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 D. Each function f∈ Co(α) maps the unit disk D onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1 -{pipe}z{pipe}2) (f″(z)/f′(z)), f ∈ 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 -{pipe}z{pipe}2) (f″(z)/f′(z)), f ∈ 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 Hp space for p < 1/α. © 2009 Springer-Verlag.