When is it true that $f(x)\int_{U} g(x) dx \le \int_U f(x) g(x)$ ($f$ non constant)?

Let $$f,g: U \to \mathbb{R}^+$$ be integrable functions, where $$U$$ is a bounded subset of $$\mathbb{R}^n$$.

Under what assumptions on $$f$$ is it true that $$f(x)\int_{U} g(x) dx \le \int_U f(x) g(x) dx \ ?$$