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Hyperbolically convex functions

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EN
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.
Twórcy
autor
  • Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
autor
  • Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
Bibliografia
  • [1] P. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York, 1983.
  • [2] B. Flinn and B. Osgood, Hyperbolic curvature and conformal mapping, Bull. London Math. Soc. 18 (1986), 272-276.
  • [3] A. W. Goodman, Univalent Functions, Vols. I and II, Mariner, Tampa, 1983.
  • [4] M. Heins, On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1-60.
  • [5] W. Ma, D. Mejia and D. Minda, Distortion theorems for hyperbolically and spherically k-convex functions, in: Proc. Internat. Conf. on New Trends in Geometric Function Theory and Applications, R. Parvatham and S. Ponnusamy (eds.), World Sci., Singapore, 1991, 46-54.
  • [6] W. Ma and D. Minda, Euclidean linear invariance and uniform local convexity, J. Austral. Math. Soc. Ser. A 52 (1992), 401-418.
  • [7] W. Ma and D. Minda, Hyperbolic linear invariance and hyperbolic k-convexity, J. Austral. Math. Soc. Ser. A to appear.
  • [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.
  • [9] D. Mejia and D. Minda, Hyperbolic geometry in hyperbolically k-convex regions, Rev. Colombiana Mat. 25 (1991), 123-142.
  • [10] D. Minda, A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129-144.
  • [11] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J. Analyse Math. 49 (1987), 90-105.
  • [12] B. Osgood, Some properties of f''/f' and the Poincaré metric, Indiana Univ. Math. J. 31 (1982), 449-461.
  • [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.
  • [14] Ch. Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 (1964), 108-154.
  • [15] E. Study, Konforme Abbildung einfachzusammenhängender Bereiche, Vorlesungen über ausgewählte Gegenstände der Geometrie, Heft 2, Teubner, Leipzig und Berlin, 1913.
  • [16] K.-J. Wirths, Coefficient bounds for convex functions of bounded type, Proc. Amer. Math. Soc. 103 (1988), 525-530.
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Bibliografia
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