There are two reasonable analogs of Euclidean convexity in hyperbolic geometry on the unit disk 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 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 𝔻 with f(𝔻) ⊆ 𝔻 is hyperbolically 1-convex if and only if f/(1-wf) is a Euclidean convex function for each w ∈ 𝔻̅. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-3-1,
author = {William Ma and David Minda and Diego Mejia},
title = {Hyperbolically 1-convex functions},
journal = {Annales Polonici Mathematici},
volume = {83},
year = {2004},
pages = {185-202},
zbl = {1070.30006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-3-1}
}
William Ma; David Minda; Diego Mejia. Hyperbolically 1-convex functions. Annales Polonici Mathematici, Tome 83 (2004) pp. 185-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-3-1/