The Schwarzian derivative of a locally injective holomorphic function f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} is Sf=f′′′/f′-(3/2)f′′/f′2\documentclass[12pt]{minimal}
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\begin{document}$$S_f= f'''/f' - (3/2)\left( f''/f'\right) ^2$$\end{document}. As is well-known, Sf=0\documentclass[12pt]{minimal}
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\begin{document}$$S_f=0$$\end{document} if and only if f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} is a Möbius transformation. Intuitively, if a locally injective holomorphic function has a small Schwarzian derivative, then it should behave roughly like a Möbius transformation. Two quantitative results of this type are established. First, if |Sf(z)|≤2t,z∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$|S_f(z)|\le 2t, z \in \Omega $$\end{document}, on a convex region Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}, then sharp upper and lower two-point distortion bounds on |f(a)-f(b)|\documentclass[12pt]{minimal}
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\begin{document}$$|f(a)-f(b)|$$\end{document} for a,b∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$a,b \in \Omega $$\end{document} are given. The upper bound is valid for all a,b∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$a,b \in \Omega $$\end{document} while the lower bound is valid for |a-b|<π/t\documentclass[12pt]{minimal}
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\begin{document}$$|a-b| < \pi /\sqrt{t}$$\end{document}. For t=0\documentclass[12pt]{minimal}
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\begin{document}$$t=0$$\end{document} the bounds are the familiar identity |f(a)-f(b)|=|a-b||f′(a)||f′(b)|\documentclass[12pt]{minimal}
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\begin{document}$$\vert f(a)-f(b) \vert = \vert a-b\vert \sqrt{|f'(a)||f'(b)|}$$\end{document} for Möbius transformations. These upper and lower two-point distortion theorems characterize locally injective holomorphic functions with bounded Schwarzian derivative. Second, if Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is a convex region with diameter D\documentclass[12pt]{minimal}
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\begin{document}$$D$$\end{document} and |Sf(z)|≤2t<π2/D2\documentclass[12pt]{minimal}
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\begin{document}$$|S_f(z)| \le 2t<\pi ^2/D^2$$\end{document} for z∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$z\in \Omega $$\end{document}, then f\documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document} is Kt(D)\documentclass[12pt]{minimal}
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\begin{document}$$K_t(D)$$\end{document}-quasi-Möbius, where the constant depends only on t\documentclass[12pt]{minimal}
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\begin{document}$$t$$\end{document} and D\documentclass[12pt]{minimal}
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\begin{document}$$D$$\end{document}. This means that 1/Kt(D)≤|f(a),f(b),f(c),f(d)|/|a,b,c,d|≤Kt(D)\documentclass[12pt]{minimal}
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\begin{document}$$1/K_t(D) \le |f(a),f(b),f(c),f(d)|/|a,b,c,d|\le K_t(D)$$\end{document} for all distinct a,b,c,d∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$a,b,c,d\in \Omega $$\end{document}, where |a,b,c,d|\documentclass[12pt]{minimal}
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\begin{document}$$|a,b,c,d|$$\end{document} denotes the absolute cross-ratio.
