Show that the inequality is true | Geometric Mean

[michonamona]

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

 

[ystael]

Use the arithmetic mean-geometric mean inequality... several times.

 

[michonamona]

Any other insights?

The prof hinted that we should use log(1+e^x) and associate r with e^x.

 

[ystael]

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].