# Hyperbolically convex functions II

Annales Polonici Mathematici (1999)

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

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topWilliam 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

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- [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] 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. Zbl1021.30009
- [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. Zbl0201.40802
- [9] T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775-777. Zbl0206.36202
- [10] S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266-267. Zbl0283.30014

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