I've been wondering about this problem for some time. This is in a way related to the proof of Jensen's inequality. Say we have a convex function $f$. Can we prove $$\frac{f(a)+f(b)+f(c)}{3}\geq f(\frac{a+b+c}{3})$$ using only the fact that $$\frac{f(a)+f(b)}{2}\geq f(\frac{a+b}{2})$$
The proof that I know for Jensen's inequality generalises the problem to a form I wouldn't have guessed on my own. The proof is given here.
Thanks in advance!
Best Answer
Just as Sanchez said, you can in fact prove that $$\frac{f(x_1)+f(x_2)+...+f(x_n)}{n} \geq f(\frac{x_1+x_2+...x_n}{n}) $$
Here is the idea of the proof.
First prove this for $n=2^k$ by induction on $k$. The induction step is just applying the induction hypothesis twice.
Second, knowing the proposition is true for $n=2^k$ prove it for each $i<2^k$. To do this apply the $2^k$ inequality for $x_1,...,x_i$ plus additional $2^k-i$ copies of $$y=\frac{x_1+x_2+...x_i}{i}$$ After easy transformations this will turn equivalent to the proposition for $n=2^k$