Note on Absolute Matrix Summability Factors

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 35, Note on Absolute Matrix Summability Factors Pragati Sinha, Hirdesh Kumar 1 and Vipin Kumar Gupta Department of Mathematics S. M. P.G.) College Chandausi , India Abstract In this paper a theorem on absolute matrix summability factors, which generalizes a theorem of Özarslan and Öğdük [5] on T, k summability methods, has been proved. Keywords: Absolute Summability, Infinite Series, Summability Factors 1 Introduction Let a n be a given infinite series with the partial sums {s n }. By wn δ ), we denote the n th Cesáro mean of order δδ > 1) of the sequence {s n }. The series a n is said to be summable C, δ k,k 1, if [3] n k 1 wn δ wδ n 1 < 1.1) In the special case for δ =1, C, δ k summability reduces to C, 1 k summability. = p v, as n, P i = p i =0,i 0) 1.2) The sequence-to-sequence transformation ϑ n = 1 p v s v 1.3) defines the sequence ϑ n ) of the N, ) means of the sequence {s n }, generated by the sequence of coefficient ) [4]. The series a n is said to be summable N, k, k 1 if [1] ϑ n ϑ n 1 k < 1.4) 1 hirdeshkumar150877@gmail.com

2 1738 P. Sinha, H. Kumar and V. K. Gupta If we take = 1 for all values of n, then N, k summability is the same as C, 1 k summability. Given a normal matrix T =t nk ), we associate two lower semimatrices T = t nk ) and ) ˆT =ˆt nk ) as follows t nk = t ni, n,k =0, 1, 2,... i=k ˆt 00 = t 00 = t 00, ˆt nk = t nk = t n 1,k n =1, 2, ) It may be noted that T and ˆT are the well-known matrices of series-tosequence and series-to-series transformation, respectively. Then, we have T n s) = ΔT n s) = t nv s v = t nv a v, ˆt nv a v 1.6) The series a n is said to be summable T, k,k 1, if [6] ΔT n s) k <. 1.7) In the special case, for t nv = pn, T, k summability is the same as N, k summability. 2 Özarslan and Öğdük [5] proved the following theorem for T, k summability. Theorem 2.1. Let k 1. Let {s n } be a bounded sequence and suppose that λ n ) is a sequence such that λ n k t nn k = O1) as m, 2.1) Δλ n = O1) as m.

3 Note on absolute matrix summability factors 1739 If 1 n 1 1 n 1 Δ v ˆt nv ) = O1) as n, 2.2) ) k 1 Pv ) ) k 1 Δ vˆt nv t nn k = O t vv k as m, 2.3) p v Δλ v ˆt n,v+1 = O1) as n, 2.4) ˆt n,v+1 t nn k 1 = O1) as m, 2.5) then the series a n λ n is summable T, k. 3 We generalizes the theorem of Özarslan and Öğdük [5] for T, k summability. Theorem 3.1. Let k 1. Let s n ) be a bounded sequence and suppose that λ n ) is a sequence such that λ n k P v t nn k = O1) as m, 3.1) Δλ n P v = O1) as m 3.2) λ n ΔP v = O1) as n. 3.3) If 1 n 1 Δ v ˆt nv ) = O1) as n, 3.4) ) k 1 Pv ) ) k 1 Δ vˆt nv t nn k 1 = O t vv k as m, 3.5) 1 n 1 p v Δλ v ˆt n,v+1 = O1) as n, 3.6) ˆt n,v+1 t nn k 1 = O1) as m 3.7) then the series a n λ n is summable T, k.

4 1740 P. Sinha, H. Kumar and V. K. Gupta 4 Proof of the Theorem Let y n ) be the T-transform of the series a n λ n. Then we have, 1.6), Y n = y n y n 1 = ˆt nv a v λ v P v 4.1) Since ˆt nn = t nn, by Abel s transformation, we get n 1 Y n = Δ v ˆt nv λ v P v )s v + ˆt nn λ n s n n 1 n 1 = Δλ vˆt n,v+1 P v s v + λ v P v Δ v ˆt nv )s v n 1 + λ v Δ v P v )ˆt nv s v + s n t nn λ n = Y n 1) + Y n 2) + Y n 3) + Y n 4). 4.2) Using Minkowski s inequality, it is sufficient to show that Y n r) k <, for r =1, 2, 3, 4 4.3) Since s n ) is bounded, when k > 1, applying Hölder s inequality with

