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2004 | 84 | 3 | 185-202
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Hyperbolically 1-convex functions

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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.
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  • School of Integrated Studies, Pennsylvania College of Technology, Williamsport, PA 17701, U.S.A.
autor
  • Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A.
autor
  • Departamento de Matemáticas, Universidad Nacional de Colombia, A.A. 3840, Medellín, Colombia
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bwmeta1.element.bwnjournal-article-doi-10_4064-ap84-3-1
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