課程名稱︰複分析導論
課程性質︰數學系大三必修
課程教師︰蔡宜洵
開課學院：理學院
開課系所︰數學系
考試日期︰2020年12月03日(四)
考試時限：15:10-15:40，共計30分鐘
試題 :
Complex Analysis
Quiz 3
Name: ID:
1. (15 points) Let a1,...,an be points on ∂D(0,1). Prove that there exists a
point z on ∂D(0,1) such that the product of the distances from z to the aj
is at least 1. (Hint: By maximum modulus principle.)
2. (15 points) Let f(z)∈Hol(D(0,1)), f(0) = 1, and Re(f(z)) ≧ 0 for
all z∈D(0,1). Show that
1 - |z| 1 + |z|
--------- ≦ |f(z)| ≦ ---------, ∀z∈D(0,1).
1 + |z| 1 - |z|
(Extra 5 points) Furthermore, prove that
|f'(0)| ≦ 2
w - 1
(Hint: Consider the function T。f, where T = -------, and applying Schwzrs'
w + 1
Lemma.)
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