I'm not sure how to do this problem using the hints. The only thing I understand is why we can assume z0=0. Other than that, I'm stuck. 9. Let be a bounded open subset of C, and 4:Prove that if there exists a point zo EN such that2 a holomorphic function.4(20) = 20 and ' (zo) = 1then is linear.[Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near0, and prove that if yk = o...o (where y appears k times), then k(z) =z+kanzn +0(z+¹). Apply the Cauchy inequalities and let k → ∞ to concludethe proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.]

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9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a holomorphic function. 4(zo) = 20 and 4' (zo) = 1 then is linear. [Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near 0, and prove that if k = o...o (where y appears k times), then k(z) = z+kanzn +0(z+¹). Apply the Cauchy inequalities and let k → ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that f(z)| ≤ C|g(z)| for some constant C as [2] → 0.]

9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a holomorphic function. 4(zo) = 20 and 4' (zo) = 1 then is linear. [Hint: Why can one assume that zo = 0? Write y(z)=z+anz" +O(z+¹) near 0, and prove that if k = o...o (where y appears k times), then k(z) = z+kanzn +0(z+¹). Apply the Cauchy inequalities and let k → ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that f(z)| ≤ C|g(z)| for some constant C as [2] → 0.]

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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