Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let $Co(α)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(α)$ 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 $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(α)$. 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 $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in Co(α)$ 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 $Co(α)$ belong to the $H^p$ space for $p<1/α$.

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