Sequences, Series, and Special Functions 577Proof Similar to that ~f Theorem 6.29.Remarks I. Some authors call Theorem 8.29 the weak form of the maxi-
mum modulus principle, and Theorem 8.30 the strong form of the principle.
This remark applies also, mutatis mutandi, to the following two theorems.
II. The region G in Theorems 8.29 and 8.30 may be simply or mul-
tiply connected. Furthermore, since the weak form of the maximum
modulus principle is a local property, it holds also for multiple-valued an-alytic functions, provided that lf(z)I is single-valued in the region under
consideration.III. If the region G in Theorem 8.30 is unbounded, the maximum mod-
ulus principle is no longer valid in general; i.e., it may or may not holddepending on the case. For instance, if G = {z: lzl > 1} and f(z) = 1/z,
the maximum of lf(z)I is attained on 8G = {z: lzl = 1}. However, if
f(z) = ez, then lf(z)I =ex, and if G = {z: Rez > O}, lf(z)I is unbounded
on G while lf(z)I = 1 on 8G = {z: Rez = O}. Here oo is not regarded
as a boundary point of G.Theorem 8.31 Let f be a nonconstant analytic (or conjugate analytic)
function in a region G, and suppose that f(z) 'f. 0 for every z E G. Thenlf(z)I does not attain a minimum value anywhere in G.
Proof Since f(z) 'f. 0 everywhere in G, the function g(z). = 1/ f(z) is
analytic (respectively, conjugate analytic) in G. By Theorem 8.29 lg(z)I =1/lf(z)I does not attain a maximum value anywhere in G. Hence it follows
that lf(z)I does not assume a minimum value anywhere in that region.
Note If f(z 0 ) = 0 for some point z 0 E G, then lf(z)I will attain a
minimum value at z 0 and the conclusion of the theorem is not valid.Theorem 8.32 Let f be a nonconstant analytic (or conjugate analytic)
function on a bounded region G. Suppose that f(z) 'f. 0 for every z E Gand that f is continuous on 8G. Then lf(z)I attains a minimum value
somewhere on {)G (minimum modulus principle).Proof Similar to that of Theorem 6.31.Example Given f(z) = z + 2 to find max lf(z)I and min lf(z)I on lzl :5 1.
Since the given function is conjugate analytic on lzl :::; 1, both max Jf(z)J
and min Jf(z)I will occur on the boundary of the unit disk. For z = eit,
0 :5 t :5 271", we have
Jf(z)J = yf(cost + 2)^2 + (-sint)^2
= v'5 + 4cost