Take Home Examination
Math 5321
March 25, 1998
Due: March 30th
100 Points
Notation:
B(a,r) = { z : * z - a * < r }
C(a,r) = { z : * z - a * = r }
D = B(0,1)
UHP = Upper-half Plane
Q1 = First Quadrant
$(G) = { f : f is analytic on G}
0(G) = { f : f is meromorphic on G}
For f 0 $(B(a,R)) and 0 < r < R
/C
T ' OCZ * H \ *
#C
T ' OCZ 4G H \ *
\ 0 %
CT
\ 0 %
CT
5. Let f 0 $(B(a,R)) be such that f(a) = 0 and |f(z)| # M on B(a,R). Show that
/
^ H \ ^ #
^ \ & C ^ with equality if and only if
4
/
. (This is a generalization of Schwarz’s
H
\ ' 8 \ & C YJGTG ^8^ '
4
Lemma.)
2. Find the number of zeros of f in G:
a. f(z) = z7 + 6z3 + 7
G = Q1
b. f(z) = z4 + 3iz2 + z - 2 + i
G = UHP
3. Let B1,1 ' { f | f 0 A(D), f(0) ' 1, and f (D) d B(1,1) } . For f 0 B1,1, let
4
H
\ ' j CP \ P on D.
P'
a. Show that if f 0 B1,1, then |an| # 1, for n = 1, 2, . . . .
b. Show that if f 0 B1,1, then ^CP^ #
& ^C^ for n = 2, 3, 4, . . . (which,
of course, implies that if |a1| = 1, then an = 0 for n = 2, 3,
4, . . .).
4. a. Show for a 0 D that
B
m
NQI * % CG K2* F2 ' b. Show for a 0 D and *a* < r < 1 that
B
NQI* C & TG K2* F2 ' NQI T
B m
c. Let f 0 0(B(0,R)) with f(0) ú 0. Let Zf denote the zero set of f and Pf the
pole set of f. Let 0 < r < R be such that Zf 1 C(0,r) = i and Pf 1 C(0,r) = i. Let
{a1, a2, . . ., an} = Zf 1 B(0,r) and {b1, b2, . . ., bm} = Pf 1 B(0,r). Show that
P
O
B
T
T
NQI * H TG K2 * F 2 ' NQI * H * % j NQI
& j NQI
*CM*
*DN*
B m
M
N
5. Let f 0 $(B(a,R)) such that f(a) = 0. Show that g defined by I
T '
/C
T
T
an increasing function of r and that if g is non-constant, then g is strictly
increasing.
6. Let f 0 $(B(a,R)) such that f(a) = 0. Show that /C
T #
Hint: Consider I
\ '
T
# 4 .
4&T C
H \
and apply Schwarz’s Lemma.
#C
4 & H \
is