HARDY TYPE INEQUALITIES 29
* vc( II T ^ r l l
2
+  (!•*• )P
I
^
r

+
 (l*p
+
4 ^ 1 )v ) S
£ vc(n
w
H
2
+
n
p v
ii
+
" 1 ^ 1
+ (1+q)1 ( l +
*
, ) v
i i
}
and
 tS
2
v  =  (l+vp')tS
2
(l+cp')"" 1 v £
S (l
+
q )  ptS2(l+cp' )~ 1 v * v d + q H I ^ r H g •
These two inequalities prove (2.44) .
(ii) To prove (2.45), we use Lemma 2.6. Under the present assumptions
on the function r\9 we may deduce from (2.22) that
(2.46)   nv2 c   nL(p)v+ cp(i+p')nSv +
+ c ( i + p )  (i+cp»)nPv + c ( i + p ) 11 [ ( i + t p ' ) 2 + cp" ] n v   .
On the other hand , if we replace in (2.44) the function v by
(l+cpf)nv , we get that
 (i+p»)nSv ^  ts(i+p»)nv ^ vc nvj
2
+
+ vc (i+tp» )nPvi[(i+cpf )n 3' v  + vc (l+tp1 )2nv
^ vc nv
2
+ vc (i+cp')nPv + vc (cp"  n + (
i+cpf)

nf
 )v +
+ vc  (i+p» )2nv .
(2.45) is now obtained by inserting this last inequality into (2.46) ,
by using the hypotheses made on cp" and on r\ , and by taking v small
enough . •