Authors
William Ma, David Minda
Publication date
1999/5/1
Journal
Journal of computational and applied mathematics
Volume
105
Issue
1-2
Pages
385-392
Publisher
North-Holland
Description
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.
Total citations
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Scholar articles
W Ma, D Minda - Journal of computational and applied mathematics, 1999