# The punishing factors for convex pairs are $2_{n−1}$

### Farit G. Avkhadiev

Kazan State University, Russian Federation### Karl-Joachim Wirths

Universität Braunschweig, Germany

## Abstract

Let $Ω$ and $Π$ be two simply connected proper subdomains of the complex plane $C$. We are concerned with the set $A(Ω,Π)$ of functions $f:Ω⟶Π$ holomorphic on $Ω$ and we prove estimates for $∣f_{(n)}(z)∣,f∈A(Ω,Π),z∈Ω$, of the following type. Let $λ_{Ω}(z)$ and $λ_{Π}(w)$ denote the density of the Poincaré metric with curvature $K=−4$ of $Ω$ at $z$ and of $Π$ at $w$, respectively. Then for any pair $(Ω,Π)$ of convex domains, $f∈A(Ω,Π),z∈Ω$, and $n≥2$ the inequality

is valid. The constant $2_{n−1}$ is best possible for any pair $(Ω,Π)$ of convex domains. For any pair $(Ω,Π)$, where $Ω$ is convex and $Π$ linearly accessible, $f,z,n$ as above, we prove

The constant $2_{n−2}$ is best possible for certain admissible pairs $(Ω,Π)$. These considerations lead to a new, nonanalytic, characterization of bijective convex functions $h:Δ→Ω$ not using the second derivative of $h$.

## Cite this article

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

DOI 10.4171/RMI/516