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The Schwarzian derivative of a locally injective holomorphic function $$f$$ f is $$S_f= f'''/f' - (3/2)\left( f''/f'\right) ^2$$ S f = f ′ ′ ′ / f ′ - ( 3 / 2 ) f ′ ′ / f ′ 2 . As is well-known, $$S_f=0$$ S f = 0 if and only if $$f$$ f 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 $$|S_f(z)|\le 2t, z \in \Omega $$ | S f ( z ) | ≤ 2 t , z ∈ Ω , on a convex region $$\Omega $$ Ω , then sharp upper and lower two-point distortion bounds on $$|f(a)-f(b)|$$ | f ( a ) - f ( b ) | for $$a,b \in \Omega $$ a , b ∈ Ω are given. The upper bound is valid for all $$a,b \in \Omega $$ a , b ∈ Ω while the lower bound is valid for $$|a-b| < \pi /\sqrt{t}$$ | a - b | < π / t . For $$t=0$$ t = 0 the bounds are the familiar identity $$\vert f(a)-f(b) \vert = \vert a-b\vert \sqrt{|f'(a)||f'(b)|}$$ | f ( a ) - f ( b ) | = | a - b | | f ′ ( a ) | | f ′ ( b ) | for Möbius transformations. These upper and lower two-point distortion theorems characterize locally injective holomorphic functions with bounded Schwarzian derivative. Second, if $$\Omega $$ Ω is a convex region with diameter $$D$$ D and $$|S_f(z)| \le 2t<\pi ^2/D^2$$ | S f ( z ) | ≤ 2 t < π 2 / D 2 for $$z\in \Omega $$ z ∈ Ω , then $$f$$ f is $$K_t(D)$$ K t ( D ) -quasi-Möbius, where the constant depends only on $$t$$ t and $$D$$ D . This means that $$1/K_t(D) \le |f(a),f(b),f(c),f(d)|/|a,b,c,d|\le K_t(D)$$ 1 / K t ( D ) ≤ | f ( a ) , f ( b ) , f ( c ) , f ( d ) | / | a , b , c , d | ≤ K t ( D ) for all distinct $$a,b,c,d\in \Omega $$ a , b , c , d ∈ Ω , where $$|a,b,c,d|$$ | a , b , c , d | denotes the absolute cross-ratio.
Computational Methods and Function Theory – Springer Journals
Published: Nov 21, 2013
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