Suppose $X\in\Omega$ is a random variable and $f:\Omega\rightarrow [0,1]$. Is the following true:
$$E[|f(X)-E[f(X)]|]^2\leq \operatorname{Var}[f(X)]?$$
This was stated without proof in a research paper (perhaps the proof is trivial, but I can't see it). I also wonder if the class of functions $f$ can be generalized (perhaps this is true for $f:\Omega\rightarrow \mathbb{R}^+$?).
Hint: By Jensen's inequality, $E(Y^2)\ge E(|Y|)^2$. Now use $Y=f-E(f)$, so that $E(Y^2)=Var(f)$.