Let $\Omega$ and $\Pi$ be two simply connected proper subdomains
of the complex plane $\mathbb{C}$. We are concerned with the set
$A(\Omega,\Pi)$ of functions $f: \Omega\longrightarrow\Pi$
holomorphic on $\Omega$ and we prove estimates for $|f^{(n)}(z)|,
f \in A(\Omega,\Pi), z \in \Omega$, of the following type. Let
$\lambda_{\Omega}(z)$ and $\lambda_{\Pi}(w)$ denote the density of
the Poincaré metric with curvature $K=-4$ of $\Omega$ at $z$ and of
$\Pi$ at $w$, respectively. Then for any pair $(\Omega,\Pi)$ of
convex domains, $f \in A(\Omega,\Pi), z \in \Omega$, and $n\geq 2$
the inequality
$$
\frac{|f^{(n)}(z)|}{n!}\leq
2^{n-1}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}
$$
is valid. The constant $2^{n-1}$ is best possible for any pair
$(\Omega,\Pi)$ of convex domains.
For any pair $(\Omega,\Pi)$, where $\Omega$ is convex and $\Pi$
linearly accessible, $f,z,n$ as above, we prove
$$
\frac{|f^{(n)}(z)|}{(n+1)!}\leq
2^{n-2}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}.
$$
The constant $2^{n-2}$ is best possible for certain admissible pairs
$(\Omega,\Pi)$.
These considerations lead to a new, nonanalytic, characterization of
bijective convex functions $h:\Delta\to\Omega$ not using the second
derivative of $h$.