Phragmen-Lindelöf theorem, question from Conway, chapter VI Page 141, Question 3:


  Let $G=\{z:|\operatorname{Im} z| &lt; \pi/2\}$  and suppose $f:G\rightarrow C$ and $\limsup|f(z)| \leq M$ on $w$ in the boundary of $G$. Also, suppose $A &lt; \infty$ and $a &lt; 1$ can be found such that $|f(z)| &lt; \exp[A \exp(a|\operatorname{Re} z|)]$ for all $z$ in $G$. Show that $|f(z)|\leqslant M$ for all $z$ in $G$.


Thanks in advance.
Consider $f\circ\log (z)$ and apply Corol. 4.2 of Chapter 6 in Conway's book.

Corollary 4.2   Let $a\geq \frac{1}{2}$ and put $G=\{z: |arg z|&lt;\frac{\pi}{2a}\}$. Suppose that $f$ is analytic on $G$ and there is a constant $M$ s.t. $\limsup_{z\to w}|f(z)|\leq M$ for all $w$ in $\partial G$. If there are positive constants $P$ and $b&lt;a$ s.t. $|f(z)|\leq P \exp(|z|^b)$ for all z with $|z|$ sufficiently large, then $|f(z)|\leq M$ for all $z$ in $G$.
Consider $f\circ\log (z)$ and apply Corol. 4.2 of Chapter 6 in Conway's book.

Corollary 4.2   Let $a\geq \frac{1}{2}$ and put $G=\{z: |arg z|&lt;\frac{\pi}{2a}\}$. Suppose that $f$ is analytic on $G$ and there is a constant $M$ s.t. $\limsup_{z\to w}|f(z)|\leq M$ for all $w$ in $\partial G$. If there are positive constants $P$ and $b&lt;a$ s.t. $|f(z)|\leq P \exp(|z|^b)$ for all z with $|z|$ sufficiently large, then $|f(z)|\leq M$ for all $z$ in $G$.