578 Chapter^8
and it is clear that the maximum is attained at t = 0 and the minimum
at t = Tr. Hence
max lf(z)I = 3
lzl:9and min lf(z)I = 1
lzl::;I
As simple consequences of the maximum and minimum modulus
theorems we prove the following properties. Others are given in
Exercises 8.4.
Corollary 8.14 If a function f is analytic (or conjugate analytic) in a
region G and if(z)I = K (a constant) for every z E G, then f(z) = C
(a constant) in G.
Proof If f were not a constant function in G, then lf(z)I could not be
constant in G in view in Theorem 8.29.
Corollary 8.15 If f is a nonconstant analytic (or conjugate analytic)
function in the disk lzl < R, then max lf(z)I = M(r), 0 ~ r < R, is a
strictly increasing function of r.
Proof Consider the circles lzl = r1, izl = r2 with 0 ~ r1 < r2 < R.
M(r1) is assumed by lf(z)I on lzl = r1 which is inside izl = r2. Hence by
Theorem 8.30 we have M(r1) < M(r 2 ).
As another application of the maximum modulus principle we shall prove
the following modification of the first part of Theorem 8.2 (Weierstrass
theorem).
Theor1::~m 8.33 Suppose:
- The function Fn(z) (n = 1, 2,. .. ) are analytic in a bounded region G
and continuous on 8G, - Fn(z):::; F(z) on 8G,
Then Ji'(z) is analytic in G.
Proof Because the sequence {Fn(z)} converges uniformly on the boundary
of G, given f > 0 there is N such that m > n > N implies
IFm(z) - Fn(z)I < f (8.11-4)for all z E 8G. By the maximum modulus principle the inequality (8.11-
4) also holds for all z E G and, in particular, on any open disk D(z)
contained in G. By the sufficiency part of the Cauchy condition it follows
that Fn(z)=tF(z) on D(z), as well as on compact subsets of D(z). Since
D(z) is a simple connected region, by Corollary 8.3 F(z) is analytic in
D(z) and thus also analytic in G (z being an arbitrary point of G).