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|Title:||Characterization and the Pre-Schwarzian Norm Estimate for Concave Univalent Functions|
Wirths, K J
|Citation:||Journal of Monatsh Math, (post print)|
|Abstract:||Let Co(®) denote the class of concave univalent functions in the unit disk D. Each function f 2 Co(®) maps the unit disk D onto the complement of an unbounded convex set. In this paper we ¯nd the exact disk of variability for the functional (1¡jzj2) (f00(z)=f0(z)), f 2 Co(®). 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 ¡ jzj2) (f00(z)=f0(z)),f 2 Co(®) whenever f00(0) is ¯xed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coe±cient inequalities, we prove that functions in Co(®) belong to the Hp space for p < 1=®.|
|Appears in Collections:||Journal Articles|
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