I would like to go back to the Phragmen-Lindelöf theorem that I presented in a previous post. Let us recall the result. In the following, we write, for all , , with and real numbers. The open set is defined as

denotes the closure of in , and its boundary. I write for the set of all holomorphic functions on and for the set of all continuous functions on .

Theorem (Phragem-Lindelöf)Let 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.

The Phragmen-Lindelöf *method* can be adapted to prove results of this type on various domains by constructing suitable families of functions (see this same previous post for context). There is however another way to obtain a similar result for another domain. If we can find an holomorphic change of variable, that is to say a conformal mapping, that maps the domain in the theorem to the domain that we are considering, we obtain the Phragmen-Lindelöf result on this last domain. Of course, the growth condition will be modified by the mapping. Let us give several examples. In the following, the original complex variable will be denoted by as before and the new variable by .

**General horizontal strip**

Let and be real numbers such that and let be the strip

Proposition 1Let be a function in such that for , 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 define by

It is obviously an holomorphic change of variable (it is even linear). We have Let us write . The function is in , and for all . Let us now consider and . We have, for the real part of ,

Since we have

we obtain

Since , the Phragmen-Lindelöf Theorem yields the desired result. *QED.*

**General vertical strip**

Let and be real numbers such that , and let be the strip

Proposition 2

Let be a function in such that for , 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.

*Proof.* We define by . The function is an holomorphic change of variable that maps to . The function satisfies the hypotheses of Proposition 1, which yields the desired result. *QED.*

**Sector**

Let be a number in and another number such that .

We define the open sector by

and we set

the closure of with the origin removed.

Proposition 3Let be a function in such that and for all . Let us assume that there exist real constants and such that

if with and

if with .

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

Let us note that this proposition implies the following weaker statement, which is often referred to as the Phragmen-Lindelöf principle.

CorollaryLet be a function in such that and for all . 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.* Since , the exponential function is a bijection from to , and furthermore . Let us consider and . We have

.

If , , and since

we obtain

If , , and since

we obtain

In both cases, we have

and we can apply Proposition 2. *QED.*

One last remark: in my previous post, I presented several results which give a precised form of the maximum principle. This allows us to see which part of the boundary has the most weight in controlling the modulus at a given interior point. I stated the results for bounded holomorphic functions, but they are not limited to them. Indeed, we can <em<first apply one of the results in this post to show that an holomorphic function that does not grow to fast at infinity is bounded, and *then* apply the corresponding result in the previous post.