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 $p=\infty$, i.e. if $$\sup_{x+iy\in\mathbb{C}}|f(x+iy)|\leqslant e^{\tau p|y|}\sup_{x\in\mathbb{R}}|f(x)|$$ when $f(z)$ is a bounded entire function of exponential type $\tau$. Or if at least with such hypothesis it is possible to control somehow $\sup|f(x+iy)|$ by $\sup|f(x)|$.

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 $C>0$.

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 $p=\infty$, i.e. if $$|f(x+iy)|\leqslant e^{\tau p|y|}|f(x)|$$ for all $x,y$ when $f(z)$ is a bounded entire function of exponential type $\tau$. Or if at least with such hypothesis it is possible to control somehow $|f(x+iy)|$ by $|f(x)|$.

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 $p=\infty$, i.e. if $$\sup_{x+iy\in\mathbb{C}}|f(x+iy)|\leqslant e^{\tau p|y|}\sup_{x\in\mathbb{R}}|f(x)|$$ when $f(z)$ is a bounded entire function of exponential type $\tau$. Or if at least with such hypothesis it is possible to control somehow $\sup|f(x+iy)|$ by $\sup|f(x)|$.