- Here are further results similar to the Phragmén–Lindelöf theorem.
- Let \(F\) be a holomorphic function in the right half-plane that extends continuously to the boundary, that is, the imaginary axis. Suppose that \(| F ( i y ) | \le 1\) for all \(y \in \mathbb { R }\) , and
\[ | F (z) | \leq C e ^ {c | z | ^ {\gamma}} \]
for some \(c , C > 0\) and \(\gamma < 1\) . Prove that \(| F ( z ) | \le 1\) for all z in the right half-plane.
- 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 \(| F ( z ) | \le 1\) on the boundary of S and
\[ | F (z) | \leq C e ^ {c | z | ^ {\alpha}} \text { for all } z \in S \]
for some \(c , C > 0\) and \(0 < \alpha < \beta\) . Prove that \(| F ( z ) | \le 1\) for all \(z \in S\)
Solution
For (a), work in the right half-plane with the branch of \(z^\delta\) determined by \[ -\frac\pi2<\arg z<\frac\pi2. \] Choose \(\delta\) so that \[ \gamma<\delta<1. \] For \(\varepsilon>0\), set \[ F_\varepsilon(z)=F(z)e^{-\varepsilon z^\delta}. \] If \(z=re^{i\theta}\) in the closed right half-plane, then \[ \Re(z^\delta)=r^\delta\cos(\delta\theta) \ge r^\delta\cos\frac{\delta\pi}{2}>0. \] Hence on the imaginary axis \[ |F_\varepsilon(iy)|\le|F(iy)|\le1. \] On the large semicircle \(|z|=R\) in the right half-plane, \[ |F_\varepsilon(z)| \le C\exp\!\left(cR^\gamma -\varepsilon R^\delta\cos\frac{\delta\pi}{2}\right), \] which tends to \(0\) as \(R\to\infty\) because \(\delta>\gamma\). Thus, for sufficiently large \(R\), the maximum principle on the right half-disc gives \(|F_\varepsilon|\le1\). Letting \(R\to\infty\) and then \(\varepsilon\downarrow0\) yields \[ |F(z)|\le1 \] throughout the right half-plane.
For (b), rotate the sector so that it is symmetric about the positive real axis: \[ S=\left\{z:\left|\arg z\right|\le\frac{\pi}{2\beta}\right\}. \] Choose \[ \alpha<\delta<\beta. \] The branch \(z^\delta=r^\delta e^{i\delta\theta}\) is holomorphic in the sector, and \[ \Re(z^\delta) =r^\delta\cos(\delta\theta) \ge r^\delta\cos\frac{\delta\pi}{2\beta}>0. \] For \[ F_\varepsilon(z)=F(z)e^{-\varepsilon z^\delta}, \] the boundary-ray bound remains at most \(1\), while on the circular arc \(|z|=R\), \[ |F_\varepsilon(z)| \le C\exp\!\left(cR^\alpha -\varepsilon R^\delta\cos\frac{\delta\pi}{2\beta}\right) \longrightarrow0. \] The maximum principle on truncated sectors therefore gives \(|F_\varepsilon|\le1\). Letting \(\varepsilon\downarrow0\) proves \[ |F(z)|\le1 \] for all \(z\in S\).