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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.

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Could you explain what you mean by 'characterize'; when I read this first I believe I totally misunderstood the intent. –  quid Aug 3 '11 at 19:21
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You seem to be asking for suggestions without much motivation. The feeling of MO is that questions should have concrete answers unless specifically asked as big list communit wiki questions. FAQ reading is recommended, and then perhaps think of a way to ask a single question with a question mark at the end. You will already have made progress if you can do this. –  Spencer Aug 3 '11 at 19:25
    
@quid, @Spencer Sorry for not being precise, I modified the request. Indeed, I am looking for a general answer and hint to literature or fields where such indexes where defined. –  VitoshKa Aug 4 '11 at 9:08
    
Convexity orderings of functions are standard concepts. Something you can find in the infamous "Inequalities" by Hardy, Littlewood, and Polya. As you are concerned mainly with functions defined on a compact interval, the following might be also of some interest (do not get distracted by the statistical flavor of the paper): * W. Chan, F. Proschan, and J. Sethuraman: Convex-ordering among functions, with applications to reliability and Mathematical Statistics. In: Topics in statistical dependence (Somerset, PA, 1987), volume 16 of IMS Lecture Notes Monogr. Ser. –  Peter Sarkoci Aug 4 '11 at 10:09
    
Thanks @Peter, that's helpful, but it's only about partial convex order. No global index as we need. –  VitoshKa Aug 4 '11 at 11:27

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