Non Constant - Complex Analysis - Exam, Exams of Mathematics

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Math 246C, Complex Analysis

Spring 2012

Final Exam

Name:

Problem 2: Let Ω ⊆ C be a region, and h : Ω → R be a harmonic function. Show that if h vanishes on a set of positive measure M in Ω, then h ≡ 0. Hint: One way to prove this is to consider the gradient ∇h of h. (12 pts)

Problem 3: If f is a non-constant holomorphic function on D, we denote by Lf ∈ (0, ∞] the radius of the “largest” disk contained in the image f (D); more precisely,

Lf := sup{r > 0 : there ex. z 0 ∈ C with B(z 0 , r) ⊆ f (D)}.

The number L := inf{Lf : f ∈ H(D) and f ′(0) = 1}

is known as Landau’s constant. Its precise numerical value is not known. The purpose of this problem is to show that L > 0 by establishing an explicit positive lower bound for L.

(a) Let B be the family of all functions f ∈ H(D) satisfying f (0) = 0, f ′(0) = 1, and |f ′(z)| ≤

1 − |z|^2

for z ∈ D.

Show that L = inf{Lf : f ∈ B}. Hint: First show that in the definition of L we may assume that the functions f are holomorphic in an open set containing D. For such a function consider a point z 0 ∈ D, where |f ′(z)|(1 − |z|^2 ) attains a maximum on D, and precompose f by a suitable map that sends 0 to z 0. (4 pts) (b) Let g : D → C be a holomorphic function with g(0) = w 0 ≥ 0 and |g(z)| ≤ 1 for z ∈ D. Show that

Re(g(z)) ≥

w 0 − |z| 1 − w 0 |z| for all z ∈ D with |z| ≤ w 0. (4 pts) (c) Find an explicit number c > 0 such that B(0, c) ⊆ f (D) for all f ∈ B. (4 pts) (d) Find an explicit positive lower bound for L. (1 pt)