MATH 113 Midterm: MATH 113 Harvard 113 Fall 01113hw9
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Throughout this assignment, will denote the open unit disk {|z| < 1}: a. [hint: use part (a). : the horizontal strip { 1 < im(z) < 1}. [hint: use part (a) and the exponential map. : the vertical strip { 1 < re(z) < 1}. [hint: use part (c). : suppose f is an analytic function on with f (0) = 0 and re(f (z)) < 1 for all z . Show that |f (0)| 2 and that |f (z)| 2|z| [hint: consider the composition of f and the mapping from exercise 2, part (b). : prove that every conformal automorphism of the punctured disk = {0 < |z| < 1} is a rotation. [hint: use riemann"s removable singularity theorem. : prove pick"s lemma: if f : is analytic, then. 1 |z|2 for all z , with equality i f is a conformal automorphism of .
