PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1999 | 71 | 3 | 273-285
Tytuł artykułu

Hyperbolically convex functions II

Treść / Zawartość
Warianty tytułu
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.
Twórcy
autor
  • School of Integrated Studies, Pennsylvania College of Technology, Williamsport, Pennsylvania 17701, U.S.A.
autor
  • 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv71z3p273bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.