Inequality to prove involving convex notions

Hi everyone,

Here's an exercice I have trouble with.

Let $\displaystyle a = (a_1,\dots,a_n)\in\mathbb{R}^n_+$ and define $\displaystyle G(a) = (a_1\dots a_n)^{\frac{1}{n}}$

I have to show that, assuming $\displaystyle a,b\in\mathbb{R}^n_+, G(a+b) \geqslant G(a) + G(b)$.

I have already tried a looooot of things. I think I'm suppose to use some convex inequalities : my first thought was to take the log of $\displaystyle G(a+b)$, but it didn't seem to end very well for me.

I've also tried to use other known inequalities, such as $\displaystyle ab \leqslant \displaystyle \frac{a^2+b^2}{2}$. But I don't think it works.

So know, it's been almost an hour I'm on it, and I would like to have a small hint : I do not want the whole answer but just a tips which can make me progress.

Thank you for your always-so-good answers,

Hugo.

Re: Inequality to prove involving convex notions

We have to show that for $\displaystyle 0\leq x_j \leq 1$, we have $\displaystyle \left(\prod_{j=1}^nx_j\right)^{\frac 1n}+\left(\prod_{j=1}^n(1-x_j)\right)^{\frac 1n}\leq 1$. Take the $\displaystyle n$-th power, and use $\displaystyle m:=\min_{1\leq j\leq n}x_j$.