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 $G$.

If $g(z)$ is not identically zero, you can find an open set $U$ on which $g(z)$ is nonzero. Thus on $U$, $\bar{f} = \bar{f} g \times {1 \over g}$ is analytic. So ${1 \over 2}(f + \bar{f})$ = 
$Re(f)$ and $Im(f) = {1 \over 2i}(f - \bar{f})$ are analytic functions on $U$. Applying the Cauchy-Riemann equations to these two functions gives that $Re(f)$ and $Im(f)$ are constant on $U$, so the same is true for $f$. Since $f$ is equal to a constant on an open subset of your region, it's equal to that constant everywhere. 
