I am considering exercise 9 from chapter 4 of Stein and Sharkarchi's complex analysis:
(a). Let $F$ be a holomorphic function in the right half-plane that extends continuously to the boundary, that is, the imaginary axis. Suppose that $|F(iy)|\leq 1$ for all $y \in \mathbb{R}$, and
$|F(z)| \leq Ce^{c|z|^\gamma}$
for some $c,C >0$ and $0 <\gamma <1$. Prove that $|F(z) |\leq 1$ for all $z $ in the right half plane.
(b). More generally, let $S$ be a sector whose vertex is the origin, and forming an angle of $\pi/\beta$. Let $F$ be a holomorphic function in $S$ that is continuous on the closure of $S$, so that $|F(z)| \leq 1$ on the boundary of $S$ and
$|F(z)|\leq Ce^{c|z|^\alpha}$
for some $c,C>0$ and $0<\alpha < \beta$. Prove that $|F(z)| \leq 1$ for all $z \in S$.
I was able to prove part (a) by considering the function $F_{\epsilon}(z) = F(z) e^{-\epsilon z^D}$, where $\gamma<D<1$ and showing that $F_{\epsilon}(z)\leq 1$ for all $z$ in the right half plane. This is the idea behind the Phragmen-Lindelof principle. I assume that a similar approach is needed for part (b), but I can't seem to find a good candidate for $F_\epsilon$ in this case.
For reference, Chapter 4 of Stein's complex analysis gives a nice demonstration of this proof tactic.
Best Answer
Let $F$ be defined in a sector with vertex at the origin and an opening angle of $\pi/\beta$. Without loss of generality assume that the sector is symmetric to the real axis, i.e. $$ S = \left\{ r e^{i\phi} \mid r > 0 , -\frac{\pi}{2\beta} < \phi < \frac{\pi}{2\beta} \right\} \, . $$ Then $G(z) = F(z^{1/\beta})$ (using the principal branch of $z^{1/\beta}$) is defined in the right half-plane. $|F(z)| \le 1$ on the boundary of $S$ implies that $|G(z)| \le 1$ on the imaginary axis, and $$ |G(z)| = |F(z^{1/\beta})| \le Ce^{c|z|^{\alpha/\beta}} \, , $$ so that the first part can be applied to the function $G$.
It follows that $|G(z)| \le 1$ in the right half-plane, and consequently $|F(z)| \le 1$ in the sector $S$.