Show that the inequality is true | Geometric Mean

[michonamona]

Homework Statement

Let $$r_{1}, r_{2}, ... , r_{n}$$ be strictly positive numbers. Show that the inequality

$$(1+R_{G})^{n} \leq V$$

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

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 $$r_j$$.