Lindelöf's theorem

Remarks

Proof idea: for $\varepsilon > 0$ let $g_\varepsilon(z) = f(z) e^{i \varepsilon (z - i y_0)}$, so that $\abs{g_\varepsilon(x + iy)} = \abs{f(x + iy)} e^{-\varepsilon (y - y_0)}$. Then $\abs{g_\varepsilon} \le \abs{f}$ on $\Omega$, and $g_\varepsilon \to 0$ as $\operatorname{Im} z \to \infty$ because $f$ grows at most polynomially. The maximum modulus principle on the rectangles $\Omega \cap \set{\operatorname{Im} z \le T}$ with $T$ large gives $\abs{g_\varepsilon} \le M$, and $\varepsilon \to 0$ gives $\abs{f} \le M$. A growth hypothesis is needed. On the half-strip $\abs{\operatorname{Re} z} \le \frac{\pi}{2}$, $\operatorname{Im} z \ge 0$, the function $\exp(e^{-iz})$ has modulus $\exp(e^{y} \cos x)$. It is bounded on the boundary, where $\cos x = 0$ or $y = 0$, but unbounded on the line $x = 0$. The theorem is a case of the Phragmén–Lindelöf principle.

Going deeper

For a student

The theorem is used in analytic number theory for functions such as the Riemann zeta function on vertical half-strips. Together with the Hadamard three-lines theorem it gives the convexity bounds for the growth of $\zeta(\sigma + it)$ as $t \to \infty$. In the proof, the factor $e^{i\varepsilon z}$ is the standard device to force decay at infinity before applying the maximum modulus principle.