Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. Further, let $ X $ be a random variable supported on $\mathbb{X}$.
I am interested in the following inequality:
$$ \frac{ f(x_n) - \mathbb{E}[f(X)] }{ \sqrt{ \mathbb{E}\left[ \left(f(X) - \mathbb{E}[f(X)]\right)^2 \right] } } \leq \frac{ x_n - \mathbb{E}[X] }{ \sqrt{ \mathbb{E}\left[ \left(X - \mathbb{E}[X]\right)^2 \right] } } $$
Is this inequality always true under the given conditions? If so, can you provide a proof or some intuition behind why this inequality holds? If not, can you provide a counterexample or conditions under which it might fail?
