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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 convex functions

100%

Ma W., Minda D.

Annales Polonici Mathematici

1994-1995

tom 60

nr 1

81-100

We investigate univalent holomorphic functions f defined on the unit disk 𝔻 such that f(𝔻) is a hyperbolically convex subset of 𝔻; there are a number of analogies with the classical theory of (euclidean) convex univalent functions. A subregion Ω of 𝔻 is called hyperbolically convex (relative to hyperbolic geometry on 𝔻) if for all points a,b in Ω the arc of the hyperbolic geodesic in 𝔻 connecting a and b (the arc of the circle joining a and b which is orthogonal to the unit circle) lies in Ω. We give several analytic characterizations of hyperbolically convex functions. These analytic results lead to a number of sharp consequences, including covering, growth and distortion theorems and the precise upper bound on |f''(0)| for normalized (f(0) = 0 and f'(0) > 0) hyperbolically convex functions. In addition, we find the radius of hyperbolic convexity for normalized univalent functions mapping 𝔻 into itself. Finally, we suggest an alternate definition of "hyperbolic linear invariance" for locally univalent functions f: 𝔻 → 𝔻 that parallels earlier definitions of euclidean and spherical linear invariance.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Ma W.

2 Minda D.

1 1999

1 1994

Wyniki wyszukiwania

Hyperbolically convex functions II

Hyperbolically convex functions