William Ma

Two-point distortion for univalent functions

Abstract We discuss two-point distortion inequalities for (not necessarily normalized) univalent functions f on the unit disk D . By a two-point distortion inequality we mean an upper or lower bound on the Euclidean distance |f(a)−f(b)| in terms of d D (a,b) , the hyperbolic distance between a and b, and the quantities (1−|a| 2 )|f′(a)|, (1−|b| 2 )|f′(b)| . The expression (1−|z|2)|f′(z)| measures the infinitesimal length distortion at z when f is viewed as a function from D with hyperbolic geometry to the complex plane C with Euclidean geometry. We present a brief overview of the known two-point distortion inequalities for univalent functions and obtain a new family of two-point upper bounds that refine the classical growth theorem for normalized univalent functions.

Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric

We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.

Two-Point Distortion Theorems for Strongly Close-to-Convex Functions

A holomorphic function f defined on the unit disk is called strongly close-to-convex of order α > 0 if there is a convex univalent function φ defined on such that For α e (0.1) this condition f is univalent in . By convention a convex univalent function is called close-to-convex of order 0. If f is close-to-convex of order α e [0, 1 ] and , then where and is the hyperbolic distance between a and b. Equality holds for distinct if and only if where S is a conformal automorphism of automorphism , T is a conformal automorphism of ; and .

Estimates for Conformal Metric Ratios

We present uniform and pointwise estimates for various ratios of the hyperbolic, quasihyperbolic and Möbius metrics. We determine when these ratios are constant. We exhibit numerous illustrative examples.

Euclidean Properties of Hyperbolic Polar Coordinates

In a simply connected hyperbolic region hyperbolic polar coordinates possess global Euclidean properties similar to those of hyperbolic polar coordinates about the origin in the unit disk if and only if the region is convex. for example, the Euclidean distance between travelers moving at unit hyperbolic speed along distinct hyperbolic geodesic rays emanating from a common point is increasing if and only if the region is convex. A consequence of this is that the ends of distinct hyperbolic geodesic rays in a convex region cannot be too close. Uniform local versions of these Euclidean properties of hyperbolic polar coordinates hold if and only if the hyperbolic region is uniformly perfect.