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Assume a discrete function $f\left(n\right)\geq 0$ for $n\in\mathcal{N}$. Can we say $f(n)$ is a convex function if $f(n+1)+f(n-1)-2f(n)\geq0$ ? I don't know why there is no such kind of expression for discrete function on Wikipedia.

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up vote 1 down vote accepted

$f(n)$ as defined by you is a sequence and such a sequence is called convex, exactly as you suggest. Have a look here for example.

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When you have a function $f:\mathbb N\to\mathbb R$ and want to apply some concept for functions defined on $\mathbb R$, the natural thing to do is to extend $f$ to be affine on each interval $[n,n+1]$, and consider the properties of this extension. In particular, the extension is a convex function if and only if $f(n+1)+f(n-1)-2f(n)\geq0$ for all $n$.

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Sorry for bothering you a century after your answer, but by any chance, do you have some reference (or some easy proof) for the last statement ? – Manuel Lafond Mar 23 '15 at 16:08

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