nB n
(2)
is convergent. If the sequence )( ,
nW defined by
1>1,=|,=| ,,
nn tW (3)
1>1,<<0|,|max= ,
1
,
v
nv
n tW
(4)
satisfies the condition
mmOW
n
n k
nk
kkm
n
as)log(=)( ,
1)(
1=
(5)
then nna is summable kC |,,,,| .
III. The main result
The aim of this paper is to generalize Theorem 2.1 to kC |,,,,| summability. We shall
prove the following theorem.
Theorem 3.1 Let n be the sequence of Complex numbers and let the sequence )( nB & )( n such that the
conditions (1), (2), (3), (4) with
mmO
n
Wn
k
k
nn
kkm
n
as)log(=
|| ,1)(
1=
are satisfied then the series nna is summable kn C |,,,,| .
IV. Lemmas
We need the following lemmas for the the proof of our theorem
Lemma 4.1 (Mazhar [9]) Under the condition on )( nB as taken in the statement of the theorem, we have
following
(1)=log OnBn n (6)
and
=1
log | |n
n
n n B
(7)
Lemma 4.2 (Bor [4]) If 1>1,<0 and nv 1 then
pppm
m
pvm
pppn
v
p
aAAaAA 1
0=1
1
0=
max
(8)
V. Proof of the theorem
Let )( ,
nT be the n
th
),,( C mean of the sequence )( nnna then we have
.
1
= 1
1=
,
vvvvn
n
vn
n vaAA
A
T
Using Abel's transformation.](https://image.slidesharecdn.com/c0751720-150428031117-conversion-gate01/85/C0751720-2-320.jpg)
![On Generalized kC |,,,,| -Summability Factor
www.iosrjournals.org 20 | Page
mBmOvBOvBvO mv
m
v
v
m
v
log||(1)log||(1)log||(1)= 1
1
1=
1
1
mO as(1)=
k
nn
kkk
n
m
n
k
nn
kkk
m
n
WnOTn ||)(||(1)= ,1)(
1=
,
,2
1)(
1
2=
k
n
k
v
kkk
n
v
n
m
n
WvO ||)(||(1)= ,1)(
1=
1
1=
k
n
k
v
kkk
m
v
m WvO ||)(||(1) ,1)(
1=
mOnO mn
m
n
log||(1)log||(1)=
1
1=
mOnBO mn
m
n
log||(1)log||(1)=
1
1=
mO as(1)=
by virtue of the hypothesis of the theorem and lemma 1. This completes the proof of the theorem.
References
[1] Balci, M.; Absolute -summability factors, Commun. Fac. Sci. Univ. Ank. Ser. A, Math. Stat. 29, 1980.
[2] Bari, N.K., Szeckin, S.B.; Best approximation of differential properties of two conjugate functions, Tomosk. Mat. obs 5, 1956.
[3] Bor, H.; On generalized absolute cesaro summability, Proc. Appl. Math. 2, 2009.
[4] Bor, H.; On a new application of quasi power increasing sequences, Proc. Est. Acad. Sci. 57, 2008.
[5] Borwein, D.; Theorems on some methods of summability Q.J. Math. 9, 1958.
[6] Das, G; A Tauberian theorem for absolute summability, Proc. Comb. Phlos. Soc. 67, 1970.
[7] Leincler, L; A new application of quasi power increasing sequence. Publ. Math. (Debar) 58, 2001.
[8] Mazhar, S.M.; On a generalized quasi-convex sequence and its applications, India J. Pure Appl. Math. 8, 1977.
[9] Tuncer A.N.; On generalized absolute Cesaro summability factors, Tuncer J. of Inequalities and Ap. 2012.](https://image.slidesharecdn.com/c0751720-150428031117-conversion-gate01/85/C0751720-4-320.jpg)
C0751720
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 5 (Jul. - Aug. 2013), PP 17-20 www.iosrjournals.org www.iosrjournals.org 17 | Page On Generalized kC |,,,,| -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 established a theorem on kC |,,,,| -summability factor, which gives some new results. I. Introduction A positive sequence )( nb is said to be almost increasing if there exist a positive sequence )( nc and two positive constants A and B such that nnn BcbAc (Bari [2]). A sequence )( n is said to be of bounded variation, denoted by BVn )( if |<||=| 11=1= nnnnn . A positive sequence )(= nXX is said to be quasi- -power increasing sequence if there exist a constant 1),(= Xkk such that 1, mnXmXkn mn (Leindler [7]). Let n be a sequence of complex numbers. Let na be a given infinite series with partial sum )( ns . We denote by , nz and , nt the n th Cesaro means of order ),( with 1> of the sequences )( ns and )( nna respectively (Borwein [5]). vvvn n vn n sAA A z 0= , 1 = vvvn n vn n vaAA A t 1= , 1 = where 0=1,=1,>),(= 0 nn AAnOA for 0>n . The series na is said to be summable 1,|,,| kC k and 1> (Das [6]) if < 1 = , 1= , 1 ,1 1= k n n k nn k n t n zzn The series na is said to be summable kC |,,,,| , 1k 1> , 0 and is a real number (Bor [4]) if <= ,1)( 1= , 1 ,1( 1= k n kkk n k nn kk n tnzzn The series na is said to be summable 1>1,,|,| kC k if (Balci [1]) <||=|)(| 1= 1 1= k n k n k nnn n tnzz And the series na is said to be summable | , , , , |kC if <=)( ,1)( 1= , 1 ,1)( 1= k nn kkk n k nnn kk n tnzzn
