I have recently come across the following result.

Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $e_k$ is the $k$-th elementary symmetric polynomial.

I think the inequality is tight exactly when $x_{d+1} = \cdots = x_{n} = 0$. I believe this result is true, but I am having problems making progress. Any references would be appreciated.