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

Abstract

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 $(1-|z{|}^2)\left(f^{\prime \prime }(z)/f^{\prime }(z)\right)$, $f\in Co(\alpha )$. 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 $(1-|z{|}^2)\left(f^{\prime \prime }(z)/f^{\prime }(z)\right)$, $f\in Co(\alpha )$ whenever $f^{\prime \prime }(0)$ 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 $p<1/\alpha$.