2017 Fall Qualifying Exam in Complex Analysis

Problem 1.

Let be a real-valued continuous function on such that is harmonic in . Prove that is constant.

Proof.

The function

is positive and harmonic on all of . A positive harmonic function on the plane is constant. Indeed, by Harnack's inequality applied on disks of radius and then letting , one obtains

for every .

Thus is constant. Since the exponential is injective on real numbers, is constant.

Problem 2.

Prove or disprove: there is a holomorphic function on such that

for all positive integers .

Proof.

No such function exists.

If such an existed, then for the sequence ,

Thus the holomorphic function

has zeros at , and these zeros accumulate at . By the identity theorem,

But then

This contradiction proves that no such holomorphic function exists.

Problem 3.

Let be entire and suppose and are real for . Prove there is an entire function such that

for all .

Proof.

Since is real for , the Schwarz reflection identity gives

for all .

For , the number is real. Hence

Thus the entire function

vanishes on the segment . By the identity theorem,

So is even.

Write the Taylor series

Since is even, all odd coefficients vanish. Therefore

Define

This is entire, and

Problem 4.

Let and

Let be holomorphic on . Prove that

is a polynomial of degree at most such that

for .

Proof.

For fixed , the expression

is a polynomial in of degree at most . Therefore is a polynomial of degree at most .

Now evaluate at . Since ,

By Cauchy's integral formula,

Problem 5.

For , prove that

has a unique solution in , and that this solution is real and positive.

Proof.

The equation is equivalent to

On ,

By Rouche's theorem, and have the same number of zeros in . Hence there is exactly one solution in the unit disk.

Now consider the real function

on . We have

Also,

for . Therefore there is a unique such that

This real positive solution must be the unique solution in .

Problem 6.

Prove that

Proof.

For , let

The beta-integral gives

Differentiating twice under the integral sign,

At ,

Since

the desired integral is

Problem 7.

Let be a family of holomorphic functions on such that

Prove that is a normal family.

Proof.

We prove local boundedness. Fix . For ,

Using Cauchy-Schwarz along the segment and then the area bound locally gives

Since both terms are bounded by the hypothesis, there is a constant such that

for all and all .

Thus is locally bounded. By Montel's theorem, is normal.

Problem 8.

Let be holomorphic on such that

defines a continuous function in and

for all . Prove that every is constant.

Proof.

For , the mean-value inequality gives

Summing over and using monotone convergence,

But everywhere, so the average on the right is at most . Hence equality holds throughout.

Therefore, for each ,

Equality in the mean-value inequality for implies that is harmonic and constant on . Thus is constant on .

Since was arbitrary, every is constant on .