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Estimates of the Derivatives of Meromorphic Maps from Convex Domains into Concave Domains

  • Farit G. AvkhadievEmail author
  • Karl-Joachim Wirths
Article

Abstract

Let ω be a proper convex subdomain of the complex plane C. Let further π1 ⊂ ℂ be a compact convex set containing more than one point and π = \({\rm}\overline C\ \Pi_1\). We denote by R Ω (z) and R π (ω) the conformal radius of Ω at z and of π at finite points ω, respectively. We are concerned with the set A(Ω, π) of functions f: Ω → π meromorphic on Ω. We prove that for n ≥ 2, fA(Ω, π), z∈Ω and f(z) finite the inequalities
$${|f^{(n)}(z)|\over n!}{(R_\Omega(z))^n\over R_\Pi(f(z))} \leq {(1+p)^{n-2}\over p^{n-1}} {\sum^n_{k=0}} p^k$$
are valid, where p is a measure for the distance between f(z) and the point at infinity.

We give examples showing that equality is possible in this estimate.

Keywords

Convex domain concave domain nth derivative conformal radius subordination 

2000 MSC

30C80 30C55 30C20 

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Copyright information

© Heldermann  Verlag 2008

Authors and Affiliations

  1. 1.Chebotarev Research InstituteKazan State UniversityKazanRussia
  2. 2.Institut für Analysis und AlgebraTU BraunschweigBraunschweigGermany

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