标准正态方差界（A standard normal variance bound）不等式证明求助

Hey folks, I recently came across this inequality that seems straightforward at first glance, but I'm really struggling to work out its proof. Let me share the problem statement clearly:

Let $Z$ be a standard normal random variable, and $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function with bounded first derivative. Prove that:
$$\mathbb{E}[f'(Z)]^2 \leq \operatorname{Var}[f(Z)] \leq \mathbb{E}\left[f'(Z)^2\right]$$

A quick side note: I saw a similar inequality called the Gaussian Poincare inequality in some online discussion before, which I believe is related to proving the right-hand side of this statement. But I'm still not confident about how to formalize that connection, and the left-hand side bound has me completely stuck so far.

If anyone has insights on either part of the proof, or can point me in the right direction, I'd really appreciate it!

备注：内容来源于stack exchange，提问作者Harry