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dc.contributor.authorBhowmik, B-
dc.contributor.authorPonnusamy, S-
dc.contributor.authorWirths, K J-
dc.identifier.citationJournal of Monatsh Math, (post print)en
dc.description.abstractLet 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=®.en
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dc.titleCharacterization and the Pre-Schwarzian Norm Estimate for Concave Univalent Functionsen
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