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~ tx dfi(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 q j2 s o tha t -W 2 e and e -\A" /»OOopo - / e~ tz dßq/2(t), Vz 0 , Jo /»OO = / e- tz2 diiq/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 Voi n _ 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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