9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a point zo EN such that (20) = 20 and y' (zo) = 1 then 4 is linear. [Hint: Why can one assume that zo = 0? Write y(z) = z+anz" +O(zn+¹) near 0, and prove that if k = o... (where appears k times), then Ök(z) = z+kanz" +0(zn+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 |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.]
9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a point zo EN such that (20) = 20 and y' (zo) = 1 then 4 is linear. [Hint: Why can one assume that zo = 0? Write y(z) = z+anz" +O(zn+¹) near 0, and prove that if k = o... (where appears k times), then Ök(z) = z+kanz" +0(zn+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 |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 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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