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Show that the inequality is true | Geometric Mean

  • #1
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Homework Statement



Let [tex] r_{1}, r_{2}, ... , r_{n} [/tex] be strictly positive numbers. Show that the inequality

[tex] (1+R_{G})^{n} \leq V [/tex]

is true. Where [tex] R_{G} = (r_{1}r_{2}...r_{n})^{1/n}[/tex] and [tex] V= \Pi_{k=1}^{n} (1+r_{k}) [/tex]

Homework Equations





The Attempt at a Solution



I've tried taking the log of both sides, as well as expanding out the term. Any insight?

Thanks,
M
 

Answers and Replies

  • #2
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Use the arithmetic mean-geometric mean inequality... several times.
 
  • #3
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Any other insights?

The prof hinted that we should use log(1+e^x) and associate r with e^x.
 
  • #4
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That's an entirely different way to approach it. The approach I was thinking of uses the fact that the terms of the right side are the elementary symmetric functions of the [tex]r_j[/tex].
 

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