Prove this inequality : Geometric Mean and Arithmetic Mean

[michonamona]

Homework Statement

let [tex] r_{1}, r_{2}, ... , r_{n}[/tex] be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to [tex]1+r_{k}[/tex] at the end of year k (so that [tex]r_{k}[/tex] 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 [tex]V= \Pi_{k=1}^{n} (1+r_{k})[/tex] at the end of this period. Prove that

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

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

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

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 [tex]r_{1} + r_{2} + ... + r_{n}[/tex]. 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?