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

Lower bound for ${\bar{\rho }}^{\mathrm{\prime }\mathrm{\prime }}(0)$, 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, ${\bar{\rho }}^{\mathrm{\prime }\mathrm{\prime }}(0)\ge 10\mathrm{/}3Z^2\rho (0)$ and ${\bar{\rho }}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)\le -14\mathrm{/}3Z^3\rho (0)$ where $Z$ is the charge of the nucleus, and $\rho (0)$ is the electron density at the nucleus. Tighter bounds for ${\bar{\rho }}^{\mathrm{\prime }\mathrm{\prime }}(0)$ and ${\bar{\rho }}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)$ 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.