If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int\_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$
then it can be proven that $$\int\_{-\infty}^{+\infty}|f(x+iy)|^pdx\leqslant e^{\tau p|y|}\int\_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$ for all $y$.


I was wondering if such result also holds when $p=\infty$. For example if something like $$\sup\_{x\in\mathbb{R},|y|\leqslant M}|f(x+iy)|\leqslant e^{M\tau}\sup\_{t\in\mathbb{R}}|f(t)|$$ holds when $f(z)$ is a bounded entire function of exponential type $\tau$. Or even if the inequality holds changing $e^{M\tau}$ factor by any other constant $C>0$.



Plancharel-Pólya inequality for functions of exponential type
Yes, this is true. This is Theorem 11 on chapter 2 (page 82) in R.Young's book (An introduction to nonharmonic Fourier analysis). The proof is based on an application of the Phragmén–Lindelöf principle.


