Two-point distortion for univalent functions

✍ Scribed by William Ma; David Minda

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 84 KB
• Volume: 105
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00021-7

✦ Synopsis

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 inÿnitesimal 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 reÿne the classical growth theorem for normalized univalent functions.