Hyperbolically convex functions II

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

Abstrakty

EN

Unlike those for euclidean convex functions, the known characterizations for hyperbolically convex functions usually contain terms that are not holomorphic. This makes hyperbolically convex functions much harder to investigate. We give a geometric proof of a two-variable characterization obtained by Mejia and Pommerenke. This characterization involves a function of two variables which is holomorphic in one of the two variables. Various applications of the two-variable characterization result in a number of analogies with the classical theory of euclidean convex functions. In particular, we obtain a uniform upper bound on the Schwarzian derivative. We also obtain the sharp lower bound on |f'(z)| for all z in the unit disk, and the sharp upper bound on |f'(z)| when |z| ≤ √2 - 1.

Słowa kluczowe

EN

hyperbolic convexity; two-variable characterization; Schwarzian derivative; distortion theorem

Kategorie tematyczne

• 30C50: Coefficient problems for univalent and multivalent functions
• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1999
• Tom: 71
• Numer: 3
• Strony: 273-285

Daty

wydano

1999

otrzymano

1998-11-03

Twórcy

autor

William Ma

• School of Integrated Studies, Pennsylvania College of Technology, Williamsport, Pennsylvania 17701, U.S.A.

autor

David Minda

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

Bibliografia

• [1] L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979.
• [2] R. Fournier, J. Ma and S. Ruscheweyh, Convex univalent functions and omitted values, preprint.
• [3] W. Ma and D. Minda, Hyperbolically convex functions, Ann. Polon. Math. 60 (1994), 81-100.
• [4] W. Ma and D. Minda, Hyperbolic linear invariance and hyperbolic k-convexity, J. Austral. Math. Soc. Ser. A 58 (1995), 73-93.
• [5] D. Mejia, Ch. Pommerenke and A. Vasil'ev, Distortion theorems for hyperbolically convex functions, preprint.
• [6] D. Mejia and Ch. Pommerenke, On hyperbolically convex functions, J. Geom. Anal., to appear.
• [7] S. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montréal, Montréal, 1982.
• [8] T. Sheil-Small, On convex univalent functions, J. London Math. Soc. 1 (1969), 483-492.
• [9] T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775-777.
• [10] S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266-267.