Real and Complex Analysis (Rudin)

ZEROS OF HOLOMORPIDC FUNCTIONS 309
and
J1.,(f) :s; J1.*(f) if 0 < r < 1. (5)
Note the following consequence: One can choose r so thatf(z) =1= 0 if I z I = r;
then J1.,(f) is finite, and so is J1.*(f), by (5). Thus log I f* I E i!(T), and f*(ei~ =1= 0
at almost every point ofT.
PROOF There is an integer m ~ 0 such thatf(z) = zmg(z), 9 E H oo , and g(O) =1=
O. Apply Jensen's formula 15.18(1) to 9 in place of f Its left side obviously
cannot decrease if r increases. Thus J1.,(g) :s; J1..(g) if r < s. Since
we have proved (3).
J1.,(f) = J1.,(g) + m log r,
Let us now assume, without loss of generality, that I f I :s; 1. Write f..(ei~
in place of f(rei~. Then f.. ...... f(O) as r ...... 0, and f.. ...... f* a.e. as r ...... 1. Since
log (1/ I f..1 ) ~ 0, two applications of Fatou's lemma, combined with (3), give
~~~ W
15.20 Zeros of Entire Functions Suppose f is an entire function,
M(r) = sup I f(rei8) I
8
(0 < r < (0), (1)
and n(r) is the number of zeros of fin D(O; r). Assume f(O) = 1, for simplicity.
Jensen's formula gives
{ If" } ,,(2,) 2r ,,(,) 2r
M(2r) ~ exp 2n _.log If(2rei~1 dO = lIn ~ ~ lIn ~ ~ 2"('),
if {IX,,} is the sequence of zeros off, arranged so that IIXII :s; 11X21 :s; .... Hence
n(r) log 2 :s; log M(2r). (2)
Thus the rapidity with which n(r) can increase (i.e., the density of the zeros of
f) is controlled by the rate of growth of M(r). Suppose, to look at a more specific
situation, that for large r
M(r) < exp {Ar"}
(3)
where A and k are given positive numbers. Then (2) leads to
I· log n(r) k
1m sup I :s;.
''''00 og r
For example, if k is a positive integer and
(4)
(5)