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$f(z), g(z)$ 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.

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    Do you know what real limits mean?2011-11-07
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    I think it cannot be true (in general) that $|f(z)|\leq c|g(z)|$ (unless $|f(z)|=c|g(z)|$).2011-11-07
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    @Lindsey: Did you mean to ask "are there any interesting non-trivial consequences of these limits using the fact $f$,$g$ are analytic" instead of merely "what do these limits mean?"2011-11-07
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    Yes, I hope there is some consequences fot this!2011-11-08

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