  This paper deals with the Neumann eigenvalue problem for the Hermite operator
defined in a convex, possibly unbounded, planar domain $\Omega$, having one
axis of symmetry passing through the origin. We prove a sharp lower bound for
the first eigenvalue $\mu_1^{odd}(\Omega)$ with an associated eigenfunction odd
with respect to the axis of symmetry. Such an estimate involves the first
eigenvalue of the corresponding one-dimensional problem. As an immediate
consequence, in the class of domains for which
$\mu_1(\Omega)=\mu_1^{odd}(\Omega)$, we get an explicit lower bound for the
difference between $\mu(\Omega)$ and the first Neumann eigenvalue of any strip.
