We investigate univalent holomorphic functions f defined on the unit disk 𝔻 such that f(𝔻) is a hyperbolically convex subset of 𝔻; there are a number of analogies with the classical theory of (euclidean) convex univalent functions. A subregion Ω of 𝔻 is called hyperbolically convex (relative to hyperbolic geometry on 𝔻) if for all points a,b in Ω the arc of the hyperbolic geodesic in 𝔻 connecting a and b (the arc of the circle joining a and b which is orthogonal to the unit circle) lies in Ω. We give several analytic characterizations of hyperbolically convex functions. These analytic results lead to a number of sharp consequences, including covering, growth and distortion theorems and the precise upper bound on |f''(0)| for normalized (f(0) = 0 and f'(0) > 0) hyperbolically convex functions. In addition, we find the radius of hyperbolic convexity for normalized univalent functions mapping 𝔻 into itself. Finally, we suggest an alternate definition of "hyperbolic linear invariance" for locally univalent functions f: 𝔻 → 𝔻 that parallels earlier definitions of euclidean and spherical linear invariance.
@article{bwmeta1.element.bwnjournal-article-apmv60z1p81bwm,
author = {Wancang Ma and David Minda},
title = {Hyperbolically convex functions},
journal = {Annales Polonici Mathematici},
volume = {60},
year = {1994},
pages = {81-100},
zbl = {0818.30010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv60z1p81bwm}
}
Wancang Ma; David Minda. Hyperbolically convex functions. Annales Polonici Mathematici, Tome 60 (1994) pp. 81-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv60z1p81bwm/
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