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Minkowski's Inequality?

  1. Aug 2, 2011 #1
    I don't understand how it is possible to show using the Minkowski's Inequality that
    [itex] (\sum x_i )^a \leq \sum x_i^a[/itex] where [itex] x_i \geq 0 \forall i [/itex] and [itex] 0<a<1 [/itex].

    I also tried to prove this without using Minkowski, but to no avail.

    This is driving me crazy although it seems to be trivial in the literature.
    I will appreciate any help
     
  2. jcsd
  3. Aug 2, 2011 #2

    micromass

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    Hi St41n! :smile:

    It seems that you must prove that [itex](x+y)^\alpha\leq x^\alpha+y^\alpha[/itex] for [itex]x,y\geq 0[/itex] and [itex]0<\alpha<1[/itex].

    For that, you must look at the function

    [tex]f:\mathbb{R}^+\rightarrow \mathbb{R}:x\rightarrow 1+x^\alpha-(1+x)^\alpha[/tex]

    Try to show that f is increasing and has its minimum in 0. It follows that [itex]f(x)\geq 0[/itex].
     
  4. Aug 2, 2011 #3
    Thank you very much for the quick reply!
     
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