Gaussian Poincare inequality for Normal Random Variables that are not Standard

Question

Let $X$ be a standard normal random variable. Then, for any differentiable f:R→R such that $\mathbb{E}f(X)^2<∞$ the Gaussian Poincare inequality states that $$Var(f(X))≤\mathbb{E}[f′(X)^2]$$. I'd like to know what is the equivalent bound for normal random variables with variance $\sigma^2 \neq 1$, if there is such an equivalent. I tried reading up on the Gaussian Poincare inequality and where the fact that $\sigma^2 = 1$ comes in, and couldn't find my way around it. I'd love some help. Thanks!

Answer 1

More generally, for differentiable $f: \mathbb{R}^d \to \mathbb{R}$, and $d$-dimensional Gaussian vector $X \sim N(0, \Sigma$), we have $$ \text{Var}(f(X)) \le \mathbb{E} \langle \Sigma \nabla f(X), \nabla f(X) \rangle $$