Mazhar, Syed Mohammad

English

Mazhar Syed Mohammad

University of Szeged

On | C, l|fc summability factors of infinite series By S. M. M A Z H A R in Aligarh (India) ia) k • :: o o . 1. Let Zan be a given infinite series with partial sums sn, and let tn = t® = nan. By <fn and t* we denote the n-th Cesàro means of order a (a = — 1) of the sequences {s„} and {i„}, respectively. The series Ia n is said to be absolutely summable (C, a) with index k, or simply summable \C,a\k (& £ 1), if ( i - i ) ' Z ^ - ' K - C i f « » ( [ ! ] ) • Summability |C, a], is the same as summability |C, a|. Since condition (1. 1) can also be written as (1.2) n If 0 - 3 ) 2 ^ = 0 ( \ 0 G R I ) , 1 v as then Ian is said to be strongly bounded by logarithmic means with index 1, or bounded [.Rj log«, 1]. 2 . Recently PATI [ 2 ] proved the following theorem concerning summability |C, 1| of a "factored infinite series. A Let {A„} be a convex sequence such that I — is convergent (then, necessarily, n )-n s 0). If Ian is bounded [7?, log n, 1], then ZanA„ is summable ]C, 1|. The object of this note is to generalize this result by obtaining a theorem for summability |C, 1 3. In what follows we shall establish the following theorem. A T h e o r e m . If {An} is a convex sequence such that I — -< and n ( 3 . 1 ) 2 W V v = 0 ( l o g " ) ( f c ë l ) , 1 then IanA„ is summable |C,l|k. 68 S. M. Mazhar It is clear that in the special case k = 1 our theorem includes the above theorem of PATI. For 1 + w e ° B S E R V E that i - ^ T = - 0 ( O o g » ) l l k ( l o g = O (logn). Thus condition ( 3 . 1 ) implies condition ( 1 . 3 ) . However the results of FLETT [1] show that summability | C, l | t and summability | C, 1| in general are independent of each other. 4. The following lemmas will be required for the proof of this theorem. A ' L e m m a 1. [2] If {X„} is a convex sequence such that I log (n -f l)AÂn < » and m log (m + \)Alm = 0(1), as m-+ L e m m a 2. [2] Under the condition of Lemma 1, we have m 2nlog(n + l)A2À„ = 0 (1 ) , as m — i 5. P r o o f of t he T h e o r e m . Let T„ denote the «-th Cesàro mean of order 1 of the sequence {na„X„}. Then we have to show that ( 5 . , I ) 2 » - M R / « » . i Now, ^ l - A - ¿ J ? -= = L'r'+Lf+L'f. n+I! n+1 By MINKOWSKI'S inequality it is therefore sufficient to prove that (5 . 2) 2 , < (5- 3 ) 2 ( 5 . 4 ) 2 n l 4 n ) l * n I L<«>|< Summability factors 69 P r o o f of (5. 2). In the sequel C j , C2 denote positive constants. We have 00 ir(n)|t ~ 1 I 1 * ~ i ( n \k « c , J ( J | s , | j " i - l ' , li.lj a 00 i n In \klk' » , n ( n \k]k' 1 " l VI J 1 " . v = 1 Vv = l / N o w £ AAv\sv\ k = Z vAl V=1 1 W l/c m— I v t Ik m \ \k 1 H=1 ft /1=1 f1 = log (v + l ) j + O [z AAV log (V + l ) j + 0(mAXnl log (m + 1)) = 0 ( 1 ) , ; by virtue of Lemmas 1 and 2. Also applying Lemma 1 we have 1 m— 1 v . .t m i . if 2 ^ M k = 2 AX. 2 ^ 1 V 1 fJ= 1 H H=l = 0^2 AK log (v + 1)J + 0(Xm log (m + 1)) = O(l). 'fi _ Hence ¿ ^ = 0(1). i n P r o o f of (5. 3). We have by virtue of the hypothesis 2 ^ = 2 i S ¿*J6E - 2 2 — = 1 n i f 1 n I /1 x f i m-1 / m \ = 2 AX knO(\ogn) + 0(Akmlogm) = 0\2 AX knlogn\ + 0 ( 1 ) = O ( l ) . 70 S. M. Mazhar: Summability factors Finally, it is clear that This completes the proof of the theorem. The author would like to express his warmest thanks to Professor B. N. PRASAD for his kind encouragement and helpful suggestions. References [1] T. M. FLETT, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113—141. [2] T. PATI, Absolute Ces&ro summability factors of infinite series, Math. Z., 78 (1962), 293—297. DEPARTMENT OF MATHEMATICS AND STATISTICS ALIGARH MUSLIM UNIVERSITY (Received May 11, 1965) 

On | C, l|k summability factors of infinite series

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On | C, l|k summability factors of infinite series

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