Sequences, Series, and Special Functions 581min_[u(x,y)-U(x,y)] = min [u(x,y)-U(x,y)] = 0
(x,y)EG (x,y)E8Gso that u(x,y) = U(x,y) for all (x,y) E G.
Corollary 8.17 If u(x, y) is harmonic in a region G, continuous on 8G,
and zero on 8G, then u(x,y) is identically zero in G.Exercises 8.4
1. Given J(z) = z^2 + 5, find max lf(z)I and min lf(z)I on lzl:::; 2.
2. Given f(z) = (z+3)^2 , find max lf(z)I and min lf(z)i on the triangular
closed region with vertices at z = 0, z = 1, and z = -i.
- Given f(z) = l/(z + 4), find max lf(z)I and min lf(z)I on 1:::; lzl :::; 2.
4. Let f be analytic in a simply connected region containing the circles
lzl = r and lzl = R (0 < r < R), and let M(R) = maxlzl=R lf(z)I,M'(r) = maxlzl=r lf'(z)I. Show that
M'(r):::; M(R)
R-r5. Let f be analytic on a closed disk D and suppose it attains its maximum
modulus at a point Zo E an. Prove that f'(zo) f 0 unless f is a
constant.6. Let f(z) = (1 - z)/(l - zn) for z f 1 and f(l) = l/n (n = 1, 2, 3, ... ).
Show that lf(z)I :::; 1 on the disk iz -^1 / 21 :::; %· [P. J. O'Hara, Amer.
Math. Monthly, 74 (1967), 603]7. Let f be a nonconstant analytic function in a region G and suppose
that C* = {z: lf(z)I = k, k > 0 a constant} is the graph of a simple
closed curve such that C* U Int C* C G. Prove that there is at least
one zero of f(z) in Int C*.8. Suppose that f is a nonconstant analytic function on the disk lzl :::; r
and let zo (lzol = r) be a point where lf(z)I attains its maximum value.Show that z 0 f'(z 0 )/ f(zo) is real and positive.
*9. Suppose that f is analytic in lzl < 1, continuous on lzl = 1, and
vanishing in some open subset of lzl = 1. Prove that f(z) vanishes
identically in lzl:::; 1. [R. Arens and I. M. Singer, Trans. Amer. Math.
Soc., 81 (1956), 279-283]10. Let f be analytic in an unbounded region G and continuous on 8G (the
point oo being excluded). Suppose that there are constants M, K > 0
such that lf(z)I :::; M for all z E aG and lf(z)I :::; ]{for all z E G.
Prove that lf(z)I :::; M for all z E G.