Hyperbolic Curvature and k-Convex Functions

Song Tai-Sung; Song Tai-Sung

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P-ISSN3059-0604
E-ISSN3059-1309
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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: Theoretical Mathematics and Pedagogical Mathematics / Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics, (P)3059-0604; (E)3059-1309

2006, v.13 no.2, pp.151-155

Song Tai-Sung

Song, T. (2006). Hyperbolic Curvature and k-Convex Functions, 13(2), 151-155.

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Abstract

Let $\gamma$ be a $C_2$ curve in the open unit disk $\mathbb{D}. Flinn and Osgood proved that $K_{\mathbb{D}}(z,\gamma){\geq}1$ for all $z{\in}{\gamma}$ if and only if the curve ${\Large f}o{\gamma}$ is convex for every convex conformal mapping $\Large f$ of $\mathbb{D}, where $K_{\mathbb{D}}(z,\;\gamma)$ denotes the hyperbolic curvature of $\gamma$ 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

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Article Contents

Vol.13 No.2

Hyperbolic Curvature and k-Convex Functions

Abstract

Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics