  Let $X$ be a standard Gaussian random variable. For any $p \in (1, \infty)$,
we prove the existence of a universal constant $C_{p}>0$ such that the
inequality $$(\mathbb{E} |h'(X)|^{p})^{1/p} \geq C_{p} \sqrt{d} (\mathbb{E}
|h(X)|^{p})^{1/p}$$ holds for all $d\geq 1$ and all polynomials $h : \mathbb{R}
\to \mathbb{C}$ whose spectrum is supported on frequencies at least $d$, that
is, $\mathbb{E} h(X) X^{k}=0$ for all $k=0,1, \ldots, d-1$. As an application
of this optimal estimate, we obtain an affirmative answer to the Gaussian
analogue of a question of Mendel and Naor (2014) concerning the growth of the
Ornstein-Uhlenbeck operator on tail spaces of the real line. We also show the
same bound for the gradient of analytic polynomials in an arbitrary dimension.
