582 Chapters
11. If f is a nonconstant entire function, show that there exists an arc ""(:
z = z(t), 0:::; t < oo, along which f(z) 4 oo i.e., such that lf(z(t))i > n
t ;::: tn, for each positive integer n.- Given the harmonic function u = x^3 - 3xy^2 on the disk lzl :::; 1, find the
points of the disk where u assumes a maximum or a minimum value,
and compute those values. - Given the harmonic function u = ex cosy on the rectangle with vertices
at (0, 0), (2, 0), (2, 71" ), and (0, 71" ), find the points of the rectangle where
u assumes a maximum or a minimum value, and compute those values.
14. (a) If u( x, y) is a real harmonic function in IR^2 and has a finite limit as
(x, y) -----+ oo, show that u is a constant function.(b) If u(x,y) is a real harmonic function in IR^2 and u(x,y);::: 0 for all
( x, y) E IR^2 , show that u is a constant function.- Suppose that u(x,y) is harmonic in a simply connected region G, and
let Go be a proper subregion of G. Prove the following. (a) If u(x,y) =
0 for all (x,y) E Go, then u(x,y) = 0 for all (x,y) E G. (b) If U(x,y)
is also harmonic in G and if u(x,y) = U(x,y) for all (x,y) E Go, then
u(x,y) = U(x,y) for all (x,y) E G.
8.13 Schwarz's Lemma
The following simple result (also depending on the maximum modulus prin-
ciple) is due to H. A. Schwarz [34], Vol. 2, p. 110. Its importance in complex
analysis, especially in the theory of conformal representation, will come to
light later.Theorem 8.37 (Schwarz's Lemma). Suppose that:
1. f is analytic in the open disk lzl < 1
2. lf(z)I :::; 1 for lzl < 1
- f(O) = 0
Then we have the estimates
lf(z)l 5 lzl for lzl < 1 and if'(O)i 5 1 (8.13-1)
The equality occurs iff f(z) = az, where a is a constant such that lal = 1.
Proof Since J(O) = 0, by the Cauchy-Taylor expansion we havef(z) = f'(O)z + f'~~O) z^2 + · · ·
valid for lzl < 1. If we define
g(z) = f'(O) + f'~~O) z + · · ·