# HYPERBOLIC CURVATURE AND K-CONVEX FUNCTIONS

• Song Tai-Sung (DEPARTMENT OF MATHEMATICS EDUCATION, PUSAN NATIONAL UNIVERSITY)
• Published : 2006.05.01

#### 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.