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Monday, November 23, 2009

Mean of Means

If a_i,b_i are positive numbers for i=1,\cdots,n, and

! M_r (a,b) = \left( \frac{a^r+b^r}{2} \right)^\frac{1}{r}

Prove that for 0 < r < s,

! \displaystyle \sum M_r(a_i,b_i) \sum M_{-r}(a_i,b_i) \leq \sum M_s(a_i,b_i) \sum M_{-s}(a_i,b_i) \leq \sum a_i \sum b_i

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