Hyperbolically convex functions

Wancang Ma; David Minda

Annales Polonici Mathematici (1994)

  • Volume: 60, Issue: 1, page 81-100
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Wancang Ma, and David Minda. "Hyperbolically convex functions." Annales Polonici Mathematici 60.1 (1994): 81-100. <http://eudml.org/doc/262411>.

@article{WancangMa1994,
abstract = {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.},
author = {Wancang Ma, David Minda},
journal = {Annales Polonici Mathematici},
keywords = {hyperbolic convexity; distortion theorem; growth thoerem; linear invariance; hyperbolically convex functions},
language = {eng},
number = {1},
pages = {81-100},
title = {Hyperbolically convex functions},
url = {http://eudml.org/doc/262411},
volume = {60},
year = {1994},
}

TY - JOUR
AU - Wancang Ma
AU - David Minda
TI - Hyperbolically convex functions
JO - Annales Polonici Mathematici
PY - 1994
VL - 60
IS - 1
SP - 81
EP - 100
AB - 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.
LA - eng
KW - hyperbolic convexity; distortion theorem; growth thoerem; linear invariance; hyperbolically convex functions
UR - http://eudml.org/doc/262411
ER -

References

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  1. [1] P. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York, 1983. 
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  6. [6] W. Ma and D. Minda, Euclidean linear invariance and uniform local convexity, J. Austral. Math. Soc. Ser. A 52 (1992), 401-418. Zbl0780.30010
  7. [7] W. Ma and D. Minda, Hyperbolic linear invariance and hyperbolic k-convexity, J. Austral. Math. Soc. Ser. A to appear. Zbl0838.30022
  8. [8] W. Ma and D. Minda, Spherical linear invariance and uniform local spherical convexity, in: Current Topics in Geometric Function Theory, H. M. Srivastava and S. Owa (eds.), World Sci., Singapore, 1993, 148-170. Zbl0982.30500
  9. [9] D. Mejia and D. Minda, Hyperbolic geometry in hyperbolically k-convex regions, Rev. Colombiana Mat. 25 (1991), 123-142. Zbl0784.53006
  10. [10] D. Minda, A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129-144. Zbl0576.30022
  11. [11] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J. Analyse Math. 49 (1987), 90-105. Zbl0638.30007
  12. [12] B. Osgood, Some properties of f''/f' and the Poincaré metric, Indiana Univ. Math. J. 31 (1982), 449-461. Zbl0503.30014
  13. [13] G. Pick, Über die konforme Abbildung eines Kreises auf eines schlichtes und zugleich beschränktes Gebiete, S.-B. Kaiserl. Akad. Wiss. Wien 126 (1917), 247-263. Zbl46.0553.01
  14. [14] Ch. Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 (1964), 108-154. Zbl0128.30105
  15. [15] E. Study, Konforme Abbildung einfachzusammenhängender Bereiche, Vorlesungen über ausgewählte Gegenstände der Geometrie, Heft 2, Teubner, Leipzig und Berlin, 1913. 
  16. [16] K.-J. Wirths, Coefficient bounds for convex functions of bounded type, Proc. Amer. Math. Soc. 103 (1988), 525-530. Zbl0655.30014

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