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Inequality proof help

  1. Oct 2, 2006 #1
    If a1 >= a2 >= ... >= a_n and b1 >= b2 >= ... >= b_n, prove that:

    ( Sum(a_k from k = 1 to n) )*( Sum(b_k from k = 1 to n) ) <= n*Sum(a_k*b_k
    from k = 1 to n).

    Hint: Sum( (a_k - a_j)*(b_k - b_j) s.t. 1 <= j <= k <= n ) >= 0

    This is what I have so far:

    0 <= Sum( (a_k - a_j)*(b_k - b_j) s.t. 1 <= j <= k <= n )

    0 <= Sum( a_k*b_k - a_k*b_j - a_j*b_k + a_j*b_j s.t. 1 <= j <= k <= n )

    0 <= Sum( a_k*b_k from k = 1 to n ) - Sum( a_k*b_j s.t. 1 <= j <= k <=
    n ) - Sum( a_j*b_k s.t. 1 <= j <= k <= n ) + Sum( a_j*b_j from j = 1 to
    n )

    0 <= 2*Sum( a_k*b_k from k = 1 to n ) - Sum( a_k*b_j s.t. 1 <= j <= k <=
    n ) - Sum( a_j*b_k s.t. 1 <= j <= k <= n )

    I hope that is correct thus far, but do not know what to do next.
     
  2. jcsd
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