We are looking for a global index of the convexity/concavity of a function.

For concreteness, how can I formalize the intuitive notion that a function $f$ is more convex than $g$ where $f,g:[0,1]\rightarrow \mathbb{R}$ are both increasing.

The only index which I am aware of and could be used for this purpose is the Gini coefficient.

Thanks.