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Added on 05/07/06
Let's prove the original inequality by induction. 1) ; we have to show that, for any and in , . But, since must be equal to 1, we can put , so that the thesis becomes , which is true by definition of a convex function. 2) Taking as true that , where...
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Let's prove the original inequality by induction. 1) ; we have to show that, for any and in , . But, since must be equal to 1, we can put , so that the thesis becomes , which is true by definition of a convex function. 2) Taking as true that , where , we have to prove that , where . First of all, let's observe that and that if all , belongs to as well. In fact, being non-negative, , and summing over , , that is . We have, by definition of a convex function: . But, by inductive hypothesis, since , we have: , so that which is the thesis. See less 
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