MONOTONICITY OF EUCLIDEAN CURVATURE UNDER LOCALLY UNIVALENT FUNCTIONS

  • Song, Tai-Sung (Department of Mathematics Education Pusan National University)
  • Published : 2001.12.30

Abstract

Let K($z,\gamma$) denote the euclidean curvature of the curve $\gamma$ at the point z. Flinn and Osgood proved that if f is a univalent mapping of the open unit disk D={z:|z|<1} into itself with f(0)=0 and |f'(0)|<1, then $K(0,\gamma){\leq}K(0,f\;o\;\gamma)$ for any $C^2$ curve $\gamma$ on D through the origin with $K(0,\gamma){\geq}4$. In this paper we establish a generalization of the Flinn-Osgood Monotonicity Theorem.

Keywords