Real and Complex Analysis (Rudin)

ZEROS OF HOLOMORPHIC FUNCTIONS 317
Prove that f(z) is independent of the choice of y(z) (although the integral itself is not), that f is
holomorphic in the complement of {z.}, that f has a removable singularity at each of the points z.,
and that the extension off has a zero of order m. at z •.
The existence theorem contained in Theorem 15.9 can thus be deduced from the Mittag-Leffler
theorem.
9 Suppose 0 < IX < 1,0 < P < l,fe H(U),f(u) c U, andf(O) = IX. How many zeros canfhave in the
disc D(O; P)? What is the answer if (a) IX = t, p = t; (b) IX = !, p = t; (e) IX = ~, p = i; (d) IX = 1/1,000,
P = 1/10?
10 For N = 1, 2, 3, ... , define
"" ( Z2)
gJ..z) = n 1- 2
n=N n
Prove that the ideal generated by {gN} in the ring of entire functions is not a principal ideal.
II Under what conditions on a sequence of real numbers Y. does there exist a bounded holomorphic
function in the open right half plane which is not identically zero but which has a zero at each point
1 + iy.? In particular, can this happen if (a) y. = log n, (b) y. = In, (e) y. = n, (d) y. = n 2 ?
12 Suppose 0 < IIX.I < 1, :E(1 - IIX.I) < 00, and B is the Blaschke product with zeros at the points IX ••
Let E be the set of all points l/,a. and let Q be the complement of the closure of E. Prove that the
product actually converges uniformly on every compact subset of Q, so that B e H(Q), and that B has
a pole at each point of E. (This is of particular interest in those cases in which Q is connected.)
13 Put IX. = 1 - n - 2, for n = 1, 2, 3, ... , let B be the Blaschke product with zeros at these points IX,
and prove that Iimr~ I B(r) = o. (It is understood that 0 < r < 1.)
More precisely, show that the estimate
N-I r-IX N-I IX - IX
• < 2e-N/ 3
I1-IX.r I I-IX.
I B(r) I < n -_. < n _N __
is valid if IXN- I < r < IXN.
14 Prove that there is a sequence {IX.} with 0 < IX. < 1, which tends to 1 so rapidly that the Blaschke
product with zeros at the points IX. satisfies the condition
lim sup I B(r)l = 1.
r~1
Hence this B has no radial limit at z = 1.
15 Let qJ be a linear fractional transformation which maps U onto U. For any z e U define the
qJ-orbit of z to be the set {qJ.(z)}, where qJo(z) = z, qJ.(z) = qJ(qJ._I(Z», n = 1, 2, 3, .... Ignore the case
qJ(z) = z.
(a) For which qJ is it true that the qJ-orbits satisfy the Blaschke condition :E(1 -I qJ.(z) I) <