On the inclusion relation between strong (i, p„) and strong...

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Now , we put

- 3

ФДХ ) =2 tnX"Pn,

/ 1 = 3

, , \p{x)lii-x) for 0<x<U I Po for x = 0.

It is clear that ф(х) and фДх) converge for 0<x<l, since lim/„ = 0.

Further we have, for 0<x< 1

( ? { x ) =p ( x ) (1-х)

/ 1=0 /1=0

In this lemma, we put

dn=Po - ^Pi - ^~P2 + ' - -^Pn ('^>0)

and

Since

c , ^t „ P „ in>3),

Pn

Pçy + Pl + ' • ' +Pn

= 1,

we get

lim i^^>=lim(„ = 0 л->1-о ф(л:) »-*«

From the assumption of the theorem, Hence the proof is complete.

Remark : Taking p„-— - in the above theorem we obtain the

n + 1 following:

Theorem 2. If 2a„ is summabie [R, log n, k] to S, Шеи it is mabie [L, k] to the same sum.