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MATH 440/508 - Assignment #4 Due on Monday November 22, 2010 Name —————————————– Student number ————————— 1 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). 2 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. 3 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. 4 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 . 5 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 .