  Exact upper and lower bounds on the ratio
$\mathsf{E}w(\mathbf{X}-\mathbf{v})/\mathsf{E}w(\mathbf{X})$ for a centered
Gaussian random vector $\mathbf{X}$ in $\mathbb{R}^n$, as well as bounds on the
rate of change of $\mathsf{E}w(\mathbf{X}-t\mathbf{v})$ in $t$, where
$w\colon\mathbb{R}^n\to[0,\infty)$ is any even unimodal function and
$\mathbf{v}$ is any vector in $\mathbb{R}^n$. As a corollary of such results,
exact upper and lower bounds on the power function of statistical tests for the
mean of a multivariate normal distribution are given.
