Conformal mappings and the Phragmen-Lindelöf Theorem

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 z\in \mathbb{C}, zx+iy, with x and y real numbers. The open set \Omega is defined as

\displaystyle \Omega=\{z\in \mathbb{C}\,;\,-\frac{\pi}{2}<y<\frac{\pi}{2}\},

\overline{\Omega} denotes the closure of \Omega in \mathbb{C}, and \partial \Omega=\overline{\Omega}\setminus\Omega its boundary. I write \mathcal{H}(\Omega) for the set of all holomorphic functions on \Omega and \mathcal{C}(\overline{\Omega}) for the set of all continuous functions on \overline{\Omega}.

Theorem (Phragem-Lindelöf)

Let f be a function in \mathcal{H}(\Omega)\cap\mathcal{C}(\overline{\Omega}) such that

\displaystyle \left|f\left(x\pm i\frac{\pi}{2}\right)\right|\le 1

for all x\in \mathbb{R}, and let us assume that there exist real constants A and \alpha <1 such that

\displaystyle |f(z)|\le \exp(A\exp(\alpha|x|))

for all z \in \Omega. Then |f(z)|\le 1 for all z \in \Omega, and, if there exists z_0\in \Omega such that |f(z_0)|=1, f 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 (g_{\varepsilon})_{\varepsilon>0} (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 \Omega 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 z=x+iy as before and the new variable by w=u+iv.

General horizontal strip

Let a and b be real numbers such that a<b and let \Omega(a,b) be the strip

\{z\in \mathbb{C}\,;\,a<y<b\}.

Proposition 1

Let f be a function in \mathcal{H}(\Omega(a,b))\cap\mathcal{C}(\overline{\Omega(a,b)}) such that |f(z)|\le 1 for z\in \partial \Omega(a,b), and let us assume that there exist real constants A and \alpha<\frac{\pi}{b-a} such that

\displaystyle |f(z)|\le \exp\left(A\exp\left(\alpha|x|\right)\right)

for all z \in \Omega(a,b).
Then |f(z)|\le 1 for all z\in \Omega(a,b), and, if there exists z_0\in \Omega(a,b) such that |f(z_0)|=1, f is a constant.

Proof. Let us define \varphi:\mathbb{C}\to \mathbb{C} by

\displaystyle \varphi(w)=\frac{b-a}{\pi}w+i\frac{a+b}{2}.

It is obviously an holomorphic change of variable (it is even linear). We have \varphi(\Omega)=\Omega(a,b). Let us write g=f\circ \varphi. The function g is in \mathcal{H}(\Omega)\cap\mathcal{C}(\overline{\Omega}), and |g(w)|\le 1 for all w \in \partial \Omega. Let us now consider w\in \Omega and z=\varphi(w). We have, for the real part of z,

\displaystyle x=\frac{b-a}{\pi}u.

Since we have

\displaystyle  |f(z)|\le \exp\left(A\exp\left(\alpha|x|\right)\right),

we obtain

\displaystyle |g(w)|\le \exp\left(A\exp\left(\frac{\alpha(b-a)}{\pi}|u|\right)\right).

Since \frac{\alpha(b-a)}{\pi}<1, the Phragmen-Lindelöf Theorem yields the desired result. QED.

General vertical strip

Let a and b be real numbers such that a<b, and let \Pi(a,b) be the strip

\{z\in \mathbb{C}\,;\,a<x<b\}.

Proposition 2
Let f be a function in \mathcal{H}(\Pi(a,b))\cap\mathcal{C}(\overline{\Pi(a,b)}) such that |f(z)|\le 1 for z\in \partial \Pi(a,b), and let us assume that there exist real constants A and \alpha<\frac{\pi}{b-a} such that

\displaystyle |f(z)|\le \exp\left(A\exp\left(\alpha|y|\right)\right)

for all z \in \Pi(a,b).
Then |f(z)|\le 1 for all z\in \Omega(a,b), and, if there exists z_0\in \Pi(a,b) such that |f(z_0)|=1, f is a constant.

