  We obtain estimates for derivative and cross--ratio distortion for
$C^{2+\eta}$ (any $\eta>0$) unimodal maps with non--flat critical points. We do
not require any `Schwarzian--like' condition.
  For two intervals $J \subset T$, the cross--ratio is defined as the value
$$B(T,J):=\frac{|T||J|}{|L||R|}$$ where $L,R$ are the left and right connected
components of $T\setminus J$ respectively. For an interval map $g$ \st $g_T:T
\to \RRR$ is a diffeomorphism, we consider the cross--ratio distortion to be
$$B(g,T, J):=\frac{B(g(T),g(J))}{B(T,J)}.$$
  We prove that for all $0<K<1$ there exists some interval $I_0$ around the
critical point \st for any intervals $J \subset T$, if $f^n|_T$ is a
diffeomorphism and $f^n(T) \subset I_0$ then $$B(f^n, T, J)> K.$$ Then the
distortion of derivatives of $f^n|_J$ can be estimated with the Koebe Lemma in
terms of $K$ and $B(f^n(T),f^n(J))$. This tool is commonly used to study
topological, geometric and ergodic properties of $f$. This extends a result of
Kozlovski.

Dynamical Systems