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

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 = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm}\overline C\ \Pi_1$\end{document}. We denote by RΩ (z) and Rπ (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, f ∈ A(Ω, π), z∈Ω and f(z) finite the inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|f^{(n)}(z)|\over n!}{(R_\Omega(z))^n\over R_\Pi(f(z))} \leq {(1+p)^{n-2}\over p^{n-1}} {\sum^n_{k=0}} p^k$$\end{document} are valid, where p is a measure for the distance between f(z) and the point at infinity.