Show either $f$ is constant or $g(z)=0$ for all $z$ in the region let $G$ be a region, and $f$ and $g$ be holomorphic function on $G$. if $\bar{f}\cdot g$ is holomorphic, show that either $f$ is a constant or $g(z)=0$ for all $z$ in $G$.

I think we have to assume $G$ is connected otherwise this does not work : let $G = G_1 \cup G_2$ with $G_1$ and $G_2$ disjoint open sets, set $f = 1_{G_1}$ (ie $f(z)$ is $1$ if $z \in G_1$ and $0$ otherwise), $g = 1_{G_2}$. Then $f$, $g$ and $\overline{f}g$ are holomorphic but $f$ is not constant, and $g$ is not zero.
We now assume $G$ is connected. Denote $h = \overline{f} g$. Let $\Omega \subset G$ be the open subset of points on which $g$ does not vanish. Assume $\Omega$ is non empty (otherwise $g = 0$ and we're done). Then $f$ and $\overline{f} = \frac{h}{g}$ are holomorphic on $\Omega$, so $f$ is constant on $\Omega$ (using the Cauchy-Riemann equation). And because $G$ is connected, the identity theorem shows that $f$ is constant on $G$. 
