  Assume $z_0$ lies in the open unit disk $\mathbb{D}$ and $g$ is an analytic
self-map of $\mathbb{D}$. We will determine the region of values of $g''(z_0)$
in terms of $z_0$, $g(z_0)$ and the hyperbolic derivative of $g$ at $z_0$, and
give the form of all the extremal functions. In particular, we obtain a smaller
sharp upper bound for $|g''(z_0)|$ than Ruscheweyh's inequality for the case of
the second derivative. Moreover, we use a different method to obtain
Sz{\'a}sz's inequality, which provides a sharp upper bound for $|g''(z_0)|$
depending only on $|z_0|$.
