# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## Hyperbolically 1-convex functions

### Tom 84 / 2004

Annales Polonici Mathematici 84 (2004), 185-202 MSC: Primary 30C45, 30C50. DOI: 10.4064/ap84-3-1

#### 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.

#### Autorzy

• William MaSchool of Integrated Studies
Pennsylvania College of Technology
Williamsport, PA 17701, U.S.A.
e-mail
• David MindaDepartment of Mathematical Sciences
University of Cincinnati
Cincinnati, OH 45221-0025, U.S.A.
e-mail