Bounds for the second and the third derivatives of the electron density at the nucleus

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.