  Let $Co(\alpha)$ denote the class of concave univalent functions in the unit
disk $\ID$. Each function $f\in Co(\alpha)$ maps the unit disk $\ID$ onto the
complement of an unbounded convex set. In this paper we find the exact disk of
variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in
Co(\alpha)$. In particular, this gives sharp upper and lower estimates for the
pre-Schwarzian norm of concave univalent functions. Next we obtain the set of
variability of the functional $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in
Co(\alpha)$ whenever $f''(0)$ is fixed. We also give a characterization for
concave functions in terms of Hadamard convolution. In addition to sharp
coefficient inequalities, we prove that functions in $Co(\alpha)$ belong to the
$H^p$ space for $p<1/\alpha$.
