holomorphic image of Jordan curve. In general, holomorphic image of Jordan curve may not be a Jordan curve. (see the obvious case of costant function). Is there any sufficient condition regarding that holomorphic function which guarantees


*

*image of Jordan curve under that function again a Jordan curve

*Winding number of that Jordan curve is preserved 


?
 A: Really nice functions will break the winding number. Like, polynomial nice.
Let's just assume that your original Jordan curve winds around $0$. Then the function $f:\mathbb{C}\to\mathbb{C}$ by $z\mapsto z^n$ will increase the winding number by a factor of $n$. Call your curve $\Gamma$. Then
$$
\int_\Gamma \frac{1}{z}\,dz = k
$$
for some $k$. Then
$$
\int_{f(\Gamma)} \frac{1}{z}\,dz = \int_{\Gamma} \frac{f'(z)}{f(z)}\,dz=\int_{\Gamma} n\frac{z^{n-1}}{z^n}\,dz =nk 
$$
A: If $f$ is a holomorphic function, then the most basic condition that will ensure that $f$ preserves Jordan curves and winding numbers is that $f$ be a homeomorphism; i.e., a continuous function with a continuous (two-sided) inverse.  Since holomorphic functions are automatically continuous, you only need to check that it has a continuous inverse.  Can you see why this will preserve Jordan curves and winding numbers?
More generally, any homotopy equivalence will preserve winding numbers (though you have to take care to define this in the right way - ask if you want the full details), though I think it will take a bit more work to get a similar condition for Jordan curves.  
