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I am wondering if there exists any one-to-one analytic function mapping annulus to punctuated disk? i.e. if we let $D_1=\{1/2<|z|<1\}, D_2=\{0<|z|<1\}$, is there a one-to-one analytic function $f$ maps $D_1$ to $D_2$?

Roy Han
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    see http://math.stackexchange.com/questions/278995/finding-a-conformal-map-between-annuli – John M Aug 22 '13 at 12:14

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I just give an elementary answer for it.

If there exist such a kind of holomorphic mapping, you can extend it to the boundary (no singular points on the boundary). There will be a correspondence from the component of boundary of $D_{1}$ to 0, which is an another component of $D_{2}$, this correspondence is holomorphic. contradiction

yaoxiao
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