On an inequality and on the related characterization of the normal distribution

We obtain the conditions on the distribution of the random variable $\xi$ under which the inequality
$$ \mathbf Dg(\xi)\le c\mathbf E(g'(\xi))^2 $$
holds for any differentiable function $g$. Some properties of the functional
$$ U_\xi=\sup_g\frac{\mathbf Dg(\xi)}{\mathbf D\xi\mathbf E(g'(\xi))^2} $$
are investigated also. It is proved that $U_\xi\ge 1$ and that $U_\xi=1$ iff the random variable $\xi$ has the normal distribution. The theorem of continuity is true as well: if $U_{\xi_n}\to 1$ as $n\to\infty$, then the distributions of $\xi_n^{(1)}=(\xi_n-\mathbf E\xi_n)/\sqrt{D\xi_n}$ converge to the normal one.