Lindelöf's theorem

Proof

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