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

58 H. BOR AND В. KUTTNER Lemma. Let &S 1. Then there are two (strictly) positive constants A, B, depend­ing only on k, such that whenever pn>0, P„=Po+Pi+Pi+ •••+Pn^co as n-+°° we have, for all vsl, Proof. To prove the lemma, it is enough to show that there are constants A, В such that (for n^l) Hence there must be two strictly positive constants +, 5 such that Asf(t)^B all 0. This completes the proof of the lemma. Proof of the Theorem. Let (Tn) be the sequence of (N,p„) mean of the series 2 anK- Then, by definition, we have Now the space of sequences (sk) with (13) (6) — .. rs V__—__ Ä B pk — p pk — pk r\-l n=v rnrn-l rv-l where A and В are independent of (pn). (7) Í 1 41- A 1IH f 1 M U*-, pt J u*-l ад For then (6) will follows on adding (7) for n=v, v+1, v+2, .... Write p„=P„-it. So that t=tn is a positive number. Then i£=J£_1(l + f). So that (8) Pn t PnPt-l = l+t _ tjl + tf-1 f i ll (l+o*-1 n РЦ-1 Pnk (1+0* Clearly, f(t) is continuous and strictly positive for 0. Also (9) /(0 -7- as t —► 0+ and /(0-*■ 1 as /С (10) Then (ID Тш = 4- 2p, 2 «Л = 4- 2(P„-Pv-l)flvAv. Гп v=0 r=0 -*n v=»0 я Sl, (F-i = 0). PnPn-l v=l The series ^ a„A„ is summable |N,pn|t so that (12) íí-^-Í V.-Ji.J* n=l ' Pn > -p~ 2Pv\SyI* =0(1), v*=0 Acta Mathematica Hungarica 54, 1989

Next