Lindelöf's theorem

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

${\displaystyle \Omega =\{z\in \mathbb {C} |x_{1}\leq \mathrm {Re} (z)\leq x_{2}\ {\text{and}}\ \mathrm {Im} (z)\geq y_{0}\}\subsetneq \mathbb {C} .}$

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

${\displaystyle |f(z)|\leq M\ {\text{for all}}\ z\in \partial \Omega }$

and

${\displaystyle |f(x+iy)|\leq By^{A}\ {\text{for all}}\ x+iy\in \Omega .}$

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

${\displaystyle |f(z)|\leq M\ {\text{for all}}\ z\in \Omega .}$

Fix a point ${\displaystyle \xi =\sigma +i\tau }$ inside ${\displaystyle \Omega }$. Choose ${\displaystyle \lambda >-y_{0}}$, an integer ${\displaystyle N>A}$ and ${\displaystyle y_{1}>\tau }$ large enough such that ${\displaystyle {\frac {By_{1}^{A}}{(y_{1}+\lambda )^{N}}}\leq {\frac {M}{(y_{0}+\lambda )^{N}}}}$. Applying maximum modulus principle to the function ${\displaystyle g(z)={\frac {f(z)}{(z+i\lambda )^{N}}}}$ and the rectangular area ${\displaystyle \{z\in \mathbb {C} \mid x_{1}\leq \mathrm {Re} (z)\leq x_{2}\ {\text{and}}\ y_{0}\leq \mathrm {Im} (z)\leq y_{1}\}}$ we obtain ${\displaystyle |g(\xi )|\leq {\frac {M}{(y_{0}+\lambda )^{N}}}}$, that is, ${\displaystyle |f(\xi )|\leq M\left({\frac {|\xi +\lambda |}{y_{0}+\lambda }}\right)^{N}}$. Letting ${\displaystyle \lambda \to +\infty }$ yields ${\displaystyle |f(\xi )|\leq M}$ as required.

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