$f(z), g(z)$ are two entire functions, both have no zeros in the closed upper half plane. What does it mean/imply that $\bigg| \lim_{y\rightarrow \infty}\frac{f(z)}{g(z)}\bigg|=c$ ($z=x+iy$) i.e. after taking the limit inside the modulus the resulting function -depends on x- have modulus c. (In fact what I have is like: $|\lim_{y\rightarrow \infty} (\dots)|=|..ce^{ix}|=c$)
(I think it implies that $|f(z)|\leq c|g(z)|$ for all $z$ in the upper half plane, is that correct, and if so how to prove it!)
Also, what does it mean/imply that
$\bigg|\lim_{y\rightarrow 0}\frac{f(z)}{g(z)}\bigg|=d$
EDIT: $c$ and $d$ are non zero.