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?