Hyperbolically convex functions

• Autorzy: Wancang Ma , David Minda
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

hyperbolic convexity; distortion theorem; growth thoerem; linear invariance

Kategorie tematyczne

• 30C50: Coefficient problems for univalent and multivalent functions
• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
• 30C25: Covering theorems in conformal mapping theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1994-1995
• Tom: 60
• Numer: 1
• Strony: 81-100

Daty

wydano

1994

otrzymano

1993-07-29

Twórcy

autor

Wancang Ma

• Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.

autor

David Minda

• Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.

Bibliografia

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• [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.
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• [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.
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