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hw1additional - Homework 1 Additional Problem 1(1 Let 1 < p...

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Homework 1, Additional Problem. (1) Let 1 < p < a real number and let q be defined by 1 = 1 p + 1 q . a. Let f ( t ) = 1 p t p + 1 q - t . Show (by means of calculus), that f ( t ) 0 for all t 0. b. Show that ab a p p + b q q for all a, b > 0. (Hint: Take t = a b q - 1 in part a. ). c. Show that | n i =1 a i b i | ≤ ( n i =1 | a i | p ) 1 p ( n i =1 | b i | q ) 1 q for all a = ( a 1 , · · · , a n ) , b = ( b 1 , · · · , b n ) R n . d. Show that n i =1 | a i + b i | p 1 p n i =1 | a i | p 1 p +
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