Math 534 Homework #6
Autumn 2008
1. Suppose f is analytic in a neighborhood of Dand that |f(z)|<1 on ∂D. Using Rouch´e’s theorem
show that there exists one and only one point z0∈Dsuch that f(z0) = z0.
2. Let f(z) = z+a2z2+a3z3+a4z4+... be analytic in D. Suppose
c=|a2|+|a3|+|a4|+. . . < 1.
Show that if |w|<1−cthen there is exactly one z∈Dwith f(z) = w.
3. How many zeros does p(z) = 3z5+ 21z4+ 5z3+ 6z+ 7 have in D? How many zeros in
{z: 1 <|z|<2}? Choose coefficients of a fourth degree polynomial randomly and find out how
many zeros in D, using the algorithm in Appendix A.
4. Prove that the number of roots of the equation
z2n+αz2n−1+β2= 0,
where α, β are real and nonzero, and nis a natural number, that have positive real part is equal
to nif nis even. If nis odd, their number is n−1 for α > 0 and n+ 1 for α < 0.
Hint: See what happens to
z2n+αz2n−1+β2
as ztraces the boundary of a large half-disk.
5. Let nbe a positive integer and a > 0. Show that there exists fanalytic in a neighborhood of
z= 1 such that f(1) = 1 and
af(z)n+1 + (1 −a)f(z)n=z
in a neighborhood of z= 1.
6. (this problem is complementary to problem 7 of HW #1). Suppose fis analytic on an open set
Uand z0∈U. Prove that there are points z1, z2∈Usuch that
f0(z0) = f(z1)−f(z2)
z1−z2
.
Hint: consider g(z) = z f 0(z0)−f(z).
