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Unformatted text preview: Some Classical Inequalities For all of these inequalities there are many methods. We give a sampling. 1. ARITHMETIC GEOMETRIC MEAN INEQUALITY If { b j } > 0, prove the following – and decide when equality holds. ( b 1 b 2 ··· b n ) 1 / n ≤ b 1 + b 2 + ··· + b n n . (1) Solution : Here are two approaches. Note that equality holds only if all the b j ’s are equal. METHOD 1. The most naive approach is probably by induction on n . The assertion is clearly true when n = 1. Let B = ( b 1 + ··· + b n ) / n . Say the desired inequality b 1 b 2 ··· b n ≤ B n holds for a certain n (our induction hypothesis). Using this we find that ( b 1 b 2 ··· b n + 1 ) 1 / ( n + 1 ) ≤ [ B n b n + 1 ] 1 / ( n + 1 ) , (2) so we will be done if we can show that ( B n b n + 1 ) 1 / ( n + 1 ) ≤ nB + b n + 1 n + 1 (3) (one can interpret this as reducing (1) for n + 1 terms to the special case when b 1 = ··· = b n = B ). At this point, we could stop since this inequality is a special case of)....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
 Fall '10
 STAFF
 Inequalities

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