1. Th e Cauchy-Schwar z inequalit y Exercise 1.6 . Prov e th e followin g relate d inequality . Le t #i,.. . , denote positiv e rea l numbers. The n (1.9) {Xl'X2 X n )n . Hint: Ther e ar e man y way s t o d o this , bu t I sugges t explorin the followin g idea . Prov e thi s inequalit y fo r n = 2 and the n exten it, usin g induction , t o n = 2 k , k = 1,2... . The n prov e tha t i f th result hold s fo r n -f- 1 , the n i t mus t hol d fo r n , thu s filling th e gap between th e power s of two. Another approach : Writ e a,j = e l o g ^) an d us e th e convexit y the exponentia l function , i.e , the fac t tha t e tiXi+t2X2-i Wn%n fie Xl + tot X2 t n with tj 0 an d h+t 2 * n = l . Exercise 1.7 . Le t x±,..., x n an d a i , . . ., an b e positive real number Then O- J + T \«i+-+« n ~a i . ~ a 2 ™a n - V^ l ^ ^ - ^ n a i a 2 # , _ , a n ai * a 2 •an
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