• Communications of the Korean Mathematical Society
• /
• v.10 no.4
• /
• pp.867-874
• /
• 1995

Linear invariance is closely related to the concept of uniform local univalence. We give a geometric proof that a holomorphic locally univalent function defined on the open unit disk is linearly invariant if and only if it is uniformly locally univalent.

• The Pure and Applied Mathematics
• /
• v.6 no.2
• /
• pp.87-93
• /
• 1999

A holomorphic function f on D = {z : │z│ ＜ 1} is called uniformly locally univalent if there exists a positive constant $\rho$ such that f is univalent in every hyperbolic disk of hyperbolic radius $\rho$. We establish a characterization of uniformly locally univalent functions and investigate uniform local univalence of holomorphic universal covering projections.

• East Asian mathematical journal
• /
• v.17 no.2
• /
• pp.303-308
• /
• 2001

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.