Lindelöf's theorem

Not to be confused with Lindelöf's lemma in topology or the Picard–Lindelöf theorem in the study of differential equations.

In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.

Statement of the theorem

Let $\mathrm{\Omega }$ be a half-strip in the complex plane:

$$\begin{gathered} \mathrm{\Omega }=\{z\in \mathbb{ℂ}|x_1\le \mathrm{R}\mathrm{e}(z)\le x_2 \\ \text{and} \\ \mathrm{I}\mathrm{m}(z)\ge y_0\}⊊\mathbb{ℂ}. \end{gathered}$$

Suppose that $f$ is holomorphic (i.e. analytic) on $\mathrm{\Omega }$ and that there are constants $M$, $A$, and $B$ such that

$$\begin{gathered} |f(z)|\le M \\ \text{for all} \\ z\in \partial \mathrm{\Omega } \end{gathered}$$

and

$$\begin{gathered} |f(x+iy)|\le B{y}^A \\ \text{for all} \\ x+iy\in \mathrm{\Omega }. \end{gathered}$$

Then $f$ is bounded by $M$ on all of $\mathrm{\Omega }$:

$$\begin{gathered} |f(z)|\le M \\ \text{for all} \\ z\in \mathrm{\Omega }. \end{gathered}$$

Proof

Fix a point $\xi =\sigma +i\tau$ inside $\mathrm{\Omega }$. Choose $\lambda >-y_0$, an integer $N>A$ and $y_1>\tau$ large enough such that $\frac{B{y}_1^A}{(y_1+\lambda {)}^N}\le \frac{M}{(y_0+\lambda {)}^N}$. Applying maximum modulus principle to the function $g(z)=\frac{f(z)}{(z+i\lambda {)}^N}$ and the rectangular area $$\begin{gathered} \{z\in \mathbb{ℂ}\mid x_1\le \mathrm{R}\mathrm{e}(z)\le x_2 \\ \text{and} \\ {y}_0\le \mathrm{I}\mathrm{m}(z)\le y_1\} \end{gathered}$$ we obtain $|g(\xi )|\le \frac{M}{(y_0+\lambda {)}^N}$, that is, $|f(\xi )|\le M{\left(\frac{|\xi +\lambda |}{y_0+\lambda }\right)}^N$. Letting $\lambda \to +\mathrm{\infty }$ yields $|f(\xi )|\le M$ as required.

References

• Edwards, H.M. (2001). Riemann's Zeta Function. New York, NY: Dover. ISBN 0-486-41740-9.