On page 4 of this paper, the author defines a convex function on $\mathbb{Z}$ as follows. This function is the ones that verifies the condition $2f(x) \leq f(x+1) + f(x-1)$. He also said that for some $x_0\in\mathbb{Z}$, $f(x_0)$ is the global minimum of $f$ if and only if $f(x_0) \leq \min\{f(x_0 - 1), ~ f(x_0 + 1)\}$. I successfully proved this statement.
Moreover, I also proved that the global minimum is reached at only one point $x_0$ if and only if $f(x_0) < \min\{f(x_0 - 1), ~ f(x_0 + 1)\}$. However this is not stated in the paper.
I want to use both statements in my personal paper and I think that those results are certainly proved somewhere (in a book for example) and I can simply cite this reference. But I do not find such a reference.