- 2. On Generalized kC |,,,,| -Summability Factor www.iosrjournals.org 18 | Page II. Known theorem Tuncer has proved the following theorem Theorem 2.1 Let 1<01, k and be a real number such that 1>1))(1( k and let the sequences )( nB and )( n such that )( nB is -quasi-monotone with | | | |, 0asn n nB n (1) =1 log < and logn n n n n nB n (2) is convergent. If the sequence )( , nW defined by 1>1,=|,=| ,, nn tW (3) 1>1,<<0|,|max= , 1 , v nv n tW (4) satisfies the condition mmOW n n k nk kkm n as)log(=)( , 1)( 1= (5) then nna is summable kC |,,,,| . III. The main result The aim of this paper is to generalize Theorem 2.1 to kC |,,,,| summability. We shall prove the following theorem. Theorem 3.1 Let n be the sequence of Complex numbers and let the sequence )( nB & )( n such that the conditions (1), (2), (3), (4) with mmO n Wn k k nn kkm n as)log(= || ,1)( 1= are satisfied then the series nna is summable kn C |,,,,| . IV. Lemmas We need the following lemmas for the the proof of our theorem Lemma 4.1 (Mazhar [9]) Under the condition on )( nB as taken in the statement of the theorem, we have following (1)=log OnBn n (6) and =1 log | |n n n n B (7) Lemma 4.2 (Bor [4]) If 1>1,<0 and nv 1 then pppm m pvm pppn v p aAAaAA 1 0=1 1 0= max (8) V. Proof of the theorem Let )( , nT be the n th ),,( C mean of the sequence )( nnna then we have . 1 = 1 1= , vvvvn n vn n vaAA A T Using Abel's transformation.
- 3. On Generalized kC |,,,,| -Summability Factor www.iosrjournals.org 19 | Page 1 , 1 =1 =1 1 = n v n v n p p p v pn T A A pa A vvvn n vn n vaAA A 1 1= pppn v p v n vn n paAA A T || 1 || 1 1= 1 1= , vvvn n vn n vaAA A 1 1= || 1 , , =1 1 | | | | n v v v v n n vn A A W W A (say)= , ,2 , ,1 nn TT Since )|||(|2|| , ,2 , ,1 , ,2 , ,1 k n k n kk nn TTTT In order to complete the proof of theorem, it is sufficient to show that 1,2=,<|,| , , 1)( 1= rTn k nrn kkk n Whenever 1>k , we can apply Hölder's inequality with indices k and k , where 1= 11 kk we get that k nn kkk m n Tn , ,1 1)( 1 2= k vvvvv n vn kkk m n WAA A n , 1 1= 1)( 1 2= 1 11 1= 1 1= 1))(1( 1 2= ||)(|| || (1)= k v n v k vv kk n v k k v m n Wvv n O 11 1= , 1 1= 1))(1( 1 2= ||)(|| || (1)= k v n v k vv kk n v k k v m n BWBvv n O k k v m vn k vv k m v n WBvO 1))(1( 1 1= ,)( 1= || )(||(1)= k k v k vv k m v x dx WBvO 1))(1(0 ,)( 1= ||)(||(1)= k v k v kkk v m v WvBO ||)(||(1)= ,11)( 1= k v k v kkk v p v m v WpBvO ||)(||)|(|(1)= ,1)( 1= 1 1= k v k v kkk m v m WvBmO ||)(||(1) ,1)( 1= mBmOvBvO mv m v log||(1)log|)|(|(1)= 1 1
- 4. On Generalized kC |,,,,| -Summability Factor www.iosrjournals.org 20 | Page mBmOvBOvBvO mv m v v m v log||(1)log||(1)log||(1)= 1 1 1= 1 1 mO as(1)= k nn kkk n m n k nn kkk m n WnOTn ||)(||(1)= ,1)( 1= , ,2 1)( 1 2= k n k v kkk n v n m n WvO ||)(||(1)= ,1)( 1= 1 1= k n k v kkk m v m WvO ||)(||(1) ,1)( 1= mOnO mn m n log||(1)log||(1)= 1 1= mOnBO mn m n log||(1)log||(1)= 1 1= mO as(1)= by virtue of the hypothesis of the theorem and lemma 1. This completes the proof of the theorem. References [1] Balci, M.; Absolute -summability factors, Commun. Fac. Sci. Univ. Ank. Ser. A, Math. Stat. 29, 1980. [2] Bari, N.K., Szeckin, S.B.; Best approximation of differential properties of two conjugate functions, Tomosk. Mat. obs 5, 1956. [3] Bor, H.; On generalized absolute cesaro summability, Proc. Appl. Math. 2, 2009. [4] Bor, H.; On a new application of quasi power increasing sequences, Proc. Est. Acad. Sci. 57, 2008. [5] Borwein, D.; Theorems on some methods of summability Q.J. Math. 9, 1958. [6] Das, G; A Tauberian theorem for absolute summability, Proc. Comb. Phlos. Soc. 67, 1970. [7] Leincler, L; A new application of quasi power increasing sequence. Publ. Math. (Debar) 58, 2001. [8] Mazhar, S.M.; On a generalized quasi-convex sequence and its applications, India J. Pure Appl. Math. 8, 1977. [9] Tuncer A.N.; On generalized absolute Cesaro summability factors, Tuncer J. of Inequalities and Ap. 2012.