A friend told me some time ago about a family of theorems that extend the maximum principle for holomorphic functions to unbounded domains. I have read about it, and I want to record some results with their proof. My main reference is the book *Real and Complex Analysis*, by Walter Rudin (I use the second edition).

**The maximum principle for holomorphic functions**

In the following, denotes an open set in the complex plane , denotes the closure of in , and its boundary. I denote by the set of all holomorphic functions on and by the set of all continuous functions on . I will use the the notation , where and are real numbers, the real and imaginary parts of . Let us recall the maximum principle.

Theorem 1Let us assume that is bounded and that belongs to . Then,

Furthermore, if there exists such that , the function is a constant.

For a proof, see for instance Chapters 10 and 12 of Rudin’s book. As Rudin points out, this principle does not hold if we do not assume that is bounded. He gives the following counter example:

and

The function is entire, so we obviously have . We also have

and thus

On the other hand,

so that

and therefore

The maximum principle therefore does not hold in this example.

**An example of the method**

In this section, as before,

In the counter-example of the preceding section, the modulus of the function grows very rapidly when tends to . We will now show that if we prevent from growing too rapidly when tends to , the maximum principle holds.

To simplify, our statement, let us consider a function such that for any . Informally, the Phragmen-Lindelöf result states that *if we know* a priori *that does not grow to rapidly when tends to , then in fact for all *. Furthermore, if there exists in such that , then is a constant.

Since the precise formulation will be a bit technical, I will first try to give an outline of the method. The basic idea is to apply the standard maximum principle on a bounded domain, and to use a family of auxiliary functions . The trick is to build such that

- ;
- ;
- tends to when tends to for any ;
- tends to when tends to , which is allowed by the
*a priori*condition on the growth of .

Then, we obtain the desired result by applying the maximum principle to the function on the bounded rectangle

and by letting tend to and tend to .

Let us now give a precise statement and a proof. It will in particular show that the counter-example quoted above is in some sense optimal.

Theorem 2Let be a function in such that

for all , and let us assume that there exist real constants and such that

for all . Then for all , and, if there exists such that , is a constant.

*Proof.* Let us first pick . We want to prove that . In accordance with the method outlined above, let us choose some such that and . For all , let us set

For the moment, the parameter is fixed. The function is entire and therefore belongs to . Furthermore, we have

This implies

with

This implies that

for all , and that

for all .

Let us now chose large enough so that and

Such an exists since . Of course, it will depend on . The bounded rectangle is defined as above. The function is in and for all in , the boundary of . According to the maximum principle (Theorem 1), since , . This last inequality has been proved for an arbitrary , and when , therefore . This allows us to conclude that for all .

Let us now assume that there exists such that . Let us pick some such that . Using our previous notation we have , and, since for all , we have in particular for all . According to the maximum principle, this implies that is a constant on . Since is an open set in , the principle of analytic continuation tells us that is a constant on . This concludes the proof.

Let us note if , the theorem is false, as shown by the counter-example of the first section. The method is obviously very flexible. With little change in the proof, we can obtain results dealing with a domain that is bounded on one side, or with a function that satisfies the inequality only for some values of the modulus. To be more explicit, I will state two results, but we can imagine much more. In the following, we consider , the open half-strip defined by

Proposition 3Let be a function in such that

for all ,

and let us assume that there exist real constants and such thatfor all . Then for all , and, if there exists such that , is a constant.

Proposition 4Let be a function in such that for all , and let us assume that there exist real constants and , and a sequence of positive numbers tending to such that, for all ,

for all such that . Then for all , and, if there exists such that , is a constant.

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