Abstract
Let \(\cal S\) denote the class of normalized schlicht functions in the unit disk. We consider for f ∊ \({\cal S}\) and λ < 0 the Taylor coefficients a n (λ, f) of (f(z)/z)λ and prove that ∣a n(λ, f)∣ ≤ ∣a n(λ, k)∣ for every f ∈ S and every 1 ≤ n ≤ − λ + 1, where k(z) = z(l − z)−2 is the Koebe function. We also give a necessary condition such that the Koebe function maximizes the functional
in the class \({\cal S}\) for given weights σk ∈ R. These results supplement and complement previous results due to de Branges, Hayman and Hummel and others. Our proofs are based on the Löwner differential equation combined with optimal control methods.
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The first author was supported by an INTAS grant (project 99-00089).
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Roth, O., Wirths, KJ. Taylor Coefficients of Negative Powers of Schlicht Functions. Comput. Methods Funct. Theory 1, 521–533 (2001). https://doi.org/10.1007/BF03321005
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DOI: https://doi.org/10.1007/BF03321005