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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 $C>0$.

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  • $\begingroup$ did you forget the sup norm in your formula? As it is written it is certainly wrong, for example when $f(x)=0$ for some real $x$. And what is $p$ doing in your formula if $p=\infty$?? $\endgroup$ – Alexandre Eremenko Mar 25 '18 at 5:42
  • $\begingroup$ You were right, I just edited the question. $\endgroup$ – pipenauss Mar 25 '18 at 13:27
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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.

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