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# HW9 - MATH 6322 Complex Analysis Fall 2011 HW#9(all about...

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MATH 6322, Complex Analysis Fall 2011, HW#9 (all about normal family) Due date: Wed., Nov. 30, 2011 Problem 1 . Page 204, #20, #24 on the textbook. The following consists a small project about the normal family. You are first asked to prove the Marty’s theorem. You are then asked to prove the so-called Zalcman’s theorem . The final goal is to prove a generalization of Montel’s theorem (which corresponds to the little picard theorem). Note that the original Montel’s theorem we learnt corresponds to the Liouville’s theorem (why?). Recall the Montels theorem we have learnt: a family of holomporphic functions on a region of C is a normal family (in the sense that every sequence of functions in the family admits a subsequence that converges uniformly on compact sets to an analytic function) if the family is locally bounded, that is, for every compact subset K there is a constant C K such that | f ( z ) | ≤ C K for all z K and every f in the family. Next, you are asked to prove Marty’s theorem, which gives a necessary and sufficient condition for a family of holomrphic functions on region of C to be a normal family in the extended sense that every sequence of functions in the family admits either a subsequence that converges uniformly on compact sets to an analytic function

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HW9 - MATH 6322 Complex Analysis Fall 2011 HW#9(all about...

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