5 Note on absolute matrix summability factors 1741 indices k and k, where 1 k + 1 k = 1, we have get Y n 1) k = O1) = O1) = O1) = O1) ) { k 1 n 1 Δλ v P v ˆt n,v+1 s v n 1 { 1 t nn n 1 } k Δλ v P v ˆt n,v+1 k 1 n 1 k 1 Δλ v P v ˆt n,v+1 s v } 4.4) Δλ v P v Δλ v P v = O1) as m, by virtue of the hypotheses of theorem. Y n 2) k = O1) = O1) = O1) = O1) n 1 { 1 Δλ v P v ˆt n,v+1 k 1 ) { k 1 n 1 λ v P v Δ vˆt nv s v t nn n 1 ˆt n,v+1 t nn k 1 } k λ v k P v Δ vˆt nv k 1 n 1 k 1 Δ vˆt n,v+1 P v } 4.5) λ v k P v Pv p v = O1) as m, by virtue of the hypotheses of theorem. λ v k P v Δ vˆt nv k 1 λ v k P v t vv k Δ vˆt nv t nn k 1

6 1742 P. Sinha, H. Kumar and V. K. Gupta Again using Hölder s inequality, we have that Y n 3) k = O1) = O1) = O1) = O1) ) { k 1 n 1 λ v ΔP v ˆt n,v+1 s v n 1 { 1 t nn n 1 } k λ v ΔP v ˆt n,v+1 k 1 n 1 k 1 λ v ΔP v ˆt n,v+1 s v } 4.6) λ v ΔP v λ v ΔP v = O1) as m, by virtue of the hypotheses of theorem. Finally, we have that λ v ΔP v ˆt n,v+1 k 1 ˆt n,v+1 t nn k 1 Y n 4) k = O1) t nn k P v λ n k 4.7) = O1) as m, by virtue of the hypotheses of theorem. Therefore, we get that Y n r) k = O1), as m, for r =1, 2, 3, ) This completes the proof of the theorem. 5 Application Now we will prove the following corollary. Corollary 1. Bor [2] Let k 1. If the sequence {s n } is bounded and λ n )is

7 Note on absolute matrix summability factors 1743 a sequence such that pn ) λ n k = O1) as m, 5.1) Δλ n P v = O1) as m 5.2) λ n ΔP v = O1) as m, 5.3) then the series a n λ n is summable N, k. Proof. In our theorem, let t nv = pv. Then to prove the Corollary 1, it is sufficient to show that the conditions of the theorem are satisfied. If t nn = pn, 3.1) are automatically satisfied. Since Δ vˆt nv = ˆt nv ˆt n,v+1 we get = t nv t n 1,v t n,v+1 + t n 1,v+1 n 1 = t ni t n 1,i t ni + +1 = 1 p i 1 n 1 p i 1 1 = p v 1 n t n 1,i 5.4) p i + n 1 +1 p i 1 n 1 Δ vˆt nv = Thus condition 3.4) is satisfied. Using Δ vˆt nv, and t nn, n 1 p v 1 = O1) as n. 5.5) p v pn Δ vˆt nv t nn k 1 = p n = p v 1 1 = p v P v = Pv p v t vv k as m, 5.6)

8 1744 P. Sinha, H. Kumar and V. K. Gupta condition 3.5) is satisfied. Since 1 n 1 ˆt nv = t nv t n 1,v n 1 = t ni Δλ v ˆt n,v+1 = and condition 3.6) is satisfied. Finally, t n 1,i = 1 p i 1 n 1 p i 1 P n 1 P n = P v ) 1 = P v 1 1 n 1 Δλ v P v 1 = 1 n 1 1 n 1 Δλ v P v = O1) Δλ v +1 p i + n 1 +1 = O1) as n, 5.7) p i ˆt n,v+1 t nn k 1 = P v p v pn p n = P v 1 1 = O1) as m, 5.8) so condition 3.7) is satisfied. This completes the proof of the corollary. References [1] H. Bor; On two summability methods, Math. Proc. Cambridge Phil. Soc. 97, no.1, 1985),

9 Note on absolute matrix summability factors 1745 [2] H. Bor; A note on N, k summability factors, Rend. Mat. Appl. 7) 12, 1992), no. 4, [3] T.M. Flett; On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London. Math. Soc. 3) 7, 1957), [4] G.H. Hardy; Divergent Series, Clarendon Press,Oxford Univ. Press, New York, 1949). [5] H.S. Özarslan and H.N. Öğdük; On absolute matrix summability methods, International Journal Mathematics and Mathematical Sciences 16, 2005), [6] W.T. Sulaiman; Inclusion theorems for absolute matrix summability methods of an infinite series IV, Indian J. Pure Appl. Math. 34, no. 11, 2003), Received: January, 2010