Proof. We define \varphi:\mathbb{C}\to \mathbb{C} by \varphi(w)=-iw. The function \varphi is an holomorphic change of variable that maps \Omega(a,b) to \Pi(a,b). The function g=f\circ \varphi satisfies the hypotheses of Proposition 1, which yields the desired result. QED.

Sector

Let \theta_1 be a number in ]-\pi,\pi] and \theta_2 another number such that 0<\theta_2-\theta_1<2\pi.
We define the open sector \Sigma(\theta_1,\theta_2) by

\displaystyle \Sigma(\theta_1,\theta_2)=\left\{re^{i\theta}\,;\,r>0\mbox{ and } \theta_1<\theta<\theta_2\right\},

and we set

\displaystyle \Sigma'(\theta_1,\theta_2)=\left\{re^{i\theta}\,;\,r>0\mbox{ and } \theta_1\le\theta\le\theta_2\right\},

the closure of \Sigma(\theta_1,\theta_2) with the origin removed.

Proposition 3

Let f be a function in \mathcal{H}(\Sigma(\theta_1,\theta_2))\cap\mathcal{C}(\Sigma'(\theta_1,\theta_2)) such that |f\left(re^{i\theta_1}\right)|\le 1 and |f\left(re^{i\theta_2}\right)|\le 1 for all r>0. Let us assume that there exist real constants A and \alpha<\frac{\pi}{\theta_2-\theta_1} such that

\displaystyle |f(z)|\le \exp\left(A|z|^{\alpha}\right)

if z \in \Sigma(\theta_1,\theta_2) with |z|\ge 1 and

\displaystyle |f(z)|\le \exp\left(\frac{A}{|z|^{\alpha}}\right)

if z \in \Sigma(\theta_1,\theta_2) with |z|\le 1.
Then |f(z)|\le 1 for all z\in \Sigma(\theta_1,\theta_2), and, if there exists z_0\in \Sigma(\theta_1,\theta_2) such that |f(z_0)|=1, f is a constant.

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

Corollary

Let f be a function in \mathcal{H}(\Sigma(\theta_1,\theta_2))\cap\mathcal{C}(\overline{\Sigma(\theta_1,\theta_2)}) such that |f\left(re^{i\theta_1}\right)|\le 1 and |f\left(re^{i\theta_2}\right)|\le 1 for all r>0. Let us assume that there exist real constants A and \alpha<\frac{\pi}{\theta_2-\theta_1} such that

\displaystyle |f(z)|\le \exp\left(A|z|^{\alpha}\right)

for all z \in \Sigma(\theta_1,\theta_2).
Then |f(z)|\le 1 for all z\in \Sigma(\theta_1,\theta_2), and, if there exists z_0\in \Sigma(\theta_1,\theta_2) such that |f(z_0)|=1, f is a constant.

Proof. Since 0<\theta_2-\theta_1<2\pi, the exponential function is a bijection from \Omega(\theta_1,\theta_2) to \Sigma(\theta_1,\theta_2), and furthermore \exp\left(\overline{\Omega(\theta_1,\theta_2)}\right)=\Sigma'(\theta_1,\theta_2). Let us consider w\in \Omega(\theta_1,\theta_2) and z=\exp(w). We have
|z|=e^u.

If u\ge 0, |z|\ge 1, and since

\displaystyle |f(z)|\le \exp\left(A|z|^{\alpha}\right),

we obtain

\displaystyle |g(w)|\le \exp(A\exp(\alpha u)).

If u\le 0, |z|\le 1, and since

\displaystyle |f(z)|\le \exp\left(A|z|^{-\alpha}\right),

we obtain

\displaystyle |g(w)|\le \exp(A\exp(-\alpha u)).

In both cases, we have

\displaystyle |g(w)|\le \exp(A\exp(\alpha|u|)),

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.

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