Acta Mathematica Academiae Scientiarum Hungaricae 54. (1989)
1989 / 1-2. szám - Bor, H. - Kuttner, B.: On the necessary conditions for absolute weighted arithmetic mean summability factors
ON THE NECESSARY CONDITIONS FOR ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY FACTORS H. BOR (Kayseri) and B. KUTTNER (Birmingham) 1. Let 2 an be a given infinite series with the sequence of partial sums (j„) and let (pn) be a sequence of positive real constants such that (1) Pn= 2Pv^°= as n-> oo (Р_г = p_x = 0). v=0 The sequence-to-sequence transformation (2) tn = -ir 2 PvSy, (p„ ^ o) rn v=0 defines the sequence (tn) of (N, p„) mean of the sequence (s„), generated by the sequence of coefficients (p„). The series 2 a„ is said to be summable \N,pn\k, k^l, if (see [1]) In the special case when pn — 1 for all values of n (resp. k=1), then \N,pn\k summability is the same as |C, \\k (resp. \N,p„\) summability. The series 2 a„ is said to be bounded [N,pn]k, A:^l, if (see [2]) (4) 2 Pv k»l* = О(P„) as n - oo. v—X. If we take k= 1 (resp. p„=l/n), then [N,p„]k boundedness is the same as [N, pn] (resp. [R, log n, l],c) boundedness. 2. There are known sufficient conditions for absolute weighted arithmetic mean summability factors (see e.g. Bor [2] and Singh [3]). In this paper we shall prove necessary conditions. Theorem. Let 2 an be bounded [N,p„]k. If 2 anK ^ summable \N,p„\k, then the following conditions are necessary: (i) A„ = 0(1) <'5') (Ü) I as n-*oc, k^l. 3. We need the following lemma for the proof of our theorem.