Real and Complex Analysis (Rudin)

310 REAL AND COMPLEX ANALYSIS
then n(r) is about 1t - 1 kr", SO that
lim log n(r) = k.
,-+00 log r
(6)
This shows that the estimate (4) cannot be improved.
Blaschke Products
Jensen's formula makes it possible to determine the precise conditions which the
zeros of a nonconstant f E H OO must satisfy.
15.21 Theorem If {cxn} is a sequence in V such that CXn =I- 0 and
if k is a nonnegative integer, and if
00
L (1 - I CXn I) < 00, (1)
n=1
(Z E V), (2)
then B E H oo , and B has no zeros except at the points CXn (and at the origin, if
k > 0).
We call this function B a Blaschke product. Note that some of the CXn may be
repeated, in which case B has multiple zeros at those points. Note also that each
factor in (2) has absolute value 1 on T.
The term "Blaschke product" will also be used if there are only finitely many
factors, and even if there are none, in which case B(z) = 1.
PROOF The nth term in the series
t 1 - cxn~z '~I
n=1 1 - CXnZ cxn
is
CXn + ~ CXn I Z 1 (1 - I CXn I) :::;; 1 + r (1 - I cxn I)
1 (1 - CXn z)cxn 1 - r
if I z I :::;; r. Hence Theorem 15.6 shows that B E H(V) and that B has only the
prescribed zeros. Since each factor in (2) has absolute value less than 1 in V,
it follows that I B(z) I < 1, and the proof is complete.
IIII
15.22 The preceding theorem shows that
00
L (1 ~ I cxn I) < 00 (1)
n=1