Here is an extract from A less than B. The author claims to imply the AM-GM inequality with this reasoning, but I can't see how. So far the author has covered AM-GM, convexity, the smoothing principle and Jensen's inequality.
"Theorem 4: Let $f$ be a twice-differentiable function on an open interval $I$. Then $f$ is convex on $I$ if and only if $f''(x)\ge 0$ for all $ x \in I$.
For example, the AM-GM inequality can be proved by noting that $f(x)=\log(x)$ is concave; its first derivative is $1/x$ and its second $-1/x^2$. In fact, one immediately deduces a weighted AM-GM inequality..."
I understand the theorem, but not at all how it applies to AM-GM. Any enlightenment would be much appreciated!