  Lower bound for ${\bar \rho}''(0)$, the second derivative of the spherically
averaged atomic electronic density at the nucleus, and upper bound for ${\bar
\rho}'''(0)$, the third derivative, are obtained respectively. It is shown
that, for the ground state, ${\bar \rho}''(0) \ge {10/3} Z^2 \rho(0)$ and
${\bar \rho}'''(0) \le -{14/3}Z^3 \rho(0)$ where $Z$ is the charge of the
nucleus, and $\rho(0)$ is the electron density at the nucleus. Tighter bounds
for ${\bar \rho}''(0)$ and ${\bar \rho}'''(0)$ are also reported which are
valid for both the ground state and excited states. Explicit illustration with
the example of one-electron atomic ions is given.
