MATH 440/508 - Assignment #4
Due on Monday November 22, 2010
Name —————————————–
Student number —————————
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Problem 1:
Let φ(z) be the Riemann map of a simply connected domain D onto
the open unit disk such that φ(z0 ) = 0 and φ0 (z0 ) > 0. Assume f (z) is
any analytic function of D into the closed unit disk.
(a) Prove that |f 0 (z0 )| ≤ φ0 (z0 ), with equality only when f (z) is a
constant multiple of φ(z).
(b) Prove that Re(f 0 (z0 )) ≤ φ0 (z0 ), with equality only when f (z) =
φ(z).
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Problem 2:
Let f (z) be a meromorphic function on the open unit disk D and
assume that f (z) has a continuous extension to the boundary ∂D.
Assume that |f (z)| = 1 whenever |z| = 1 and also that f (z) has only
a finite number of poles in D. Prove that f (z) is a rational function.
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Problem 3:
Let {Dm ; m ∈ N} be a decreasing sequence of simply connected domains, and suppose w0 ∈ Dm for all m. Let gm (z) be the conformal
map of the open unit disk D onto Dm , such that gm (0) = w0 and
0
(0) > 0. Show that gm (z) converges uniformly on all compact subgm
sets of the open unit disk D to a function g(z). If the distance from w0
to the boundary of Dm tends to zero, then g(z) is the constant function w0 , and otherwise, g(z) maps D conformally onto some simply
connected domain D. Describe D in terms of the Dm ’s.
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Problem 4:
Let {fn (z); n ∈ N} be a sequence of uniformly bounded analytic functions on an open set U . Assume that {fn (z)} converges pointwise on
U , prove that {fn (z)} converges uniformly on compact subsets of U .
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Problem 5:
Let U 6= C be a simply connected domain, which is symmetric with
respect to the real axis, and suppose that φ(z) be a conformal mapping
of U onto the unit disk D, with φ(z0 ) = 0 and φ0 (z0 ) > 0, for some real
number z0 ∈ U . Prove that f (z) = f (z) for all z ∈ U .