If $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Omega$ prove that $f$ and $g$ have the same number of zeros in $\Omega$ 
Let , $f$ and $g$ be analytic on $\Omega$ and continuous on $\Omega\cup \partial \Omega$.
If $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Omega$ prove that $f$ and $g$ have the same number of zeros in $\Omega$ counting multiplicities.

I want to apply Rouche's theorem to show $f$ and $g$ have the same number of zeros in $\Omega$ but I can not construct such functions so that Rouche's theorem can be apply.
 A: First, you stated the problem wrong. As has been pointed out, your hypothesis implies that neither $f$ nor $g$ has a zero in $\Omega$; hence the result follows because $0=0$.
But the real result is not quite that trivial. The actual hypothesis is that $|f+g|<|f|+|g|$ on the boundary of $\Omega$.
I doubt that you'll be able to derive this from the version of Rouche's theorem that you're trying to use. You can however derive it by modifying the proof of that theorem! The hypothesis is weaker, but it's strong enough to do whatever the hypothesis $|f-g|<|f|$ does in the traditional proof.
For example: Note first that if $h=-g$ then $|h|=|g|$ and $h$ and $g$ have the same number of zeroes. So you can assume that $|f-g|<|f|+|g|$ on the boundary instead.
Now there is a neighborhood $V$ of $\partial\Omega$ in $\overline\Omega$ such that $|f-g|<|f|+|g|$ in $V$. Let $$W=\mathbb C\setminus(-\infty,0].$$We have $$f(z)/g(z)\in W\quad(z\in V).$$Let $L$ be a branch of the logarithm in $W$, and define $F\in H(V)$ by $F=L\circ(f/g)$. Now since the integral of a derivative over a closed curve is always $0$ it follows that if $\Gamma$ is a cycle in $V$ then $$\int_\Gamma F'(z)\,dz=0.$$Figure out what $F'$ is, and then figure out what this says about the zeroes of $f$ and $g$.
Or see Ullrich Complex Made Simple for a proof by homotopy. (That's the only complex text I know that has this version of Rouche...)
