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Hyperbolically convex functions II

100%

Ma W., Minda D.

Annales Polonici Mathematici

1999

tom 71

nr 3

273-285

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.

Hyperbolically 1-convex functions

81%

Ma W., Minda D., Mejia D.

Annales Polonici Mathematici

2004

tom 84

nr 3

185-202

There are two reasonable analogs of Euclidean convexity in hyperbolic geometry on the unit disk 𝔻. One is hyperbolic convexity and the other is hyperbolic 1-convexity. Associated with each type of convexity is the family of univalent holomorphic maps of 𝔻 onto subregions of the unit disk that are hyperbolically convex or hyperbolically 1-convex. The class of hyperbolically convex functions has been the subject of a number of investigations, while the family of hyperbolically 1-convex functions has received less attention. This paper is a contribution to the study of hyperbolically 1-convex functions. A main result is that a holomorphic univalent function f defined on 𝔻 with f(𝔻) ⊆ 𝔻 is hyperbolically 1-convex if and only if f/(1-wf) is a Euclidean convex function for each w ∈ 𝔻̅. This characterization gives rise to two-variable characterizations of hyperbolically 1-convex functions. These two-variable characterizations yield a number of sharp results for hyperbolically 1-convex functions. In addition, we derive sharp two-point distortion theorems for hyperbolically 1-convex functions.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Ma W.

2 Minda D.

1 Mejia D.

1 2004

1 1999

Wyniki wyszukiwania

Hyperbolically convex functions II

Hyperbolically 1-convex functions