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

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.

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Correspondence to Farit G. Avkhadiev.

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Dedicated to W. K. Hayman on the occasion of his 80th birthday

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Avkhadiev, F.G., Wirths, KJ. Estimates of the Derivatives of Meromorphic Maps from Convex Domains into Concave Domains. Comput. Methods Funct. Theory 8, 107–119 (2008). https://doi.org/10.1007/BF03321674

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Keywords

  • Convex domain
  • concave domain
  • nth derivative
  • conformal radius
  • subordination

2000 MSC

  • 30C80
  • 30C55
  • 30C20