Abstract
A conformal map f of the unit disk D of the complex plane into itself is called hyperbolically convex if the hyperbolic segment between any two points of f (D) also lies in f (D). These functions form a non-linear space invariant under Moebius transformations of D onto itself. The fact that this space is non-linear makes it impossible to use many of the standard methods.
This survey talk will concentrate on
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Analytic characterizations of h-convex functions
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Inequalities for h-convex functions
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Hausdorff dimension of image sets
A few proofs will be sketched.
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Mejía, D., Pommerenke, C. (2003). Hyperbolically Convex Functions. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_6
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DOI: https://doi.org/10.1007/978-1-4757-3741-7_6
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