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Unformatted text preview: Key to Homework 9, Thansk to Da Zheng for providing the texfile December 7, 2011 1. Page 204, #20 Let { f α } be a normal family of holomorphic functions on a domain U . Prove that { f α } is a normal family. Proof. Pick any sequence { f n } j { f α } , then by the given condtion, for the sequence { f n } , there is a subsequence which converges uniformly on compact sets of U , say { f n k } . By Corollary 3.5.2, { f n k } converges uniformly on compact subset of U . Hence, for any sequence of { f n } , we can find a subsequence which converges uniformly on compact subsets, which implies that { f α } is a normal family by definition. 2 2. Page 205, #24 Let Ω j C be a bounded domain and let { f j } be a sequence of holomorphic functions on Ω. Assume that Z Ω  f j ( z )  2 dxdy < C < ∞ where C does not depend on j . Prove that { f j } is a normal family. Proof. First we note that, as the hint given in the problem, you can use the hint in Problem 8, chapter 4 (Page 146), which you need to show that, if f is holomorphic on D ( Q, ), then  F ( Q )  2 ≤ 1 π 2 Z D ( Q, )  F ( z )  2 dxdy. We will follow the hint given on Page 146 in proving the above. Below is the proof of problem #24 1 Solution : We will show that { f j } is unformly bounded on every compact sub sets of Ω, then the Montel theorem in the book will imply that { f j } is nor mal on Ω. To do so, let K ⊂ Ω be compact. By the Lebesgue number lemma (see Munkers: Topology P. 175176, or a better proof can be found at http://mathblather.blogspot.com/2011/07/lebesguenumberlemmaandcorollary.html), there exists r K > 0 such that for each z ∈ K , D ( z,r K )) ⊂ Ω. Now fix Q ∈ K . by the Cauchy integral formula, for every 0 ≤ r ≤ r K , f 2 j ( Q ) = 1 2 πi I ∂D ( Q,r ) f 2 j ( ζ ) ζ Q dζ So, if we parameterize ∂D ( Q,r ) as re iθ , where θ ∈ [0 , 2 π ].  f j ( Q )  2 = 1 2 πi I ∂D ( Q,r ) f 2 j ( ζ ) ζ Q dζ ≤ 1 2 π I ∂D ( Q,r ) f 2 j ( ζ ) ζ Q  dζ  = 1 2 π Z 2 π  f j ( Q + re iθ )  2 r rdθ = 1 2 π Z 2 π  f j ( Q + re iθ )  2 dθ Now, use the Fubini theorem (and the polar coordinates) as well as the above inequality, Z D ( Q,r K )  f j ( x,y )  2 dxdy = Z r K Z 2 π  f j ( Q + re iθ )  2 rdθdr ≥ 2 π Z r K  f j ( Q )  2 rdr ≥ 2 π r 2 K 2  f j ( Q )  2 . Thus, by the assumption, C > Z Ω  f j ( z )  2 dxdy ≥ Z D ( Q,r K )  f j ( x,y )  2 dxdy = πr 2 K  f j ( Q )  2 . Hence, for every Q ∈ K ,  f j ( Q )  2 ≤ C πr 2 K which proves our claim. 2 3.( Marty’s Theorem ) 2 Let F be a family of holomorphic functions on a region U on C . Prove that F is normal (in the general sense) if and only if for every compact subset K of U , there is a constant C K such that f # ( z ) ≤ C K for all Z ∈ K and f ∈ F , where f # ( z ) :=  f ( z )  1 +  f ( z )  2 Proof....
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