2021 Fall Qualifying Exam in Complex Analysis

Problem 1.

Compute

Proof.

Use the standard formula

Here and , so

Problem 2.

Let be a polynomial of degree . Prove that if

then

Proof.

This is the Gauss-Lucas theorem: the zeros of lie in the convex hull of the zeros of .

Since all zeros of lie in the unit disk , their convex hull is contained in . Therefore all zeros of also lie in .

Problem 3.

Let

be entire and suppose

for all . Prove that

Proof.

By Cauchy's estimate, for any ,

Choose . Then

Problem 4.

Let be holomorphic with

(i) Prove that has at most zeros in , counting multiplicities.

(ii) Provide an example of such which has zeros in , counting multiplicities.

Proof.

(i) Let be the zeros of in , counted with multiplicity. Jensen's formula, together with , gives

Since ,

But

Therefore

In particular, .

(ii) With the usual convention that is open, the sharp bound is not attained. Indeed, the argument above gives , so one cannot have exactly zeros strictly inside .

If the intended disk in part (ii) is the closed disk , then an extremal example is the finite Blaschke product

It maps to itself, has zeros at , and satisfies

Problem 5.

Prove or disprove: there is a holomorphic function in

such that

Proof.

Let

The function is holomorphic on the exterior domain . A primitive exists on this domain exactly when the period around the hole is zero, equivalently when the sum of residues at the enclosed poles is zero.

The residues are

and

Their sum is

Therefore the period around the hole is zero, and has a holomorphic primitive on . Hence such an exists.

Problem 6.

Find a conformal map from

onto the unit disk .

Proof.

First set

Since and , this maps onto the upper half of the unit disk:

Next define

This maps the upper half-disk onto the first quadrant. Squaring maps the first quadrant onto the upper half-plane:

Finally, the Cayley map

maps the upper half-plane onto the unit disk.

Thus one conformal map is

Problem 7.

Prove that all zeros of

lie in the annulus

Proof.

If , then

Thus

for .

If , then

Therefore

for .

Hence every zero satisfies

which is exactly under the usual open-disk convention.

Problem 8.

Find all entire holomorphic functions satisfying

for all positive integers .

Proof.

The points

tend to . The estimate gives

so by continuity,

Suppose is not identically zero. Then has a zero of finite order at , so

Thus, for large ,

But

is eventually much larger than , contradicting the hypothesis.

Therefore must vanish to infinite order at , hence

Problem 9.

Let be holomorphic in and satisfy

on . Prove that

Proof.

Let

If , then by the maximum modulus principle is constant, and the result is immediate. Assume .

Apply Schwarz-Pick to :

Let and . The inequality says that the pseudo-hyperbolic distance from to is at most . The corresponding Euclidean bounds on are

Substituting back and gives the desired inequalities.