1.1. LECTUR E 1 5
PROOF. A functio n / i s calle d completely monotonic o n [0 , oo) i f i t i s
infinitely differentiabl e o n (0 , oo) an d fo r al l k G N U {0} and x 0
(-i)
f c
/
( f c )
(*)o.
The celebrate d Bernstein' s theore m (se e [Fei , Ch.18, Section 4] ) assert s
that ever y completel y monotoni c continuou s a t zer o function i s the Laplac e
transform o f a finite measur e o n [0 , oo), i.e. ther e exist s a finite Bore l mea -
sure o n [0 , oo) suc h tha t
/»OO
f(x) = / e~
txdfi(t),
V x 0 .
Jo
/o
One can check that e~
z*
i s completely monotoni c fo r 0 a 1. There -
fore fo r ever y 0 q 2 there exist s a finite measur e \i qj2 s o tha t
-W
2
e
and
e-\A"
/»OOopo
- / e~
tzdßq/2(t),
Vz 0 ,
Jo
/»OO
= / e-
tz2diiq/2{t),
VzeR .
Jo
Taking th e Fourie r transfor m o f bot h side s b y z (th e functio n e \
z\q
i s
integrable s o we can appl y th e Fubin i theorem) , w e get
/»oo
7,(0 = V ^ / r ^ e - ^ d ^ W ,
Jo
which i n particula r implie s tha t 7 9(£) 0 . Usin g th e Cauchy-Schwart z
inequality w e get tha t fo r al l £i, £ 2 0
7,2
/»OO /»OO
T T / t-^e-^d^it) / t-We-Wd^ftit) = -y q(y/Ti) 7,(/6)-
Jo J o
D
We ar e read y t o prov e th e resul t o n the extrema l sections .
THEOREM 1.5. For 0 q 2 and any £ G
5n _ 1
Voin_i(Ä? n (-L..., -^)
x
) Voin-i(^ n ^ )
n v n
Voin_1(^n(i,o,...,o)±).
PROOF. Fo r al l 0 £ 1 77 1 77 2 £ 2 such tha t £% + f | = 77 ^ + ry | an d
all £ 0 , the convexit y resul t fro m Lemm a 1.4 implie s
7g(*fi) ' 7(*6) 7 9(tyi) * 7z(*%)-
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