Let and be two simply connected proper subdomains of the complex plane . We are concerned with the set of functions holomorphic on and we prove estimates for , of the following type. Let and denote the density of the Poincaré metric with curvature of at and of at , respectively. Then for any pair of convex domains, , and the inequality
is valid. The constant is best possible for any pair of convex domains. For any pair , where is convex and linearly accessible, as above, we prove
The constant is best possible for certain admissible pairs . These considerations lead to a new, nonanalytic, characterization of bijective convex functions not using the second derivative of .
Cite this article
Farit G. Avkhadiev, Karl-Joachim Wirths, The punishing factors for convex pairs are . Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 847–860DOI 10.4171/RMI/516