Hyperbolically 1-convex functions

William Ma, David Minda, and Diego Mejia. "Hyperbolically 1-convex functions." Annales Polonici Mathematici 84.3 (2004): 185-202. <http://eudml.org/doc/280913>.

@article{WilliamMa2004,
abstract = {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.},
author = {William Ma, David Minda, Diego Mejia},
journal = {Annales Polonici Mathematici},
keywords = {hyperbolic 1-convexity; two-variable characterization; two-piont distortion theorems},
language = {eng},
number = {3},
pages = {185-202},
title = {Hyperbolically 1-convex functions},
url = {http://eudml.org/doc/280913},
volume = {84},
year = {2004},
}

TY - JOUR
AU - William Ma
AU - David Minda
AU - Diego Mejia
TI - Hyperbolically 1-convex functions
JO - Annales Polonici Mathematici
PY - 2004
VL - 84
IS - 3
SP - 185
EP - 202
AB - 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.
LA - eng
KW - hyperbolic 1-convexity; two-variable characterization; two-piont distortion theorems
UR - http://eudml.org/doc/280913
ER -