Weak Schwarzians, Bounded Hyperbolic Distortion, and Smooth Quasisymmetric Functions M. Chuaqui University of Pennsylvania B. Osgood Stanford University and IHES 1 Introduction In this paper we wish to 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 increas- ing homeomorphism, where I is an open interval that may be the whole real line. The quasisymmetry quotient of f is kf (x, h)= f (x + h) − f (x) f (x) − f (x − h) (1.1) for x, x + h, x − h ∈ I . The function is called quasisymmetric if kf (x, h) is bounded below away from zero and above away from ∞. 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 k ≤ kf (x, h) ≤ k. A similarity is 1-quasisymmetric, and the functions f and g = af + b, a, b ∈ R, have kf = kg. When f is monotonic and three times differentiable its Schwarzian derivative is Sf =  f ′′ f ′  ′ − 1 2  f ′′ f ′  2 = f ′′′ f ′ − 3 2  f ′′ f ′  2 . (1.2) 1