Prove this inequality : Geometric Mean and Arithmetic Mean

[michonamona]

Homework Statement

let $$r_{1}, r_{2}, ... , r_{n}$$ be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to $$1+r_{k}$$ at the end of year k (so that $$r_{k}$$ is the "return on investment" in year k). Then the value of an investment of one dollar at the start of an n-year period is worth $$V= \Pi_{k=1}^{n} (1+r_{k})$$ at the end of this period. Prove that

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

where $$R_{G} = (r_{1}r_{2}...r_{n})^{1/n}$$ and $$R_{A} = (r_{1} + r_{2} + ... + r_{n})/n$$ are, respectively, the Geometric and Arithmetic mean of returns.

(Hint: For one inequality, consider the function $$log(1+e^x)$$, and associate r with $$e^x$$).

Homework Equations

The Attempt at a Solution

So far I managed to recognize that the term on the right hand side is almost like (as we take n to infinity) the exponential raised to $$r_{1} + r_{2} + ... + r_{n}$$. But then I got stuck. I just need something to work with that'll get me going. I'm not looking for answers, just some insights.

I'm also not sure how I can use the hint to solve this problem.

 

[michonamona]

I figured out the inequality in the right hand side. Any hints for the one on the left?