Given a complex function $G(z)$, $z=x+iy, x,y\in \mathbb R$ which is analytic and bounded in the upper half-plane, i.e.,
$$|G(z)|\leq C, \forall z\in \mathbb C^{+}$$ does it imply that $G(x)$ is bounded on the real line (if we know that $G(x)$ has no poles on $\mathbb R$), i.e., $$|G(x)|\leq C_{o}, \forall x\in \mathbb R$$
EDIT: $\mathbb C^{+}=\{z\in \mathbb C, \text{Im}(z)>0\}$.