# Complex Analysis/Extremum principles, open mapping theorem, Schwarz' lemma

We continue our quest of proving general properties of holomorphic functions, this time even better equipped, since we have the theorems from last chapter.

## Extremum principles Edit

Under certain circumstances, holomorphic functions assume their maximal resp. minimal absolute value on the boundary. Before making this precise, we need a preparatory lemma.

**Lemma 8.1**:

Let be holomorphic, where and are arbitrary, and assume that it even satisfies for a constant . Then itself is constant.

**Proof**:

In case in , we may conclude in and are done. Otherwise, we proceed as follows:

If is constant, so is . We write . Then for all . Thus, taking partial derivatives, we get

- and .

From the Cauchy–Riemann equations we may further infer

- and ,

from which follow (after some algebra) that

- , , and ,

that is .

Now we are ready to explicate the extremum principles in the form of the following two theorems.

**Theorem 8.2 (the maximum principle)**:

Let be a function which is holomorphic on the interior of . If is such that

- ,

then either or is constant on the connected component of .

**Proof**:

Assume , that is, . Let be arbitrary such that . Then Cauchy's integral formula implies

- .

If now for some , then by the continuity of

- ,

a contradiction. Hence, on all of , and since was arbitrary (provided that ), in a small ball around . From lemma 8.1, it follows that is constant there, and hence the identity theorem implies that is constant on the whole connected component containing .

Similarly, we have:

**Theorem 8.3 (the minimum principle)**:

Let be a function which is holomorphic on the interior of . If is such that

- ,

then one of three things happens:

- ,
- is constant on the connected component of or
- has a zero inside .

**Proof**:

If does not have a zero inside , the chain rule implies that the function

is holomorphic in . Hence, the maximum principle applies and either has no maximum in the interior (and thus has no minimum in the interior) or is constant (and hence is constant as well).

## The open mapping theorem Edit

**Theorem 8.4 (open mapping theorem)**

Let be a holomorphic function. If is an open set, then is also open.

That is, as topologists would say, is an *open map*.

**Proof**:

Let . We prove that there exists a ball around which is contained within . To this end, we pick (due to the openness of ) a such that and furthermore on (by the identity theorem) and set

- ;

since is compact, assumes a minimum there and it's not equal to zero by choice of , which is why . Now for every we define the function

- .

In , the absolute value of this function is less than by choice of . However, for , we have

- .

Hence, the minimum principle implies that the function has a zero in , and this proves (since was arbitrary) that assumes every value in .

## Schwarz' lemma Edit

**Theorem 8.5 (Schwarz' lemma)**

Let be a holomorphic function such that

- and
- .

Then , and further, if either or for one particular , then we can find a such that .

**Proof**:

First, we consider the following function:

- .

Since this map is bounded, continuous and holomorphic everywhere except in , it is even holomorphic in due to Riemann's theorem (the extension in must be uniquely chosen s.t. continuity is satisfied). Furthermore, we have

for all by assumption; in particular, if , then , and thus, by the maximum principle, also for . Taking gives in , and hence for .

For the second part, if either or for a , then somewhere inside , and hence, again by the maximum principle, must be constant, from which follows , that is, we may pick .