Suppose we have a random variable $x \sim N(\mu,\sigma^2)$ and a funtion $f:R\rightarrow R$ such that $f'>0$ and $f''<0$. What do we know about the variance of $f(x)$? Can we possibly express it in terms of $\mu,\sigma,f$, etc.?
If not, can we if having extra restrictions?
It seems to me that it stops at: $$ \mathbb{Var}(f(x))=\mathbb{E}[(f(x))^2]-(\mathbb{E}[f(x)])^2 \\ $$
I'm not sure if an equivalent result holds in the case where $\mu \neq 0$ since the proof relies on the rotational invariance of a gaussian. But if $\mu = 0, \sigma = 1$ then we have that $f(X)$ is subgaussian with parameter at most $L^2$ which is always an upper bound on $\text{Var}(f(X))$. See High Dimensional Statistics by Wainwright Theorem 2.26.