Sequerices, Series, and Special Functions 575(b) Show that the distance from the origin to the nearest zero off is not
rlaol
less than M( ) I I' where 0 < r <Rand M(r) =max lf(z)I.
r + ao lzl=r6. Show that there is no analytic function f in lzl < 1 such that
f ( 2
1
k ) = f ( 2k ~ 1 ) = 21
kfor k = 1, 2, 3,7. Show that there is no analytic function f that takes on the values 1, 1,
1/s, 1/s, 1/s, 1/s, ... at the points 1,^1 / 2 ·, 1/ 3 ,^1 / 4 , 1/ 5 , ••• respectively.
8.11 THE MAXIMUM AND MINIMUM MODULUS
PRINCIPLES FOR ANALYTIC (OR CONJUGATE
ANALYTIC) FUNCTIONSThese principles have already been discussed for functions of class 'D inSection 6.20. If the analyticity of f in a region G is assumed, then the
hypothesis J1(z) =f. 0 in Theorems 6.28 to 6.31 implies that fz = f'(z) =/. 0
everywhere in G, and if f is suppose to be conjugate analytic, then J1(z) =f.
0 implies that fz = ]'(z)-:/= 0 everywhere in G. Either condition [in fact,
J1(z) =f. 0 everywhere in G for f E 'D(G)] implies that f is not constant in
G. This last property suffices to ensure the validity of the maximum and
minimum modulus principles for analytic or conjugate analytic functions[together with f(z) =f. 0 in Gin the case of the minimum modulus], as the
following theorems show.
The reader will note that a nonconstant analytic function in G, such
as f(z) = z^3 , may have a derivative with zeros at some isolated points of
G. In other words, the condition f not constant in G is weaker than the
condition f'(z) =f. 0 everywhere in G.Theorem 8.29 Let f be a nonconstant analytic (or conjugate analytic)
function in a region G. Then lf(z)I does not attain a maximum value
anywhere in G.Proof Suppose that f is analytic in G and let z 0 be a point of G. By the
Cauchy-Taylor expansion theorem we havef(z) = ao + ai(z - zo) + az(z - z 0 )^2 + · · · (8.11-1)