What is the "least varying" subgaussian random variable?

Question

Consider $X \sim \mu$ to be a mean $0$ $1$-subgaussian random variable. Define $f(\mu) = \frac{\mathbb{E} [|X|]^2}{\mathbb{E} [X^2]}$. What is the density which minimizes this metric of "variation"? And is there a constant lower bound on $f$?

Answer 1

Take a law with mass $\exp(-t^2)$ on $\pm t$, and the rest on $0$. This satisfies the $1$-subGaussian condition. We have $$ \mathbb{E}[X^2] = 2t^2 \exp(-t^2), \mathbb{E}[|X|] = 2t\exp(-t^2),$$ and so $$ \mathbb{E}[|X|]^2/\mathbb{E}[X^2] = 2\exp(-t^2),$$ which goes to $0$ as $t \to \infty$. Of course in this case $\mathbb{E}[X^2]$ is tiny, and it may well be possible that you can get a nontrivial lower bound if this is bounded away from $0$.