Computational Methods and Function Theory

, Volume 1, Issue 2, pp 521–533 | Cite as

Taylor Coefficients of Negative Powers of Schlicht Functions

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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
$$\sum_{k=1}^n \sigma_k \mid a_k(\lambda,f) \mid^2$$
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.

Keywords

Taylor coefficients univalent functions Löwner’s method optimization 

2000 MSC

30C75 49K15 
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Copyright information

© Heldermann Verlag 2001

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität WürzburgAm HublandGermany
  2. 2.Institut für AnalysisTU BraunschweigBraunschweigGermany

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