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Unformatted text preview: equation  z + c  =  1 + cz  . 6. (problem 20, Sec. 1.1) Let B and C be nonnegative real numbers and A a complex number. Suppose that B2 Re ( A )+   2 C for all complex numbers . Show that  A  2 BC . (Hint: If C=0, show that A = 0 . If C 6 = 0 , then choose = A/C .) 7. (problem 21, Sec. 1.1) Let a 1 , . . . , a n and b 1 , . . . , b n be complex numbers. Prove the Schwartz inequality : n X j =1 a j b j 2 n X j =1  a j  2 ! n X j =1  b j  2 ! . (Hint: For all complex numbers , one has that n j =1  a jb j  2 . Expand this and apply the problem 6. with A = n j =1 a j b j , B = n j =1  a j  2 , and c = n j =1  b j  2 .)...
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 Spring '09

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