Hyperbolically 1-convex functions

William Ma¹ ; David Minda² ; Diego Mejia³

¹ School of Integrated Studies Pennsylvania College of Technology Williamsport, PA 17701, U.S.A.
² Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 45221-0025, U.S.A.
³ Departamento de Matemáticas Universidad Nacional de Colombia A.A. 3840, Medellín, Colombia

Annales Polonici Mathematici, Tome 84 (2004) no. 3, pp. 185-202.

Résumé

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.

Détail

DOI : 10.4064/ap84-3-1

Keywords: there reasonable analogs euclidean convexity hyperbolic geometry unit disk mathbb hyperbolic convexity other hyperbolic convexity associated each type convexity family univalent holomorphic maps mathbb subregions unit disk hyperbolically convex hyperbolically convex class hyperbolically convex functions has subject number investigations while family hyperbolically convex functions has received attention paper contribution study hyperbolically convex functions main result holomorphic univalent function defined mathbb mathbb subseteq mathbb hyperbolically convex only wf euclidean convex function each overline mathbb characterization gives rise two variable characterizations hyperbolically convex functions these two variable characterizations yield number sharp results hyperbolically convex functions addition derive sharp two point distortion theorems hyperbolically convex functions