The Schwarzian derivative and conformally natural quasiconformal extensions from one to two to three dimensions
نویسندگان
چکیده
If2t is replaced by 2 in (1.1) then one obtains Nehari's original injectivity criterion. Nehari's result started what is by now a considerable amount of work in this area, see, e.g. ILl . Each of the present authors has been interested in theorems of this type, expecially in generalizations to higher dimensions, lOS1,2, C1,2]. What about one dimension? For a smooth real valued function on an interval, the Schwarzian can again be defined by (1.2), but there is no question of proving an injectivity criterion since one must already assume that f ' 4= 0 to define Sf. Remarkably, what does persist is the phenomenon of quasiconformal extension. We shall prove the following.
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