Real and Complex Analysis (Rudin)

250 REAL AND COMPLEX ANALYSIS
6 SupposefE H(U), where U is the open unit disc,fis one-to-one in U, 0 =f(U), andf(z) = L c.z·.
Prove that the area of 0 is
Hint: The Jacobian offis 1 f' 12.
7 (a) Iff E H(O),f(z) #' 0 for z E 0, and - 00 < IX < 00, prove that
by proving the formula
in which t/J is twice differentiable on (0, 00) and
rp(t) = W'(t) + t/J'(t).
(b) Assume f E H(O) and ClI is a complex function with domain f(O), which has continuous
second-order partial derivatives. Prove that
Show that this specializes to the result of (a) if ClI(w) = ClI( 1 wi).
8 Suppose 0 is a region,!. E H(O) for n = 1, 2, 3, ... , u. is the real part off., {u.} converges uniformly
on compact subsets of 0, and U.(zj} converges for at least one z E O. Prove that then {f.}
converges uniformly on compact subsets of O.
9 Suppose u is a Lebesgue measurable function in a region 0, and u is locally in IJ. This means that
the integral of 1 u lover any compact subset of 0 is finite. Prove that u is harmonic if it satisfies the
following form of the mean value property:
u(a) = n: 2 II u(x, y) dx dy
D(G;,.)
whenever D(a; r) c: O.
10 Suppose 1 = [a, b] is an interval on the real axis, rp is a continuous function on I, and
1 J.b rp(t)
f(z) = --: - dt
2m. t-z
(z ti I).
Show that
lim [f(x + i£) - f(x - i£)]
.~O
(£> 0)
exists for every real x, and find it in terms of rp.
How is the result affected if we assume merely that rp E IJ? What happens then at points x at
which rp has right- and left-hand limits?
II Suppose that 1 = [a, b], 0 is a region, 1 c: 0, f is continuous in n, and f E H(O -
actually f E H(O).
Replace 1 by some other sets for which the same conclusion can be drawn.
I). Prove that
12 (Harnack's Inequalities) Suppose 0 is a region, K is a compact subset of 0, Zo E O. Prove that
there exist positive numbers IX and P (depending on zo, K, and 0) such that
for every positive harmonic function u in 0 and for all z E K.