Real and Complex Analysis (Rudin)

308 REAL AND COMPLEX ANALYSIS
This is known as Jensen'sformula. The hypothesisf(O) "# 0 causes no harm in
applications, for if f has a zero of order k at 0, the formula can be applied to
f(z)/z k •
PROOF Order the points a.j so that a.l' ... , a. m are in D(O; r) and la. m+ll =
... = I a.N I = r. (Of course, we may have m = N or m = 0.) Put
m r2 _ iii z N a.
g(z) =f(z) n n n _n_.
n= 1 r(a.n - z) n=m+ 1 a.n - z
Then g E H(D), where D = D(O; r + €) for some € > 0, g has no zero in D,
hence log I g I is harmonic in D (Theorem 13.12), and so
log I g(O) I = 21t 1 f" _..tog I g(rei~ I dO. (3)
(2)
By (2),
m
r
I g(O) I = I f(O) I}I I a. n
I . (4)
For 1 :::; n :::; m, the factors in (2) have absolute value 1 if I z I = r. If a. n = rei9n
for m < n :::; N, it follows that
log I g(rei~ I = log I f(rei~ I -
N
L log 11 - e l (9-9 n ) I. (5)
n=m+l
Lemma 15.17 shows therefore that the integral in (3) is unchanged if g is
replaced by f. Comparison with (4) now gives (1).
IIII
Jensen's formula gives rise to an inequality which involves the boundary
values of bounded holomorphic functions in U (we recall that the class of these
functions has been denoted by H'D):
15.19 Theorem Iff E HOC),f not identically 0, define
and
Jl.*(f) = -
1 f"
21t -"
log I f*(ei9) I dO
(0 < r < 1)
wheref* is the radial limit function off, as in Theorem 11.32. Then
Jl.r(f) :::; Jl..(f) if 0 < r < s < 1,
Jl.r(f)-+log If(O) I as r-+O,
(1)
(2)
(3)
(4)