The aim of this paper is to prove that the envelope of hyperbolically convex functions is hyperbolicallyconvex function. Furthermore, we study the standard functional operations of hyperbolically convex functions and...
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The aim of this paper is to prove that the envelope of hyperbolically convex functions is hyperbolicallyconvex function. Furthermore, we study the standard functional operations of hyperbolically convex functions and introduce a class BH[a;b] of functions representable as the difference of two hyperbolically convex functions.
We describe extremal functions for the generalized Fekete-Szego functional vertical bar ta(3) + a(2)(2)vertical bar over the class of hyperbolically convex functions. We apply the Julia variational formula to reduce t...
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We describe extremal functions for the generalized Fekete-Szego functional vertical bar ta(3) + a(2)(2)vertical bar over the class of hyperbolically convex functions. We apply the Julia variational formula to reduce the problem to mappings onto hyperbolic polygons having no more than two proper sides. In general, this is the best result possible. We show that both one-sided and two-sided maps can be extremal for different sub-classes. (C) 2011 Elsevier Inc. All rights reserved.
In this article we recall our variational method, based on Julia's formula for the Hadamard variation, for hyperbolicallyconvex polygons. We use this variational method to prove a general theorem for solving extr...
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In this article we recall our variational method, based on Julia's formula for the Hadamard variation, for hyperbolicallyconvex polygons. We use this variational method to prove a general theorem for solving extremal problems for hyperbolically convex functions. Special cases of this theorem provide independent proofs for controlling growth and distortion for hyperbolically convex functions. [ABSTRACT FROM AUTHOR]
In the paper we study the class of univalent hyperbolicallyconvex (h-convex) functions in the unit disc. Two types of normalization are considered. The classKh(α) is defined by the inner normalization. Another natur...
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In the paper we study the class of univalent hyperbolicallyconvex (h-convex) functions in the unit disc. Two types of normalization are considered. The classKh(α) is defined by the inner normalization. Another natural normalization is the so-called boundary normalization. We consider allh-convexfunctionsfwith fixed distance>0 from the boundary ofto the origin. This class is denoted byH(c). The main result for both normalizations is the sharp upper estimate of. For the boundary normalization we use the extremal length method and derive some general two-point distortion theorems. For the classKh(α) we use the fact that the mapf:2/zis starlike in.
Let double-struck D sign;be the unit disk of the complex plane. A conformal map of double-struck D sign;into itself is called hyperbolicallyconvex if the non-Euclidean segment between any two points of f (double-stru...
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