For a random variable $X$ and bounded and non-decreasing functions $f,g : \mathbb{R} \to \mathbb{R}$, we have: $$ \mathbb{E} [f(X) g(X)] \ge \mathbb{E} [f(X)] \mathbb{E} [g(X)]$$
In case I am given coordinate-wise non-decreasing (bounded) functions $f,g : \mathbb{R}^{d} \to \mathbb{R}$ and a random vector $\vec{X}$, does: $$ \mathbb{E} [f(\vec{X}) g(\vec{X})] \ge \mathbb{E} [f(\vec{X})] \mathbb{E} [g(\vec{X})]$$ hold true?