Math 535 Homework #5
Winter 2008
1. Let {fn}be a sequence of analytic functions on a region Ω with |fn| ≤ 1 on Ω. Let Kbe
compact and contained in Ω. Suppose {fn}converges at infinitely many points in K. Then is it
true or false that {fn}necessarily converges at every point of Ω?
2. Let FMbe the set of functions analytic on the (open) unit disk Dand continuous on the closed
unit disk which satisfy
Z2π
0
|f(eiθ)|dθ ≤M.
Show FMis a normal family on D with respect to the Euclidean metric.
3. Let Bbe the set of functions fwhich are analytic on the unit disk Dand satisfy both f(0) = 0
and f(D)∩[1,2] = ∅.Prove B is a normal family (as maps from D into the complex plane with the
Euclidean metric) which contains all of its non-constant limit functions.
4. Given c > 0, prove there exists an r > 0 (depending upon c) such that if fis analytic on D,
with |f(z)| ≤ 1 for z∈Dand f(0) = 0 and |f0(0)|> c, then f(D) contains a disk centered at 0 of
radius r > 0. Show the conclusion fails if c= 0.
5. a. Prove that a family Fof analytic functions on a region ωis normal if and only if the family
F0={f0:f∈ F} is normal and for some z0∈Ω, the set {f(z0) : f∈ F } is bounded.
b. Find an example of a sequence {fn}which is normal on Cusing the spherical metric, but
{f0
n}is not normal on Cusing the spherical metric.
6. (a) Prove that there is a constant L1>0 so that if fis analytic on Dwith f0(0) = 1 and |f0| ≤ 2
on Dthen f(D) contains a disk centered at f(0) of radius L1.
(b) Prove that there is a constant L > 0 so that if fis analytic on Dwith f0(0) = 1 then f(D)
contains a disk of radius L. Hint: Set M(r) = maxrD|f0(z)|and set r0= max{r: (1−r)M(r) = 1}.
Choose awith |a| ≤ r0and renormalize fon a disk centered at awith radius (1 −r0)/2 so that it
satisfies the hypotheses of part (a).
(c) Deduce that a non-constant entire function covers disks of arbitrarily large radii.
(d) If gis analytic on a simply connected region Ω such that gomits the two values 1 and
−1, then prove that we can define h(z) = −i
πlog[g(z) + pg(z)2−1] so as to be analytic on Ω.
Furthermore show that homits the integers.
