Hyperbolically 1-convex functions

Volume 84 / 2004

William Ma, David Minda, Diego Mejia Annales Polonici Mathematici 84 (2004), 185-202 MSC: Primary 30C45, 30C50. DOI: 10.4064/ap84-3-1


There are two reasonable analogs of Euclidean convexity in hyperbolic geometry on the unit disk ${{\mathbb D}}$. 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 ${{\mathbb D}}$ 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 ${{\mathbb D}}$ with $f({{\mathbb D}})\subseteq {{\mathbb D}}$ is hyperbolically 1-convex if and only if $f/(1-wf)$ is a Euclidean convex function for each $w \in \overline {{{\mathbb D}}}$. 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.


  • William MaSchool of Integrated Studies
    Pennsylvania College of Technology
    Williamsport, PA 17701, U.S.A.
  • David MindaDepartment of Mathematical Sciences
    University of Cincinnati
    Cincinnati, OH 45221-0025, U.S.A.
  • Diego MejiaDepartamento de Matemáticas
    Universidad Nacional de Colombia
    A.A. 3840, Medellín, Colombia

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