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On two summability methods. (English) Zbl 0554.40008

Let a n be given infinite series with partial sums s n , and n=na n . By U n α and t n α we denote the nth Cesàro means of order α (α>-1) of the sequences (s n ) and (r n ), respectively. The series a n is said to be |C,α| k , k1, if n=1 n k-1 |U n α -U n-1 α | k < [see T. M. Flett, Proc. Lond. Math. Soc. 7, 113-141 (1957; Zbl 0109.044)]. Since t n α =n(U n α -U n-1 α ), we can also write n=1 (1/n)|t n α | k <. Let (p n ) be a sequence of positive real constants such that P n =p 0 +p 1 +p 2 +···+p n , as n, (P -1 =p -1 =0). Let T n denote the (N ¯,p n ) mean of the series a n . The series a n is said to be |N ¯,p n | k , k1, if

n=1 (P n /p n ) k-1 |T n -T n-1 | k <,

the author [see J. Univ. Kuwait, Sci. 10, 37-42 (1983; Zbl 0519.40011)]. We prove the following theorem. Theorem. Let (p n ) be a sequence of positive real constants such that, as n (i) np n =O(P n ), (ii) p n =O(np n ). If a n is summable |C|| k , then it is also summable |N ¯,p n|k, k1.


MSC:
40F05Absolute and strong summability
Keywords:
Cesàro means