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  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

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Vol.13 No.2

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Hyperbolic Curvature and k-Convex Functions

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2006, v.13 no.2, pp.151-155
Song Tai-Sung
Song, T. (2006). Hyperbolic Curvature and k-Convex Functions. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 13(2), 151-155.

Abstract

Let <TEX>$\gamma$</TEX> be a <TEX>$C_2$</TEX> curve in the open unit disk <TEX>$\mathbb{D}</TEX>. Flinn and Osgood proved that <TEX>$K_{\mathbb{D}}(z,\gamma){\geq}1$</TEX> for all <TEX>$z{\in}{\gamma}$</TEX> if and only if the curve <TEX>${\Large f}o{\gamma}$</TEX> is convex for every convex conformal mapping <TEX>$\Large f$</TEX> of <TEX>$\mathbb{D}</TEX>, where <TEX>$K_{\mathbb{D}}(z,\;\gamma)$</TEX> denotes the hyperbolic curvature of <TEX>$\gamma$</TEX> at the point z. In this paper we establish a generalization of the Flinn-Osgood characterization for a curve with the hyperbolic curvature at least 1.

keywords
hyperbolic metric, hyperbolic curvature, k-convex region

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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