Inequality.pdf - REAL ANALYSIS INEQUALITIES The Inequality...

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REAL ANALYSIS INEQUALITIES The Inequality of Arithmetic and Geometric Means Given x 1 0 x 2 0 . . . x n 0 A = x 1 + x 2 + . . . + x n n G = ( x 1 x 2 . . . x n ) 1 n Then G < A unless x 1 = x 2 = . . . = x n , when G = A . Proof Suppose without loss of generality that x 1 is a maximal x ν and x 2 is a minimal x ν . If x 1 = x 2 the r ’s are all equal and there is nothing to prove. Suppose then that x 1 > x 2 . We form a new set of numbers x 11 x 21 . . . x n 1 by writing x 11 = A x 21 = - A + x 1 + x 2 x r 1 = x r r = 3 , . . . , n. Let A 1 , G 1 be the A.M and G.M of the x r 1 ’s. A 1 = A since x 11 + x 21 = x 1 + x 2 . However x 11 x 2 x - x 1 x 2 = A ( x 1 + x 2 - A ) - x 1 x 2 = ( x 1 - A )( A - x 2 ) > 0 since x 1 > A > x 2 . Therefore G 1 > G If the x ν 1 are not all equal we can again take a largest x α 1 and a smallest x β 1 and replace them by A, x α 1 + x α 2 - A . A 2 = A 1 G 2 > G 1 After at most n - 1 steps all the x ’s are equal. G < G 1 < . . . < G K = A k = A therefore G < A . Cauchy’s Inequalities Given x 1 . . . x n y 1 . . . y n real. Then 1
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n X r =1 x r y r ˆ n X r =1 a 2 r !
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