7 (DUE SEPTEMBER 13TH) 1) a) Suppose D is a connected domain in C and B (z0 ) = {|z z0 | } is a subset in D. Let u be any smooth real-valued function. Referring back to proof of the Mean Value Property theorem of harmonic function, show that for any 0 < r < , we have the equality \u0012Z 2 \u0013 ZZ i u(z0 + re )d . u dxdy = r r 0 Br (z0 ) (You can assume that you can interchange /r with integration) b) Show that if u 0, then \u0012Z 2 \u0013 1 u(z0 ) u(z0 + rei )d . 2 0 c) Show that for u which satisfies u 0, the Maximal Principle also holds, i.e. if z0 is in the interior of D such that u(z0 ) assumes the maximal value of u in D, then u must be a constant function. 2) Show that if u is real-valued harmonic function in D, and f : R R is a convex function, i.e. f 00 (t) > 0 for all t R, then the function u (z) = f (u(z)) satisfies u 0. 3) Gamelin III.4 problem 1 4) Gamelin III.5, problem 3 5) Gamelin III.5, problem 4 6) Gamelin IV.2, problem 1 7) Gamelin IV.3, problem 4. 1