I have an analytic complex function $f(z)$ in the upper half-plane, $z=x+iy$, where $\int_{-\infty}^{\infty}|f(x)|^{2}dx<\infty$, and $f$ is continuous on the real axis. Is it true that $f$ is bounded on the real axis, and why?
( i.e, there exists $M>0$, such that $|f(x)|\leq M$ for all $x\in \mathbb R$).