 Solution

Write $X=\mu +\sigma Z$ with $Z$ standard normal and set $g(z)=f(\mu +\sigma z)$. The Gaussian Poincaré inequality and the chain rule giveHence $N(\mu ,{\sigma }^2)$ satisfies a $\sigma$-Poincaré inequality in the question's convention, where the constant is squared on the right-hand side.