Two-point distortion for univalent functions

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 dD(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.