Geodesic
Hyperbolically convex functions II
Annales Polonici Mathematici, Tome 71 (1999) no. 3, pp. 273-285 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/ap-71-3-273-285
Keywords: hyperbolic convexity, two-variable characterization, Schwarzian derivative, distortion theorem

William Ma 1 ; David Minda 1

1 
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William Ma; David Minda. Hyperbolically convex functions II. Annales Polonici Mathematici, Tome 71 (1999) no. 3, pp. 273-285. doi: 10.4064/ap-71-3-273-285

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