Probability | Part IA, 2006

What is a convex function? State Jensen's inequality for a convex function of a random variable which takes finitely many values.

Let p1p \geqslant 1. By using Jensen's inequality, or otherwise, find the smallest constant cpc_{p} so that

(a+b)pcp(ap+bp) for all a,b0.(a+b)^{p} \leqslant c_{p}\left(a^{p}+b^{p}\right) \text { for all } a, b \geqslant 0 .

[You may assume that xxpx \mapsto|x|^{p} is convex for p1p \geqslant 1.]

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