Hyperbolically convex functions II

William Ma; David Minda

Annales Polonici Mathematici (1999)

  • Volume: 71, Issue: 3, page 273-285
  • ISSN: 0066-2216

Abstract

top
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.

How to cite

top

William Ma, and David Minda. "Hyperbolically convex functions II." Annales Polonici Mathematici 71.3 (1999): 273-285. <http://eudml.org/doc/262813>.

@article{WilliamMa1999,
abstract = {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.},
author = {William Ma, David Minda},
journal = {Annales Polonici Mathematici},
keywords = {hyperbolic convexity; two-variable characterization; Schwarzian derivative; distortion theorem; hyperbolically convex},
language = {eng},
number = {3},
pages = {273-285},
title = {Hyperbolically convex functions II},
url = {http://eudml.org/doc/262813},
volume = {71},
year = {1999},
}

TY - JOUR
AU - William Ma
AU - David Minda
TI - Hyperbolically convex functions II
JO - Annales Polonici Mathematici
PY - 1999
VL - 71
IS - 3
SP - 273
EP - 285
AB - 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.
LA - eng
KW - hyperbolic convexity; two-variable characterization; Schwarzian derivative; distortion theorem; hyperbolically convex
UR - http://eudml.org/doc/262813
ER -

References

top
  1. [1] L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979. Zbl0395.30001
  2. [2] R. Fournier, J. Ma and S. Ruscheweyh, Convex univalent functions and omitted values, preprint. Zbl0912.30007
  3. [3] W. Ma and D. Minda, Hyperbolically convex functions, Ann. Polon. Math. 60 (1994), 81-100. Zbl0818.30010
  4. [4] W. Ma and D. Minda, Hyperbolic linear invariance and hyperbolic k-convexity, J. Austral. Math. Soc. Ser. A 58 (1995), 73-93. Zbl0838.30022
  5. [5] D. Mejia, Ch. Pommerenke and A. Vasil'ev, Distortion theorems for hyperbolically convex functions, preprint. 
  6. [6] D. Mejia and Ch. Pommerenke, On hyperbolically convex functions, J. Geom. Anal., to appear. Zbl1021.30009
  7. [7] S. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montréal, Montréal, 1982. 
  8. [8] T. Sheil-Small, On convex univalent functions, J. London Math. Soc. 1 (1969), 483-492. Zbl0201.40802
  9. [9] T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775-777. Zbl0206.36202
  10. [10] S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266-267. Zbl0283.30014

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.