Phragmén–Lindelöf principle

Remarks

Proof idea: pick $\gamma$ with $\beta < \gamma < \alpha$ and put $g_\varepsilon(z) = f(z) \exp(-\varepsilon z^{\gamma})$ for $\varepsilon > 0$. On $\bar S$ one has $\operatorname{Re} z^{\gamma} \ge \cos\left(\frac{\gamma \pi}{2\alpha}\right) \abs{z}^{\gamma}$ with a positive cosine, so $g_\varepsilon \to 0$ at infinity. The maximum modulus principle on bounded pieces of $S$ gives $\abs{g_\varepsilon} \le M$, and $\varepsilon \to 0$ gives the claim. The growth condition cannot be relaxed to $\beta = \alpha$. The function $\exp(z^{\alpha})$ has modulus $\exp(r^{\alpha} \cos(\alpha \theta))$ at $z = r e^{i\theta}$, which is $1$ on the boundary rays and unbounded on the positive axis. There are similar statements for strips, half-planes and other unbounded domains; the Hadamard three-lines theorem is one of them.

Going deeper

For a student

The principle is used throughout the theory of entire functions and in analytic number theory. It gives convexity bounds for $L$-functions in vertical strips, and it underlies results such as Lindelöf's theorem and Carlson's theorem.