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 â‰¤ B2 Re ( Î»A )+  Î»  2 C for all complex numbers Î» . Show that  A  2 â‰¤ BC . (Hint: If C=0, show that A = 0 . If C 6 = 0 , then choose Î» = 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 : Â± Â± Â± Â± Â± n X j =1 a j b j Â± Â± Â± Â± Â± 2 â‰¤ Â² n X j =1  a j  2 !Â² n X j =1  b j  2 ! . (Hint: For all complex numbers Î» , one has that â‰¤ âˆ‘ n j =1  a jÎ»b j  2 . Expand this and apply the problem 6. with A = âˆ‘ n j =1 a j b j , B = âˆ‘ n j =1  a j  2 , and c = âˆ‘ n j =1  b j  2 .)...
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 Spring '09
 Mean Value Theorem, Complex Numbers, Complex number, nonnegative real numbers

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