Hyperbolically 1-convex functions
Tom 84 / 2004
Streszczenie
There are two reasonable analogs of Euclidean convexity in hyperbolic geometry on the unit disk ${{\mathbb D}}$. One is hyperbolic convexity and the other is hyperbolic 1-convexity. Associated with each type of convexity is the family of univalent holomorphic maps of ${{\mathbb D}}$ onto subregions of the unit disk that are hyperbolically convex or hyperbolically 1-convex. The class of hyperbolically convex functions has been the subject of a number of investigations, while the family of hyperbolically 1-convex functions has received less attention. This paper is a contribution to the study of hyperbolically 1-convex functions. A main result is that a holomorphic univalent function $f$ defined on ${{\mathbb D}}$ with $f({{\mathbb D}})\subseteq {{\mathbb D}}$ is hyperbolically 1-convex if and only if $f/(1-wf)$ is a Euclidean convex function for each $w \in \overline {{{\mathbb D}}}$. This characterization gives rise to two-variable characterizations of hyperbolically 1-convex functions. These two-variable characterizations yield a number of sharp results for hyperbolically 1-convex functions. In addition, we derive sharp two-point distortion theorems for hyperbolically 1-convex functions.
