Sharp Distortion Theorems Associated with the Schwarzian Derivative

These are well known and important theorems of Nehari, Ahlfors and Weill, and Krauss. We refer to Lehto's book [8] for a discussion of these results and for the properties of the Schwarzian that we shall need. The constants 2 in (1.1) and 6 in (1.3) are sharp. An example for the latter is the Koebe function k(z) = z(l — z)~ which has Schwarzian Sk(z) = — 6/(1 —z). We also remark that the class of univalent functions satisfying (1.1) is quite large; for instance, it contains the class of convex mappings. Since S(Mof) = SfTor any Mobius transformation M, the inequalities above are independent of any such normalization M o / o f / and this is an interesting feature of these results. However, if we require the normalization /(0) = 0, /'(0) = 1, and /"(0) = 0 then we can obtain, rather simply, sharp and explicit upper and lower bounds on | / | and | / ' | for functions which satisfy (1.1) or (1.2). We introduce the following functions. Let