# Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$

Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that

$$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le a_1*x_1+a_2*x_2+...+a_n*x_n$$

I have been hinted to show that $-ln(x)$ is a strictly convex, decreasing function on $(0,\infty)$, which I have done. I am having trouble making it to this step, though.

• do you know the Jensen inequality? – Exodd Sep 29 '14 at 15:25
• @Exodd no, so I will look it up – user178831 Sep 29 '14 at 15:40
• – Macavity Sep 30 '14 at 12:40