Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

Question

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)} $$ where $\|\cdot \|_{H^1}$ is the standard Sobolev norm of the order $1$ and $\|f(x)\|_{H^1} = \sum_{k=1}^3 \|f_k (x)\|_{H^1}$, $\|f(x)\|_{L^\infty} = \sum_{k=1}^3 \|f_k (x)\|_{L^\infty}$.