Abstract
Letf be a one-to-one analytic function in the unit disc withf′(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions(f′) p, wherep<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative off. We use this to improve known estimates for integral means of the functions |f′|p for integersp⪯−2.
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Bertilsson, D. Coefficient estimates for negative powers of the derivative of univalent functions. Ark. Mat. 36, 255–273 (1998). https://doi.org/10.1007/BF02384769
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DOI: https://doi.org/10.1007/BF02384769