  Fleming's inequality is generalized to the decay function of mixed states. We
show that for any symmetric hamiltonian $h$ and for any density operator $\rho$
on a finite dimensional Hilbert space with the orthogonal projection $\Pi$ onto
the range of $\rho$ there holds the estimate $\Tr(\Pi \rme^{-\rmi ht}\rho
\rme^{\rmi ht}) \geq\cos^{2}((\Delta h)_{\rho}t) $ for all real $t$ with
$(\Delta h)_{\rho}| t| \leq\pi/2.$ We show that equality either holds for all
$t\in\mathbb{R}$ or it does not hold for a single $t$ with $0<(\Delta
h)_{\rho}| t| \leq\pi/2.$ All the density operators saturating the bound for
all $t\in\mathbb{R},$ i.e. the mixed intelligent states, are determined.
