VC-EAK SCH'WARZIANS, BOUNDED HYPERBOLIC D ISTORTION, AND SMOOTH QUASISYMMETRIC FUNCTIONS

In this paper we show how some techniques based on a Sturm comparison theorem for the differential equation associated with the Schwarzian derivative can be used to study two problems. First, to estimate the quasisymmetry quotient of a function in terms of bounds on its Schwarzian. Here, the bounds on the Schwarzian are much like those one finds in the theory of univalent functions, and the result is a sufficient condition for a function to be quasisymmetric. This is discussed in Section 3. Second, to study how much mappings of an interval distort distances in the hyperbolic metric. These results are Schwarz-Pick type lemmas and are discussed in Section 4. Apart from the differential equations arguments there are interesting issues having to do with smoothness. In Section 5 we combine the estimates for hyperbolic distances with those for quasisymmetry quotients to obtain a result expressing a quasisymmetric function of the type we have been considering as a composition of functions whose quasisymmetry quotients are arbitrarily close to 1. Finally, in Section 6 we construct some examples to show that there is no obvious necessary condition for a function to be quasisymmetric corresponding to the sufficient conditions in Section 3. We work with real valued functions of a real variable. Let f: I-- * R be an increasing homeomorphism, where I is an open interval that may be the whole real line. The quasisymmetry quotient of f is (1.1) kf(x, h) = f (x + h)- f (x) f(x)- f (x- h) for x, x + h, x- h E I. The function is called quasisymmetric if kf(x, h) is bounded below away from zero and above away from oo. Because of kf(x,-h) = kf(x, h)-1 we may assume that h> 0 for this definition. One says that f is k-quasisymmetric, k> 1, if 1 < _ kf(x, h) < _ k