JournalsrmiVol. 23, No. 3pp. 847–860

The punishing factors for convex pairs are 2n12^{n-1}

  • Farit G. Avkhadiev

    Kazan State University, Russian Federation
  • Karl-Joachim Wirths

    Universität Braunschweig, Germany
The punishing factors for convex pairs are $2^{n-1}$ cover
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Abstract

Let Ω\Omega and Π\Pi be two simply connected proper subdomains of the complex plane C\mathbb{C}. We are concerned with the set A(Ω,Π)A(\Omega,\Pi) of functions f:ΩΠf: \Omega\longrightarrow\Pi holomorphic on Ω\Omega and we prove estimates for f(n)(z),fA(Ω,Π),zΩ|f^{(n)}(z)|, f \in A(\Omega,\Pi), z \in \Omega, of the following type. Let λΩ(z)\lambda_{\Omega}(z) and λΠ(w)\lambda_{\Pi}(w) denote the density of the Poincaré metric with curvature K=4K=-4 of Ω\Omega at zz and of Π\Pi at ww, respectively. Then for any pair (Ω,Π)(\Omega,\Pi) of convex domains, fA(Ω,Π),zΩf \in A(\Omega,\Pi), z \in \Omega, and n2n\geq 2 the inequality

f(n)(z)n!2n1(λΩ(z))nλΠ(f(z))\frac{|f^{(n)}(z)|}{n!}\leq 2^{n-1}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}

is valid. The constant 2n12^{n-1} is best possible for any pair (Ω,Π)(\Omega,\Pi) of convex domains. For any pair (Ω,Π)(\Omega,\Pi), where Ω\Omega is convex and Π\Pi linearly accessible, f,z,nf,z,n as above, we prove

f(n)(z)(n+1)!2n2(λΩ(z))nλΠ(f(z)).\frac{|f^{(n)}(z)|}{(n+1)!}\leq 2^{n-2}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))}.

The constant 2n22^{n-2} is best possible for certain admissible pairs (Ω,Π)(\Omega,\Pi). These considerations lead to a new, nonanalytic, characterization of bijective convex functions h:ΔΩh:\Delta\to\Omega not using the second derivative of hh.

Cite this article

Farit G. Avkhadiev, Karl-Joachim Wirths, The punishing factors for convex pairs are 2n12^{n-1}. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 847–860

DOI 10.4171/RMI/516