ON A NEW ABSOLUTE SUMMABILITY METHOD

A theorem concerning some new absolute summability method is proved. Many other 
results, some of them known, are deduced.

INTRODUCTION and Let E am be an infinite series with partial sums sn Let an rb denote the nth Ceshro mean of order ,5(6 > 1) of the sequences {s,,} and {na} respectively.The series E o is said to be summable [C, 6[ Let {p,} be a sequence of positive numbers such that P=:ooo as --,oo(P_,=r,_,=O,i>_ 1.

It is clear that
We assume {a,}, {} and {q} be sequences of positive numbers such that Q , q oo.v--0 We prove the following.
THEOREM 1.Let t. denote the C,p)-mean of the series Ea and write To -l/At-l.
(I) and then the series E a e. is summable l, q,,, c.l k _> 1.
To prove the theorem, by Minkowski's inequality, it is sufficient to show that a.
PROOF.Applying Theorem 2 with a, Q,.,/q,.,,lg, P,,/p,.,.Clearly an 0(/) and (I) is satisfied.Therefore I,p[k I, qnlk" The result is still valid if we interchange {p,} and COROLLARY 3. Suppose that (I) is satisfied for p and q, (II) is also satisfied and that {%/p,) is nonincreasing, then the series E a is summable PROOF.Applying Theorem 2 with a, , n.It is clear that [R, p,[ = ]R, qlk.For the other direction, it needs to be shown that (I) is satisfied if we are replacing q,, by p Since {%/p } is nonincreasing, we have, using (II), _k-l_k n=v+ PnkPn-1 It may be mentioned that Corollary 3 gives an alternative proof to the sufficiency part of the theorem in [2].