MATH 304 Final, Fall 2003

tthe eR Y OF BEIRUT AMERICAN UNIVERSITY OF BEIRUT The Department of Mathematics Math 304 Final Examination, January 26, 2004 1. Consider the annular region A = A(Ri,R2) = {z € C:: Ry < |z| < Ro} where 0 < R; < R2. Define p on A by p(z) = aoe Suppose that /: A + A is holomorphic and c € Aisa fixed point of f i.e. f(c) = c. By formulating and proving, an analogue of the curvature version of Schwarz’s lemma for (A, p), show that jf'(c)i < 1. 2LetQ= {2=x+yix>0,y > Op and U= {z: [Zz < 1}. Iff: Q + Vis holomorphic, how large can |f (e'**)! be? Find the extremal functions. 3. 1f¢ : R > Ris continuous and bounded, and Q = {z =x+ip : x > 0,y > 0} show how to define a function u harmonic in 2 and equal to @ on the boundary. i.e. limxo,u(x,y) = o(y), for each y > 0, and limyo, u(x,y) = o(x), for each x > 0. Justify all your assertions. 4. Let flz) = <2 ~ Oe GF sin?nz ) (a) Prove that fis an entire function. (b) Prove that fis bounded on the interval {z =x: - x (c) Prove that fis bounded on {z = iy : -c0 < y < 0}, the y-axis. (d) Prove that fis bounded on C and that in fact f= 0. < +} of the real axis. 5. Prove that — lan pve a _ ett a-bye“?a] 1+ se}. e ( ) *Li 4n272 } 6. Let £ be a compact set in a region Q of the complex plane, and u a positive harmonic function in Q. Prove that there exists a constant VM, depending only on £ and Q, such that u(z2) < Mu(z1) for any two points 21,22 € E.