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 principlesEdit

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:

  1.  ,
  2.   is constant on the connected component of   or
  3.   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 theoremEdit

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' lemmaEdit

Theorem 8.5 (Schwarz' lemma)

Let   be a holomorphic function such that

  1.   and
  2.  .

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  .